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Multicomponent long-wave–short-wave resonance interaction system: Bright solitons, energy-sharing collisions, and resonant solitons
Abstract
We consider a general multicomponent (2+1)-dimensional long-wave-short-wave resonance interaction (LSRI) system with arbitrary nonlinearity coefficients, which describes the nonlinear resonance interaction of multiple short waves with a long wave in two spatial dimensions. The general multicomponent LSRI system is shown to be integrable by performing the Painlevé analysis. Then we construct the exact bright multisoliton solutions by applying the Hirota's bilinearization method and study the propagation and collision dynamics of bright solitons in detail. Particularly, we investigate the head-on and overtaking collisions of bright solitons and explore two types of energy-sharing collisions as well as standard elastic collision. We have also corroborated the obtained analytical one-soliton solution by direct numerical simulation. Also, we discuss the formation and dynamics of resonant solitons. Interestingly, we demonstrate the formation of resonant solitons admitting breather-like (localized periodic pulse train) structure and also large amplitude localized structures akin to rogue waves coexisting with solitons. For completeness, we have also obtained dark one- and two-soliton solutions and studied their dynamics briefly.
... To be precise, our enKP model (7) can be reduced from the above equation (8) for the choice γ 0 = 4α, γ 1 = γ 3 = 4, γ 2 = γ 4 = 0, γ 5 = γ 6 = 9, γ 7 = γ 8 = 3, γ 9 = −4, and γ 10 = 4β. In order to solve our intended enKP model (7), we adopt Hirota's bilinearization method [50,51,52,53,54], which is one of the efficient analytical techniques available for solving different classes of nonlinear equations. For this purpose, we consider the following superposed variable transformation: ...
... Note that the superposed bilinearization (9) is quite different from the standard bilinearization procedure [50,51,52,53,54], and there are a few recent reports on similar superposed techniques in the literature [37,38,39,46]. ...
... The evolution of asymmetric (P = Q) periodic solitons in the enKP system (7) arising from two soliton solution for A = 0.5, α = −0.5, β = 1.0, k 1 = i, k 2 = −i, l 1 = −1.5, l 2 = 1.5, and p 0 = q 0 = r 0 = 1.0 at t = 0 (top panels) and t = −0.5 (bottom panels).N -soliton solution, one can investigate the dynamics of different pattern formations leading to T, Y, M, H, and complex web-like wave structures through the resonant mechanism of multiple solitons with long-time interactions. Further, the creation and analysis of other localized waves like breathers, lumps, and rogue waves in the intermediate interaction regime are also of considerable future interest[52,56,57,58]. ...
The study of nonlocal nonlinear systems and their dynamics is a rapidly increasing field of research. In this study, we take a closer look at the extended nonlocal Kadomtsev-Petviashvili (enKP) model through a systematic analysis of explicit solutions. Using a superposed bilinearization approach, we obtained a bilinear form of the enKP equation and constructed soliton solutions. Our findings show that the nature of the resulting nonlinear waves, including the amplitude, width, localization, and velocity, can be controlled by arbitrary solution parameters. The solutions exhibited both symmetric and asymmetric characteristics, including localized bell-type bright solitons, superposed kink-bell-type and antikink-bell-type soliton profiles. The solitons arising in this nonlocal model only undergo elastic interactions while maintaining their initial identities and shifting phases. Additionally, we demonstrated the possibility of generating bound-soliton molecules and breathers with appropriately chosen soliton parameters. The results of this study offer valuable insights into the dynamics of localized nonlinear waves in higher-dimensional nonlocal nonlinear models.
... The LSRI literature originally starts from the theoretical investigation on Langmuir waves in plasma where the generalized Zakharov equations were derived [4]. After this pioneering work by Zakharov, there have been several experimental and theoretical research activities based on the LSRI phenomenon in different contexts ranging from lower dimensions [12][13][14] to higher dimensions [15][16][17][18], with single component [19] to multi-component [20][21][22][23][28][29][30]. These studies also report the existence of several types of nonlinear localized wave structures [19-21, 24-27, 31], namely bright soliton with a single-hump structure [19][20][21]31] and bright soliton with a double-hump structure [32], dark soliton [21][22][23], breathers [33], and roguewaves [34][35][36][37][38], and their novel properties have also been exhibited there. ...
... After this pioneering work by Zakharov, there have been several experimental and theoretical research activities based on the LSRI phenomenon in different contexts ranging from lower dimensions [12][13][14] to higher dimensions [15][16][17][18], with single component [19] to multi-component [20][21][22][23][28][29][30]. These studies also report the existence of several types of nonlinear localized wave structures [19-21, 24-27, 31], namely bright soliton with a single-hump structure [19][20][21]31] and bright soliton with a double-hump structure [32], dark soliton [21][22][23], breathers [33], and roguewaves [34][35][36][37][38], and their novel properties have also been exhibited there. The main focus of this paper is to present the soliton, both bright and dark, solutions and breather solution for the recently introduced generalized long-wave short-wave resonance interaction (LSRI) system iS t + S xx + (iαL x + α 2 L 2 − βL − 2α|S| 2 )S = 0, ...
... After this pioneering work by Zakharov, there have been several experimental and theoretical research activities based on the LSRI phenomenon in different contexts ranging from lower dimensions [12][13][14] to higher dimensions [15][16][17][18], with single component [19] to multi-component [20][21][22][23][28][29][30]. These studies also report the existence of several types of nonlinear localized wave structures [19-21, 24-27, 31], namely bright soliton with a single-hump structure [19][20][21]31] and bright soliton with a double-hump structure [32], dark soliton [21][22][23], breathers [33], and roguewaves [34][35][36][37][38], and their novel properties have also been exhibited there. The main focus of this paper is to present the soliton, both bright and dark, solutions and breather solution for the recently introduced generalized long-wave short-wave resonance interaction (LSRI) system iS t + S xx + (iαL x + α 2 L 2 − βL − 2α|S| 2 )S = 0, ...
In this paper, a generalized long-wave shortwave resonance interaction system, which describes the nonlinear interaction between a shortwave and a long-wave in fluid dynamics, plasma physics and nonlinear optics, is considered. Using the Hirota bilinear method, the general N-bright and N-dark soliton solutions are deduced and their Gram determinant forms are obtained. A special feature of the fundamental bright soli-ton solution is that, in general, it behaves like the Korteweg-deVries soliton. However, under a special condition, it also behaves akin to the nonlinear Schrödinger soliton when it loses the amplitude dependent velocity property. The fundamental dark-soliton solution admits anti-dark, grey, and completely black soliton profiles, in the shortwave component, depending on the choice of wave parameters. On the other hand, a bright soliton like profile always occurs in the long-wave component. The asymptotic analysis shows that both the bright and dark solitons undergo an elastic collision with a finite phase shift. In addition to these, by tuning the phase shift regime, we point out the existence of resonance interactions among the bright solitons. Furthermore, 1 Springer Nature 2021 L A T E X template 2 Soliton and breather solutions of the generalized LSRI system under a special velocity resonance condition, we bring out the various types of bright and dark soliton bound states. Also, by fixing the phase factor and the system parameter β, corresponding to the interaction between long and short wave components, the different types of profiles associated with the obtained breather solution are demonstrated.
... The LSRI literature originally starts from the theoretical investigation on Langmuir waves in plasma where the generalized Zakharov equations were derived [4]. After this pioneering work by Zakharov, there have been several experimental and theoretical research activities based on the LSRI phenomenon in different contexts ranging from lower dimensions [12][13][14][15] to higher dimensions [16][17][18][19][20], with single component [21] to multi-component [22][23][24][25][30][31][32][33]. These studies also report the existence of several types of nonlinear localized wave structures [21][22][23][26][27][28][29]34], namely bright soliton with a single-hump structure [21][22][23]34] and bright soliton with a double-hump structure [35], dark soliton [23][24][25], breathers [36], and roguewaves [37][38][39][40][41], and their novel properties have also been exhibited there. ...
... After this pioneering work by Zakharov, there have been several experimental and theoretical research activities based on the LSRI phenomenon in different contexts ranging from lower dimensions [12][13][14][15] to higher dimensions [16][17][18][19][20], with single component [21] to multi-component [22][23][24][25][30][31][32][33]. These studies also report the existence of several types of nonlinear localized wave structures [21][22][23][26][27][28][29]34], namely bright soliton with a single-hump structure [21][22][23]34] and bright soliton with a double-hump structure [35], dark soliton [23][24][25], breathers [36], and roguewaves [37][38][39][40][41], and their novel properties have also been exhibited there. The main focus of this paper is to present the soliton, both bright and dark, solutions and breather solution for the recently introduced generalized long-wave short-wave resonance interaction (LSRI) system ...
... After this pioneering work by Zakharov, there have been several experimental and theoretical research activities based on the LSRI phenomenon in different contexts ranging from lower dimensions [12][13][14][15] to higher dimensions [16][17][18][19][20], with single component [21] to multi-component [22][23][24][25][30][31][32][33]. These studies also report the existence of several types of nonlinear localized wave structures [21][22][23][26][27][28][29]34], namely bright soliton with a single-hump structure [21][22][23]34] and bright soliton with a double-hump structure [35], dark soliton [23][24][25], breathers [36], and roguewaves [37][38][39][40][41], and their novel properties have also been exhibited there. The main focus of this paper is to present the soliton, both bright and dark, solutions and breather solution for the recently introduced generalized long-wave short-wave resonance interaction (LSRI) system ...
In this paper, we consider a generalized long-wave short-wave resonance interaction system, which describes the nonlinear interaction between a short-wave (SW) and a long-wave (LW). The general N -bright and N -dark soliton solutions are derived using the Hirota bilinear method and they are written in a compact way using Gram determinants. Very interestingly, the fundamental bright soliton solution of the generalized LSRI system, in general, behaves like Korteweg-deVries (KdV) soliton. However, under a special condition, it also acts like the nonlinear Schro¨dinger (NLS) soliton. The fundamental dark-soliton solution admits anti-dark, grey, and complete black soliton profiles depending on the choice of wave parameters. The results of asymptotic analysis show that both bright and dark solitons undergo elastic collision with a finite phase shift. In addition to these, we demonstrate the existence of resonance interactions among the bright solitons by tuning the phase shift regime. Furthermore, we illustrate the various types of bright and dark soliton bound states. Also, we demonstrate the different types of profiles associated with the obtained breather solution.
... It is important to point out that there are several nonlinear wave solutions which have been reported in the literature for the integrable long-wave-short-wave resonance interaction model and its variants [21][22][23][24][25][26][27][28][29][30][31][33][34][35][36][37][38][39][40]. For the one-dimensional single component YO system, both bright and dark soliton solutions were derived in Ref. [21]. ...
... This system is shown to be integrable through Painlevé analysis and the dromion solutions were obtained using Painlevé truncation method [27]. Very interestingly, one of the present authors (M.L.) and his collaborators demonstrated the energy-sharing collisions of bright solitons in the two-dimensional integrable versions of the multicomponent LSRI system by deriving their explicit solutions through the Hirota bilinear method [28,29] and they have also shown that the formation of resonant solitons in this higher-dimensional system [29]. Mixed bright-dark soliton solutions and their collision dynamics for this (2 + 1)-dimensional system have been studied in Refs. ...
... This system is shown to be integrable through Painlevé analysis and the dromion solutions were obtained using Painlevé truncation method [27]. Very interestingly, one of the present authors (M.L.) and his collaborators demonstrated the energy-sharing collisions of bright solitons in the two-dimensional integrable versions of the multicomponent LSRI system by deriving their explicit solutions through the Hirota bilinear method [28,29] and they have also shown that the formation of resonant solitons in this higher-dimensional system [29]. Mixed bright-dark soliton solutions and their collision dynamics for this (2 + 1)-dimensional system have been studied in Refs. ...
In this paper, we study the dynamics of an interesting class of vector solitons in the long-wave-short-wave resonance interaction (LSRI) system. The model that we consider here describes the nonlinear interaction of long wave and two short waves and it generically appears in several physical settings. To derive this class of nondegenerate vector soliton solutions we adopt the Hirota bilinear method with the more general form of admissible seed solutions with nonidentical distinct propagation constants. We express the resultant fundamental as well as multisoliton solutions in a compact way using Gram-determinants. The general fundamental vector soliton solution possesses several interesting properties. For instance, the double-hump or a single-hump profile structure including a special flattop profile form results in when the soliton propagates in all the components with identical velocities. Interestingly, in the case of nonidentical velocities, the soliton number is increased to two in the long-wave component, while a single-humped soliton propagates in the two short-wave components. We establish through a detailed analysis that the nondegenerate multisolitons in contrast to the already known vector solitons (with identical wave numbers) can undergo three types of elastic collision scenarios: (i) shape-preserving, (ii) shape-altering, and (iii) a shape-changing collision, depending on the choice of the soliton parameters. Here, by shape-altering we mean that the structure of the nondegenerate soliton gets modified slightly during the collision process, whereas if the changes occur appreciably then we call such a collision as shape-changing collision. We distinguish each of the collision scenarios, by deriving a zero phase shift criterion with the help of phase constants. Very importantly, the shape-changing behavior of the nondegenerate vector solitons is observed in the long-wave mode also, along with corresponding changes in the short-wave modes, and this nonlinear phenomenon has not been observed in the already known vector solitons. In addition, we point out the coexistence of nondegenerate and degenerate solitons simultaneously along with the associated physical consequences. We also indicate the physical realizations of these general vector solitons in nonlinear optics, hydrodynamics, and Bose-Einstein condensates. Our results are generic and they will be useful in these physical systems and other closely related systems including plasma physics when the long-wave-short-wave resonance interaction is taken into account.
... In this paper, we showed that in general, as seen in the figures, which two solitons capture each other always lead to similar breathing oscillating structures after collisions. However, these breathing oscillating structures have been seen in many different nonlinear systems in 1 + 1 and 2 + 1 dimensions similarly [19,[23][24][25]. We think that this similarity, just is referred to the non-linearity nature of them. ...
... But, in general there is not any other meaningful relationship between them. For example, the special cases which were seen in [24] are happened when some special conditions are fulfilled, while there are not such restrictive conditions in our model and other kink-bearing models to see such breathing oscillating structures. Moreover, the kink-bearing systems are essentially relativistic, while the other systems [23,24] are non-relativistic and this is another difference. ...
... For example, the special cases which were seen in [24] are happened when some special conditions are fulfilled, while there are not such restrictive conditions in our model and other kink-bearing models to see such breathing oscillating structures. Moreover, the kink-bearing systems are essentially relativistic, while the other systems [23,24] are non-relativistic and this is another difference. ...
Inspired by the well known sine-Gordon equation, we present a symmetric coupled system of two real scalar fields in $1+1$ dimensions. There are three different topological soliton solutions which be labelled according to their topological charges. These solitons can absorb some localized non-dispersive wave packets in collision processes. It will be shown numerically, during collisions between solitons, there will be an uncertainty which originates from the amount of the maximum amplitude and arbitrary initial phases of the trapped wave packets.
... In this paper, we showed that in general, as seen in the figures, which two solitons capture each other always lead to similar breathing oscillating structures after collisions. However, these breathing oscillating structures have been seen in many different nonlinear systems in 1 + 1 and 2 + 1 dimensions similarly [19,[23][24][25] . We think that this similarity, just is referred to the non-linearity nature of them. ...
... But, in general there is not any other meaningful relationship between them. For example, the special cases which were seen in [24] are happened when some special conditions are fulfilled, while there are not such restrictive conditions in our model and other kink-bearing models to see such breathing oscillating structures. Moreover, the kink-bearing systems are essentially relativistic, while the other systems [23,24] are non-relativistic and this is another difference. ...
... For example, the special cases which were seen in [24] are happened when some special conditions are fulfilled, while there are not such restrictive conditions in our model and other kink-bearing models to see such breathing oscillating structures. Moreover, the kink-bearing systems are essentially relativistic, while the other systems [23,24] are non-relativistic and this is another difference. ...
Inspired by the well known sine-Gordon equation, we present a symmetric coupled system of two real scalar fields in 1+1 dimensions. There are three different topological soliton solutions which be labelled according to their topological charges. These solitons can absorb some localized non-dispersive wave packets in collision processes. It will be shown numerically, during collisions between solitons, there will be an uncertainty which originates from the amount of the maximum amplitude and arbitrary initial phases of the trapped wave packets.
... where σ = ±1, S ( ) and L indicate the th short-wave and long-wave components, respectively. The general multi-component LSRI system is shown to be integrable by performing the Painlevé analysis and its exact bright multi-soliton solution was constructed in [27]. In Ref. [25], the mixed (bright-dark) one-and two-soliton solutions of a particular 2D multi-component LSRI system, namely the general LSRI system with same nonlinearity coefficients, were also obtained and their propagation properties and collision dynamics were discussed. ...
... However, the general representation of the mixed multi-soliton solution for this particular LSRI system is missing. More importantly, as the detailed analysis provided in [27] involving the dynamics of the bright soliton, the arbitrariness of nonlinearity coefficients σ gives an additional freedom resulting in rich soliton dynamics. Thus, it is still worth seeking for the explicit formulation of the mixed multi-soliton solution for the general 2D multi-component LSRI system. ...
... To obtain nonsingular soliton solution, we always choose p 1R q 1R > 0 and μ 1 = μ 2 = 1. Then onesoliton solutions can be classified into the following three cases (only for σ 1 and σ 2 ): (i) positive nonlinearity coefficients [(σ 1 , σ 2 ) = (1, 1), then ν 1 = ν 2 = 1]: This case have been discussed in Ref. [25]; (ii) negative nonlinearity coefficients [(σ 1 , σ 2 ) = (−1, −1), then ν 1 = ν 2 = −1]: For the choice, the LSRI system admits a similar type of propagation and even collision behaviors as that of the positive nonlinearity coefficients [26,27]; (iii) mixed-type nonlinearity coefficients: Without loss of generality, we take (σ 1 , σ 2 ) = (1, −1) and fix σ 3 = 1, then ν 1 = −ν 2 = 1 and tau functions in (12)-(14) can be rewritten as ...
In this paper, we derive a general mixed (bright–dark) multi-soliton solution to a two-dimensional (2D) multi-component long-wave–short-wave resonance interaction (LSRI) system, which include multi-component short waves (SWs) and one-component long wave (LW) for all possible combinations of nonlinearity coefficients including positive, negative and mixed types. With the help of the KP hierarchy reduction method, we firstly construct two types of general mixed N-soliton solution (two-bright–one-dark soliton and one-bright–two-dark one for SW components) to the 2D three-component LSRI system in detail. Then by extending the corresponding analysis to the 2D multi-component LSRI system, a general mixed N-soliton solution in Gram determinant form is obtained. The expression of the mixed soliton solution also contains the general all bright and all dark N-soliton solution as special cases. In particular, for the soliton solution which include two-bright–one-dark soliton for SW components in three-component LSRI system, the dynamics analysis shows that solioff excitation and solioff interaction appear in two SW components which possess bright soliton, while V-type solitary and interaction take place in the other SW component and LW one.
... n, and will study other combinations in the future. An elegant feature of Eq. (25) is that the intensity of each component summed over the entire spatial domain is conserved [22]: ...
... Dynamics and properties of solitons in such long waveshort wave systems in two spatial dimensions have been studied in the literature [22,23], where the Hirota bilinear transform is also employed. Painlevé analysis has Fig. 5 An example of the unstable nature of rogue waves: initial noise disturbances can significantly distort the 'growth phase' of the rogue wave: ρ = 1, σ = 1, k = 1.25, 0 = 0.552 − 1.430i been performed to confirm the 'integrability' of the system [22]. ...
... Dynamics and properties of solitons in such long waveshort wave systems in two spatial dimensions have been studied in the literature [22,23], where the Hirota bilinear transform is also employed. Painlevé analysis has Fig. 5 An example of the unstable nature of rogue waves: initial noise disturbances can significantly distort the 'growth phase' of the rogue wave: ρ = 1, σ = 1, k = 1.25, 0 = 0.552 − 1.430i been performed to confirm the 'integrability' of the system [22]. Head-on and overtaking collisions, as well as energy distribution properties, are investigated [22,23]. ...
A resonance between long and short waves will occur if the phase velocity of the long wave matches the group velocity of the short wave. In this paper, a system with two distinct packets of short waves in resonance with a common long wave is studied. Breather solutions are calculated by the Hirota bilinear method, and rogue wave modes (unexpectedly large displacements from an otherwise calm background state) are obtained from the breathers through a long wave limit. The location and magnitude of the maximum displacement are determined quantitatively. Remarkably this coupling enables a rogue wave to attain a larger magnitude than that in a configuration with just one single short wave component. Furthermore, as the wavenumber varies, a transition from an elevation rogue wave to a depression rogue wave is possible. This transformation of the wave profile is elucidated in terms of the properties of the carrier envelope. The connection with the modulation instability of the background plane wave is investigated. Some numerical simulations are performed to demonstrate both the robust nature and unstable behavior for these rogue waves, depending on the parameters of the system. Dynamics and properties of rogue waves with three or more short wave components are also considered.
... By Riemann-Hilbert approach, the soliton solutions and breather type soliton have been obtained [29]. In addition, studies pertaining to the LW-SW resonant model from various perspectives can be visible in [30][31][32][33]. Base on these advancements, the nonlocal LW-SW equation presents a broader scope for research and suggests a substantial potential for further exploration. ...
... In addition, due to the nonlocal reduction condition, the solution of nonlocal LW-SW equation will be different from the case of local. Firstly, there are Y-type solitons, V-type solitons and bound state soliton solutions in local equation [28,33], but the solitons we obtain are strictly symmetric at x = 0. Secondly, these novel mixed four-soliton solutions do not exist in local LW-SW equation and the collision of two sets of solitons is elastic, such as Y-type soliton collides with bound state soliton. ...
This article’s purpose is to investigate the inverse scattering transform of the nonlocal long wave-short wave (LW-SW) equation and its multi-soliton solutions via Riemann-Hilbert (RH) approach. By using spectral analysis to the Lax pair of LW-SW equation, the RH problem can be constructed. However, we consider spectral analysis from the time part rather than the usual space part, since it is hard to obtain the analyticity of the space part. Then the RH problem can be solved and the formula of the soliton solutions can be given. We provide several special soliton solutions including Y-shaped solitons, V-shaped solitons, bound-state solitons and mixed four-soliton solutions. Compared with the local case, the solutions of nonlocal LW-SW equation exhibit distinct characteristics that (i) these soliton solutions are strictly symmetric with respect to x = 0 under special parameter conditions, (ii) the mixed four-soliton solution, which combines Y-type and bound-state solitons, is novel.
... There are many types of coupling systems apart from the Mel'nikov system. In addition to the Mel'nikov system, many scholars have also studied the general M-LSRI, Yajima-Oikawa (YO) system and other long-wave-short-wave coupling systems [43][44][45][46][47][48][49][50][51]. ...
... Scrutinizing the structure of the solution (56), we observe that these six breathers , , , − , − , and − are symmetrical to each other. The parameter relationships in Eq. (41)- (44) are not changed whether the resonant collision is separate or mutual, so we take advantage of them in the following analysis. According to the different choices of the parameter constraints, two waveform patterns demonstrated by the solution of the resonant collisions among three breathers can be observed, and are classified by the number of breathers with infinite length. ...
... To be precise, our enKP model Eqs 7a, 7b can be reduced from the above Eqs 8a, 8b for the choice γ 0 = 4α, γ 1 = γ 3 = 4, γ 2 = γ 4 = 0, γ 5 = γ 6 = 9, γ 7 = γ 8 = 3, γ 9 = −4, and γ 10 = 4β. In order to solve our intended enKP model Eqs 7a, 7b, we adopt Hirota's bilinearization method [50][51][52][53][54], which is one of the efficient analytical techniques available for solving different classes of nonlinear equations. For this purpose, we consider the following superposed variable transformation: ...
... However, we utilize the superposed bilinear transformation with A ≠ 0 to deduce the required bilinear equation(s). Note that the superposed bilinearization Eqs 9a, 9b is quite different from the standard bilinearization procedure [50][51][52][53][54], and there are a few recent reports on similar superposed techniques in the literature [37][38][39]46]. On applying the above bilinearizing transformation to Eqs 7a, 7b and after its simplification, we get the following bilinear equation in a standard Doperator form: ...
The study of nonlocal nonlinear systems and their dynamics is a rapidly increasing field of research. In this study, we take a closer look at the extended nonlocal Kadomtsev–Petviashvili (enKP) model through a systematic analysis of explicit solutions. Using a superposed bilinearization approach, we obtained a bilinear form of the enKP equation and constructed soliton solutions. Our findings show that the nature of the resulting solitons, such as the amplitude, width, localization, and velocity, can be controlled by arbitrary solution parameters. The solutions exhibited both symmetric and asymmetric characteristics, including localized bell-type bright solitons, superposed kink-bell-type and antikink-bell-type soliton profiles. The solitons arising in this nonlocal model only undergo elastic interactions while maintaining their initial identities and shifting phases. Additionally, we demonstrated the possibility of generating bound-soliton molecules and breathers with appropriately chosen soliton parameters. The results of this study offer valuable insights into the dynamics of localized nonlinear waves in higher-dimensional nonlocal nonlinear models.
... Experimentally, such resonance has been demonstrated in a laboratory for a layered fluid [18]. In terms of the theoretical aspects, analytical descriptions of rogue waves in both one [19,20] and two [21][22][23][24][25][26] spatial dimensions have been given. ...
... (2a), (2b)] − this case is discussed in Appendix B; (ii) The inclination angle between the two "crossing sea pattern" wave trains becomes small [ψ ≈ 0 in Eqs. (2a), (2b)] − this case is discussed in Appendix C. While "long-short resonances" involving multiple short waves have been treated in the literature, existing works either ignore the second horizontal spatial dimension or focus on the solitons [19][20][21][22][23][24][25][26]. We go beyond the regime of modulation instability and the calculations of solitons in the present study. ...
Free surface waves with two or more spectral peaks propagating at an oblique angle to each other are commonly termed “crossing sea states”. Such crossing patterns have been suggested as possible causes for rogue waves and maritime accidents. Modulation instabilities of plane waves using coupled Schrödinger or Zakharov equations have been adopted as theoretical models in the literature. Here, extensions to layered and stratified fluids are conducted. For a two-layer fluid with long-wave–short-wave resonance, crossing patterns with two short waves will enhance instability compared with the single-wave case. Analytical treatment beyond the linear instability regime is elucidated by a cascading mechanism. Growth of the higher-order harmonics eventually leads to finite-amplitude pulsating modes or breathers. Breathers subsequently exhibit a Fermi-Pasta-Ulam-Tsingou type recurrence. The time for the first formation of breathers predicted by the cascading mechanism attains excellent agreements with the full numerical simulations. A similar study is performed for a continuously stratified fluid with constant buoyancy frequency. Triad resonance with two components as a pair of oblique waves also produces enhanced instability and a preferred inclination of maximum growth rate. These crossing patterns will likely play critical roles in many wave-propagation configurations in fluid mechanics.
... The higher-dimensional version of system (1) was proposed in the study of dynamics of binary diskshaped BECs [31], and also in the investigations of the dynamics of certain solitons in spinor BECs [13], with the aid of a multiscale expansion procedure. Furthermore, the multi-component generalizations of system (1) for two spatial dimensions were introduced to investigate the nonlinear interactions of dispersive waves on three channels [32], and interactions of a LW with multiple SWs in two spatial dimensions [33]. Therefore, studies of multi-component LS models are physically important in nonlinear media. ...
... In panel (d), all the components (1) , (2) , and are single-hump solitons. Here we have to note that all the parameters in panels (a) and (c) are the same except for the nonlinearity parameter 2 , but the profiles of the solitons in the SW components and the LW component are completely changed; this result can be confirmed by the relation (33). ...
The focus of this paper is on the dynamics of degenerate and non-degenerate vector solitons and their collision scenarios in the two-component long-wave–short-wave (2-LS) model of Newell type. The model equation that we study here governs the resonance interactions between two capillary waves (short waves) with a common gravity wave (long wave) for the surface wave of deep water. These degenerate and non-degenerate soliton solutions of the 2-LS model of Newell type are constructed via the bilinear KP-hierarchy reduction method and are expressed in forms of determinants. The degenerate solitons having only single-hump or single-valley profiles can undergo both elastic and inelastic collisions. Two particular types of degenerate solitons are also obtained, i.e., bound state solitons and V/Y-shaped solitons. The non-degenerate solitons and their collision scenarios have more interesting properties in contrast with the degenerate ones. The obvious one is that the non-degenerate one solitons possess symmetric or asymmetric double-hump and double-valley profiles if they propagate with identical velocities in all short-wave (SW) and long-wave (LW) components. Additionally, the soliton wavenumbers in the SW components and the LW component are unequal if the ones in the two SW components propagate with different velocities. The non-degenerate two solitons can undergo two different types of elastic collision scenarios in both SW and LW components: shape-preserving and shape-changing scenarios. We also study the collisions of non-degenerate solitons with degenerate ones, and find some intriguing properties. For instance, if the degenerate one soliton in one of the two SW components vanishes into background, then there is only one fundamental non-degenerate soliton in the corresponding SW component. However, this fundamental non-degenerate soliton still alters its profile from a symmetric double-hump waveform to an asymmetric double-hump one during its propagation process without interacting with other solitons. Furthermore, it is found a new class of soliton solutions referred to as (1,2,3) solitons. In these new types of soliton solutions, one SW component has only a single soliton, the other SW component has two solitons, whereas the LW component has three solitons.
... Resonant soliton is a special kind of soliton existing in integrable system, which has been widely studied. As we all know, If phase shift of the colliding solitons becomes infinite or converges to infinity, solitons will resonate [17]. The well-known soliton fusion and fission is a resonance phenomenon [18,19]. ...
... where θ j = K j x + R j y + αR 2 j 36K j t. Then, bring a function (17) into the logarithmic transformation (2), we can gain interaction between two lumps and one 2-resonant Y-shape soliton of u. ...
Under the well-known bilinear method of Hirota, the specific expression for N-soliton solutions of (2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada(gCDGKS) equation in fluid mechanics is given. By defining a noval restrictive condition on N-soliton solutions, resonant Y-type and X-type soliton solutions are generated. Under the previous new constraints, combined with the velocity resonance method and module resonant method, the mixed solutions of resonant Y-type solitons and line waves, breather solutions are found. Finally, with the support of long wave limit method, the interaction between resonant Ytype solitons and higher-order lumps is shown, and the motion trajectory equation before and after the interaction between lumps and resonant Y-type solitons is derived.
... A specific resonant behavior has been observed during the interaction process in the long wave-short wave resonance interaction system (LSRI system) [45]. The resonant behavior occurs exactly in the place at which the phase shift occurs during the collision process. ...
... In addition to this, we point out that change in the imaginary part of the wave numbers leads to a switching of the Type-I collision into Type-II collision. We also note that the resonant pattern appearing during the collision process is not same as the one appear in the higher dimensional integrable systems [45]. In the local Manakov case, one does not observe such behavior and this occurs only due to the manifestation of nonlocal nature of the system. ...
In this paper, by considering the degenerate two bright soliton solution of the nonlocal Manakov system, we bring out three different types of energy-sharing collisions for two different parametric conditions. Among the three, two of them are new which do not exist in the local Manakov equation. By performing an asymptotic analysis to the degenerate two-soliton solution, we explain the changes which occur in the quasi-intensity/quasi-power, phase shift and relative separation distance during the collision process. Remarkably, the intensity redistribution reveals that in the new types of shape-changing collisions, the energy difference of soliton in the two modes is not preserved during collision. In contrast to this, in the other shape-changing collision, the total energy of soliton in the two modes is conserved during collision. In addition to this, by tuning the imaginary parts of the wave numbers, we observe localized resonant patterns in both the scenarios. We also demonstrate the existence of bound states in the nonlocal Manakov equation during the collision process for certain parametric values.
... Soliton dynamical interaction is a significant part in soliton theory and applications (Wazwaz 2007(Wazwaz , 2008bLi and Ma 2017c). Considering various traveling directions, incidental energy, speeds and shapes, there exist rich soliton interaction patterns, such as elastics collisions (shapes and energies remain unchanged before and after collisions) Vijayajayanthi et al. 2009;Ling et al. 2016;Zhaqilao 2017), inelastic collisions (shapes and energies change before and after collisions) (Vijayajayanthi et al. 2009), resonant interactions , lump-line-solitons interactions , bound vector collisions (Jiang et al. 2012), energy-sharing and resonant collisions (Sakkaravarthi and Kanna 2014), periodical collisions (Liu et al. 2017b) and so on. ...
... Rights reserved. 2014;Hioe 1999;Kanna and Lakshmanan 2001). The solitons resonant phenomenon in a pulse-width-tunable and mode-locked fiber laser has been found in optical engineering (Lee et al. 2016). ...
A nonlinear waveguide directional coupler system in optical fibers is investigated. The system can be modelled as coupled nonlinear Schrödinger equations. Applying Hirota bilinear transformation method, analytic one- and two-soliton solutions are constructed. Furthermore, a novel class of solitons resonant behavior is observed. The solitons are split into multiple wave peaks around colliding point. The impacts of main parameters on soliton collisions are systematically discussed. Group velocity dispersion parameter and group velocity mismatch parameter can impact and control the resonant effects. Meanwhile, the complex parameters in the soliton solutions can determine resonant peaks intensity.
... A specific resonant behaviour has been observed during the interaction process in the long wave-short wave resonance interaction system (LSRI system) [40]. The resonant behaviour occurs exactly in the place at which the phase shift occurs during the collision process. ...
... In addition to this, we point out that change in the imaginary part of the wave numbers leads to a switching of the Type-I collision into Type-II collision. We also note that the resonant pattern appearing during the collision process is not same as the one appear in the higher dimensional integrable systems [40]. In the local Manakov case, one does not observe such behaviour and this occurs only due to the manifestation of nonlocal nature of the system. ...
In this paper, by considering the degenerate two bright soliton solutions of the nonlocal Manakov system, we bring out three different types of energy sharing collisions for two different parametric conditions. Among the three, two of them are new which do not exist in the local Manakov equation. By performing an asymptotic analysis to the degenerate two-soliton solution, we explain the changes which occur in the quasi-intensity/quasi-power, phase shift and relative separation distance during the collision process. Remarkably, the intensity redistribution reveals that in the new types of shape changing collisions, the energy difference of soliton in the two modes is not preserved during collision. In contrast to this, in the other shape changing collision, the total energy of soliton in the two modes is conserved during collision. In addition to this, by tuning the imaginary parts of the wave numbers, we observe localized resonant patterns in both the scenarios. We also demonstrate the existence of bound states in the CNNLS equation during the collision process for certain parametric values.
... In nonlinear optics, electromagnetic wave propagation in a single mode optical fiber media can be observed using the NLS equation Tchio et al. 2019). The optical soliton propagation in the slowly changing optical fiber is described by the NLS equation (Agrawal and Haus 2002;Sakkaravarthi et al. 2014). The widely recognized totally integrable nonlinear Schrödinger equation, which takes the form, controls the propagation of optical solitons in a nonlinear dispersive optical fiber. ...
In an optical fiber, we report on our study of pulse compression using an analytical method. A significant role for the eighth order nonlinear Schrödinger (NLS) equation may be seen in the study of ultra-short pulses, in particular extremely nonlinear optical phenomena. As a result of our study, which plays a significant role in reviving potent mathematical procedures, such as the extended rational sinh–cosh method for obtaining the dark soliton solution and the cosine method for solving the NLS equation to attain M-shaped, W-shaped soliton structure, dark and bright solitonic structure. Solitons are shown to have a strictly chirp-free structure, which facilitates effective compression. According to the results, wavevector may effectively regulate the shape and propagation behavior of soliton.
... Furthermore, in scenarios where the FWM nonlinearity is absent, the soliton dynamics for focusing and mixedtype NLS models have been explored. This investigation involved the derivation of explicit soliton solutions and employed asymptotic analysis to investigate phenomena like energy-sharing and elastic collisions among bright solitons [59]. Furthermore, there have been numerous efforts dedicated to constructing optical soliton solutions for the system described in Eq. (1) and exploring their dynamic characteristics in the presence of different dispersion and nonlinearity effects [60]. ...
... The above two-dimensional M-LSRI system (1.1) is completely integrable and admits the Lax pair [20] and can be solved by various analytical methods like the Daboux transformation [22] and Hirota's bilinear method [17][18][19]23,24]. Regarding its solutions, the M-LSRI system admits a rich solution structure such as line soliton solutions (of bright or dark types), rational solitary wave solutions (line rogue waves and lumps) and doubly periodic (elliptic soliton) solutions [22][23][24][25][26][27][28][29][30][31][32]. Very recently, the higher-order rogue wave solutions to the one-dimensional LSRI system and one-dimensional LSRI of Newell type have been studied in [33,34]. ...
This study focuses on the occurrence and dynamics of resonant collisions of breathers among themselves and with rational solitary waves in a two-dimensional multi-component long-wave–short-wave resonance system. These nonlinear coherent structures are described by a family of semi-rational solutions encompassing almost all known solitary waves with non-zero boundary conditions. In the case of resonant collision of parallel breathers, the original breathers split up into several new breathers or combine into a new breather. They also exhibit a mixed behaviour displaying a mixture of splitting and recombination processes. The lumps and line rogue waves—two distinct forms of rational solitary waves—can undergo partial and complete resonant collisions with breathers. In the partial resonant collisions, the lumps do not exit well before interaction but suddenly emerge from the breathers and finally keep existing for an infinite time after interaction, or they alter from existence to annihilation as one moves from t → − ∞ to t → + ∞ . The line rogue waves arise and decay along ray waves instead of line waves. In the complete resonant collisions, the lumps first detach from breathers and then fuse into other breathers, while the line rogue waves appear and disappear along line segment waves of finite length, namely, both of the lumps and rogue waves are doubly localized in two-dimensional space as well as in time.
... Owing to the mixed-type nonlinearity coefficients, it has been reported that solitons display a different type of energy-sharing collision (resonance solitons) in the multi-component long-wave-short-wave resonance interaction system [31,32]. Besides, this special soliton has also been reported in the multi-component twodimensional long-wave-short-wave resonance interaction system [29] and Maccari system [30]. ...
Based on the KP-hierarchy reductionmethod, we construct the novel bright-dark mixed N-soliton for the (\(3+1\))-component Mel’nikov system including 3-component short waves (SWs) and one-component long wave (LW) for all possible combinations of nonlinearity coefficients. It is verified that dark or bright solitons can exist in the SW components, but only bright solitons appear in the LW component. According to different combinations of solitons in three different SW components, the bright-dark mixed N-soliton for the (\(3+1\))-component Mel’nikov system is mainly discussed into two types that two-bright-one-dark and one-bright-two-dark solitons. Finally, the (\(3+1\))-component Mel’nikov system can be directly extended to (\(M+1\))-component (\(M\ge 3\)) case comprised of M-component SWs and one-component LW, and the bright-dark mixed N-soliton for the multi-component generalization is also given in Gram determinant form.
... Zhu and Yang studied weak interactions of solitary waves which have analogous fractal dependency [7]. Some researchers observed resonant solitons which resemble the scenes of fission and fusion [8]. In 2016, Wang et al [9] found the non-elastic interplay between a soliton and a rogue wave. ...
According to N-soliton solutions generated by the Hirota’s bilinear method, the resonant collisions among diverse solitary waves can be obtained when the phase shifts of involved solitary waves tend to infinity. Under this constraint, the resonant collisions among a lump and dark line solitons can be derived from that of a breather and dark line solitons by means of an ingenious limiting approach. This paper takes the (2 + 1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation as an example to introduce how to utilize this constraint to derive the resonant collisions among different solitary waves containing breather, lump and dark line solitons in detail. At the same time, the dynamic behaviors of the interacting waveforms are exhibited visually by some figures which include intriguing phenomena. And the characteristics and properties of these interacting waveforms are discussed analytically. In terms of feasibility and practicability, the procedures of analysis in this paper can be exploited widely to study resonant collisions of different waveforms in other integrable systems. The results obtained would enhance the completeness of nonlinear localized wave theory.
... In this paper, the Painlevé analysis technique [40][41][42] is being adopted to test the integrability of the (2+1)-dimensional long wave-short wave resonance interaction (LSRI) system. Then, the truncated Painlevé approach (TPA) [43][44][45] is utilized to obtain localized solutions of LSRI system [46][47][48][49][50][51][52][53][54][55] in closed form. Here, the system considered is an extension of the two-dimensional LSRI system [56] which describes the two-dimensional resonant interaction between an interfacial gravity wave and two surface gravity packets propagating in directions symmetric about the interfacial wave propagation direction in a two-layer fluid. ...
In this article, Painlevé analysis is employed to test the integrability of (2+1)-dimensional long wave–short wave resonance interaction system using the Weiss–Tabor–Carnevale method. From the analysis, it is seen that the long wave–short wave resonance interaction system satisfies Painlevé property and the system is expected to be integrable. Then, the long wave–short wave resonance interaction system is investigated by adopting the truncated Painlevé approach. The solutions are obtained in terms of arbitrary functions in the closed form. By selecting appropriate arbitrary functions present in the solutions, localized solutions such as rogue waves, lump, one-dromion and two-dromion wave patterns are constructed. The results are also expressed graphically to illustrate the physical behavior of the long wave–short wave resonance interaction system.
... The 2D M-LSRI system (1) is completely integrable since it admits the Lax pair and can be solved by means of the inverse scattering transformation [5], the Darboux transformation [8,9], and the bilinear method [6,7,10,11]. It has been proved that this system has very rich solitary waves [9][10][11][12][13][14][15][16][17][18][19][20][21][22], among them the coherent structures of lumps and periodic solitons or dark line solitons are very interesting categories of solitary waves. ...
The resonant collisions of lumps and homoclinic orbits are studied for the two-dimensional (2D) multi-component long-wave–short-wave resonance interaction (M-LSRI) system. Compared with a usual lump, which keeps its shape and velocity unchanged in the course of the evolution from t→−∞ to t→+∞, the lumps in the resonant collision with homoclinic orbits have completely different dynamical behaviours. The resonant collisions are classified into partial resonant collisions and complete resonant collisions, depending upon the changes in the dynamical behaviours of the lumps. In the partial resonant collisions, the lumps are partially localized in time, and mainly exhibit two opposite behaviours: (i) they do not exist at t→−∞ but suddenly emerge from the background and remain existing at t→+∞, and (ii) they exist at t→−∞ and unexpectedly merge into the background and disappear at t→+∞. In the complete resonant collisions, the lumps are localized in time as well as in the two-dimensional space; they disappear at t→±∞, and at the intermediate collision process they first emerge from the background and then they merge into the background again after living for a very short period of time.
... The deterministic long-short-wave interaction system (1)-(2) with integer-order derivatives (i.e = 0 and = 1) has been investigated by many authors to get its analytical solutions by different approaches such as the Hirota's bilinearization method [41], the first integral [42], the ( ′ )-expansion [43], the simplest equation approach [44], the new modified exp(− ( ))-expansion method [45], He's semi-inverse variational principle and generalized tan( ∕2)-expansion [46], the ansatz and the exp-function [47], the complete discrimination system for polynomial [48], and the simplified the extended sinh-Gordon equation expansion [49], methods. ...
In this article, we consider the stochastic fractional-space long-shortwave interaction system (SFS-LSWIs) forced by multiplicative Brownian motion. To obtain a new exact stochastic fractional-space solutions, we apply two different methods such as sin-cos method and the Riccati-Bernoulli sub-ODE method. These solutions are essential for explaining some difficult and complicated physical phenomena. Because this system has never been investigated using a combination of multiplicative noise and fractional space, we will generalize certain previously obtained results as special cases. Furthermore, we use Matlab to plot 3D surfaces of analytical solutions produced in this study to show the impact of Brownian motion on the analytical solutions of the SFS-LSWIs.
... Resonant soliton is a special kind of soliton existing in integrable system, which has been widely studied. It is known that the phase shift of colliding solitons becomes infinite or converges to infinity, and solitons will resonate [23]. The well-known soliton fusion and fission is a resonance phenomenon [24,25]. ...
Under the well-known bilinear method of Hirota, the specific expression for N-soliton solutions of (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada(gCDGKS) equation in fluid mechanics is given. By defining a novel restrictive condition on N-soliton solutions, resonant Y-type and X-type soliton solutions are generated. Under the new proposed constraint, combined with the velocity resonance method and module resonant method, the mixed solutions of resonant Y-type solitons and line waves and breather solutions are found. Finally, with the support of long-wave limit method, the interaction between resonant Y-type solitons and higher-order lumps is shown, and the motion trajectory equation before and after the interaction between lumps and resonant Y-type solitons is derived. These new results greatly extend the exact solution of (2+1)-dimensional gCDGKS equation already available in the literature and provide new ideas for studying the dynamical behaviors of fluid mechanic, soliton and shallow water wave and so on.
... Benney has also derived a single-component LSRI system for modelling the dynamics of short capillary gravity waves and gravity waves in deep water [120]. After these works, there has been a large amount of work in the direction of LSRI involving (1 + 1) and (2 + 1)-dimensional single component and multi-component cases [135][136][137][138][139][140][141][142][143][144][145][146][147][148][149][150][151][152]. In nonlinear optics, the single component LSRI system was deduced from the coupled nonlinear Schrödinger equations describing the interaction of two optical modes under small amplitude asymptotic expansion [121]. ...
Nonlinear dynamics of an optical pulse or a beam continue to be one of the active areas of research in the field of optical solitons. Especially, in multi-mode fibers or fiber arrays and photorefractive materials, the vector solitons display rich nonlinear phenomena. Due to their fascinating and intriguing novel properties, the theory of optical vector solitons has been developed considerably both from theoretical and experimental points of view leading to soliton based promising potential applications. In the recent past, many types of vector solitons have been identified both in the integrable and non-integrable coupled nonlinear Schr\"{o}dinger (CNLS) equations framework. In this article, we review some of the recent progress in understanding the dynamics of the so called nondegenerate vector bright solitons in nonlinear optics, where the fundamental soliton can have more than one propagation constant. We address this theme by considering the integrable two CNLS family of equations, namely Manakov system, mixed 2-CNLS system, coherently CNLS system, generalized CNLS system and two-component long-wave short-wave resonance interaction (LSRI) system. In these models, we discuss the existence of nondegenerate vector solitons and their associated novel multi-hump geometrical profile nature by deriving their analytical forms through the Hirota bilinear method. Then we reveal the novel collision properties of the nondegenerate solitons in the Manakov system as an example. The asymptotic analysis shows that the nondegenerate solitons, in general, undergo three types of elastic collisions without any energy redistribution among the modes. Further, we show that the energy sharing collision exhibiting vector solitons arises as a special case of the newly reported nondegenerate vector solitons. Finally, we point out the possible further developments in this subject and potential applications.
... Benney has also derived a single-component LSRI system for modelling the dynamics of short capillary gravity waves and gravity waves in deep water [120]. After these works, there has been a large amount of work in the direction of LSRI involving (1 + 1) and (2 + 1)-dimensional single component and multi-component cases [135][136][137][138][139][140][141][142][143][144][145][146][147][148][149][150][151][152]. In nonlinear optics, the single component LSRI system was deduced from the coupled nonlinear Schrödinger equations describing the interaction of two optical modes under small amplitude asymptotic expansion [121]. ...
Nonlinear dynamics of an optical pulse or a beam continue to be one of the active areas of research in the field of optical solitons. Especially, in multi-mode fibers or fiber arrays and photorefractive materials, the vector solitons display rich nonlinear phenomena. Due to their fascinating and intriguing novel properties, the theory of optical vector solitons has been developed considerably both from theoretical and experimental points of view leading to soliton-based promising potential applications. Mathematically, the dynamics of vector solitons can be understood from the framework of the coupled nonlinear Schr\"{o}dinger (CNLS) family of equations. In the recent past, many types of vector solitons have been identified both in the integrable and non-integrable CNLS framework. In this article, we review some of the recent progress in understanding the dynamics of the so called nondegenerate vector bright solitons in nonlinear optics, where the fundamental soliton can have more than one propagation constant. We address this theme by considering the integrable two coupled nonlinear Schr\"{o}dinger family of equations, namely the Manakov system, mixed 2-CNLS system (or focusing-defocusing CNLS system), coherently coupled nonlinear Schr\"{o}dinger (CCNLS) system, generalized coupled nonlinear Schr\"{o}dinger (GCNLS) system and two-component long-wave short-wave resonance interaction (LSRI) system. In these models, we discuss the existence of
nondegenerate vector solitons and their associated novel multi-hump geometrical profile nature by deriving their analytical forms through the Hirota bilinear method. Then we reveal the novel collision properties of the nondegenerate solitons in the Manakov system as an example. The asymptotic analysis shows that the nondegenerate solitons, in general, undergo three types of elastic collisions without any energy redistribution among the modes. Furthermore, we show that the energy sharing collision exhibiting vector solitons arises as a special case of the newly reported nondegenerate vector solitons. Finally, we point out the possible further developments in this subject and potential applications.
... For example, the bright solitons in the Manakov system [12], a prototypical system for the multicomponent nonlinear Schrödinger (MCNLS) family of equations, undergo fascinating shape-changing collisions [13]; the partially coherent solitons in such MCNLS equations exhibit variable shape [14,15]; the symbiotic (bright-dark) solitons in the 2CNLS system can display boomeronic behavior (coherent structures that reverse their dynamics spontaneously) [16]; and multicomponent rogue waves with exotic profiles appear [17,18]. In higher dimensions the variants of such multicomponent CNLS systems display still more novel phenomena such as spatiotemporal multimode optical solitons [19], fission of solitons into a lump coherent structure and fusion of a lump into solitons [20], expanding necklace ring solitons [21], and resonant solitons with intricate structures [22] (for a detailed review one may refer to [23] and references therein). ...
Multiple double-pole bright-bright and bright-dark soliton solutions for the multicomponent nonlinear Schrödinger (MCNLS) system comprising three types of nonlinearities, namely, focusing, defocusing, and mixed (focusing-defocusing) nonlinearities, arising in different physical settings are constructed. An interesting type of energy-exchanging phenomenon during collision of these double-pole solitons is unraveled. To explore the objectives, we consider the general solutions of a set of generalized MCNLS equations and by taking the long-wavelength limit with proper parameter choices of single-pole bright-bright and bright-dark soliton pairs, the multiple double-pole bright-bright and bright-dark soliton solutions are constructed in terms of determinants. The regular double-pole bright-bright solitons exist in the focusing and focusing-defocusing MCNLS equations and undergo a particular type of energy-sharing collision for M≥2 in addition to the usual elastic collisions. A striking feature observed in the process of energy-sharing collisions is that the double-pole two-soliton possessing unequal intensities before collision indeed exactly exchange their intensities after collision. Further, the existence of double-pole bright-dark solitons in the MCNLS equations with focusing, defocusing, and mixed (focusing-defocusing) nonlinearities is analyzed by constructing explicit determinant form solutions, where the double-pole bright solitons exhibit elastic and energy-exchanging collisions while the double-pole dark solitons undergo mere elastic collision. The double-pole bright-dark solitons possess much richer localized coherent patterns than their counterpart double-pole bright-bright solitons. For particular choices of parameters, we demonstrate that the solitons would degenerate into the background, resulting in a lower number of solitons. Another important observation is the formation of doubly localized rogue waves with extreme amplitude, in the case of double-pole bright-dark four-solitons. Our results should stimulate interest in such special multipole localized structures and are expected to have ramifications in nonlinear optics.
... Painlevé singularity structure analysis is one of the efficient tools to understand the integrability nature of any dynamical (ordinary/partial differential) equation in both one-and higherdimensions [57][58][59][60][61][62][63]. This includes three important steps such as the identification of leading order, determination of resonances and arbitrary analysis to ensure the availability of required number of free parameters. ...
Considering the importance of ever-increasing interest in exploring localized waves, we investigate a generalized (3+1)-dimensional Hirota-Satsuma-Ito equation describing the unidirectional propagation of shallow-water waves and perform Painlev\'e analysis to understand its integrability nature. We construct the explicit form of higher-order rogue wave solutions by adopting Hirota's bilinearization and generalized polynomial functions. Further, we explore their dynamics in detail, depicting different pattern formation that reveal potential advantages with available arbitrary constants in their manipulation mechanism. Particularly, we demonstrate the existence of singly-localized line-rogue waves and doubly-localized rogue waves with multiple (single, triple, and sextuple) structures generating triangular and pentagon type geometrical patterns with controllable orientations that can be altered appropriately by tuning the parameters. The presented analysis will be an essential inclusion in the context of rogue waves in higher-dimensional systems.
... Recently, energy transfer between solitons or multicomponent nonlinear systems has attracted much attention [29,[36][37][38]. Vijayajayanthi et al. [37] investigated the interaction of bright-dark solitons in the N-coupled nonlinear Schrödinger equations, and the interaction would give rise to energy transfer between the solitons and the multi-components. ...
Herein, we employ the bilinear method and the KP hierarchy reduction technique for obtaining a hierarchy of semi-rational solutions to the Mel’nikov equation. These semi-rational solutions are given in terms of determinants whose matrix elements have plain algebraic expressions. Under suitable parametric conditions, these semi-rational solutions reduce to rational solutions of the Mel’nikov equation. These semi-rational solutions reveal two opposite types of excitation phenomena: fusion and fission, which are determined by a input parameter \(\gamma \). The fundamental (first-order) semi-rational solutions describe a process of a lump originating from a dark soliton and then coexisting with the dark soliton as \(t{\gg }0\), or a process of a lump coexisting with the dark soliton as \(t{\ll }0\) and then fusing into the dark soliton. The higher-order semi-rational solutions exhibit \(n_i\,(n_{i}\ge 2)\) lumps splitting from or annihilating into one dark soliton. The multi-semi-rational solutions describe \(N\,(N \ge 2)\) lumps fissuring from or fusing into N-dark solitons.
... The nonlinear long-short wave interaction systems with considering a general theory for interactions between short and long waves first introduced by Benney (1977). Describes of the nonlinear resonance interaction of multiple short waves with a long wave in two spatial dimension have been investigated by Sakkaravarthi et al. (2014) by applying the Hirota's bilinearization method. The entangled mapping approach based on the general reduction theory was investigated by Dai and Liu (2012), in which they have derived new type of variable separation solution for the considered model. ...
This paper deals with exact soliton solutions of the nonlinear long–short wave interaction system, utilizing two analytical methods. The system of coupled long–short wave interaction equations is investigated with the help of two analytical methods, namely, the generalized \(\tan (\phi /2)\)-expansion method and He’s semi-inverse variational method. Moreover, in this paper we generalize two aforementioned methods which give new soliton wave solutions. As a consequence, solutions are including solitons, kink, periodic and rational solutions. Moreover, dark, bright and singular solition solutions of the coupled long–short wave interaction equations have been found. All solutions have been verified back into its corresponding equation with the aid of maple package program. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, we believe that the executed methods are robust and efficient than other methods and the obtained solutions in this paper can help us to understand the soliton waves in the fields of physics and mechanics.
... This work deals with the following general M-component (2 + 1)-dimensional LSRI system derived by Kanna et al. [23] by employing a multiple-scale perturbation method, in the waveguide geometry t is the propagation direction, while x and y denote the transverse coordinates. System (1.1) has been shown to be integrable by the Painlevé analysis and exact bright multisoliton solutions displaying fascinating energy-sharing (shape-changing) collisions are obtained in [24]. In addition, other solutions including dark-dark and bright-dark solitons as well as rogue waves have been studied by Chen et al. [25][26][27] in one-dimensional and in two-dimensional settings. ...
General semi-rational solutions of an integrable multi-component (2+1)-dimensional long-wave-short-wave resonance interaction system comprising multiple short waves and a single long wave are obtained by employing the bilinear method. These solutions describe the interactions between various types of solutions, including line rogue waves, lumps, breathers and dark solitons. We only focus on the dynamical behaviours of the interactions between lumps and dark solitons in this paper. Our detailed study reveals two different types of excitation phenomena: fusion and fission. It is shown that the fundamental (simplest) semi-rational solutions can exhibit fission of a dark soliton into a lump and a dark soliton or fusion of one lump and one dark soliton into a dark soliton. The non-fundamental semi-rational solutions are further classified into three subclasses: higher-order, multi- and mixed-type semi-rational solutions. The higher-order semi-rational solutions show the process of annihilation (production) of two or more lumps into (from) one dark soliton. The multi-semi-rational solutions describe N(N≥2) lumps annihilating into or producing from N-dark solitons. The mixed-type semi-rational solutions are a hybrid of higher-order semi-rational solutions and multi-semi-rational solutions. For the mixed-type semi-rational solutions, we demonstrate an interesting dynamical behaviour that is characterized by partial suppression or creation of lumps from the dark solitons.
... Certain is it that nonlinear long-short-wave interaction systems, (NLSWIS), do mask inherently significant nonlinear processes, predictive of highly complex physical phenomena. NLSWIS processes do model nonlinear dynamical interaction between low-frequency long waves, and high-frequency short waves [1]. Highly motivating is uncovering basic physical interactions leading to further study and investigation of various nonlinear interactions underlying the general solution structure including: analytical, dark, and approximate solutions. ...
In this paper, we investigate and use the new modified exp(−Ω (ξ ))-expansion method,
(MEM). We apply the new MEM to nonlinear long–short-wave interaction systems
(NLSWIS). Among our findings are sets of solutions including, but not limited to, new
hyperbolic, complex, and dark soliton solutions. Not only is MEM shown to be highly
adaptable for partial differential equations with strong nonlinearities, but also, it turns out to be highly efficient, despite its ease.
... Certain is it that nonlinear long-short-wave interaction systems, (NLSWIS), do mask inherently significant nonlinear processes, predictive of highly complex physical phenomena. NLSWIS processes do model nonlinear dynamical interaction between low-frequency long waves, and high-frequency short waves [1]. Highly motivating is uncovering basic physical interactions leading to further study and investigation of various nonlinear interactions underlying the general solution structure including: analytical, dark, and approximate solutions. ...
... Certain is it that nonlinear long-short-wave interaction systems, (NLSWIS), do mask inherently significant nonlinear processes, predictive of highly complex physical phenomena. NLSWIS processes do model nonlinear dynamical interaction between low-frequency long waves, and high-frequency short waves [1]. Highly motivating is uncovering basic physical interactions leading to further study and investigation of various nonlinear interactions underlying the general solution structure including: analytical, dark, and approximate solutions. ...
... Certain is it that nonlinear long short-wave interaction systems, (NLSWIS), do mask inherently significant nonlinear processes, predictive of highly complex physical phenomena. NLSWIS processes do model nonlinear dynamical interaction between low-frequency long waves, and high-frequency short waves [1]. Highly motivating is uncovering basic physical interactions leading to further study and investigation of various nonlinear interactions underlying the general solution structure including: analytical, dark, and approximate solutions. ...
In this paper, we investigate and use the new modified exp -Ω ξ -expansion method, (MEM). We apply the new MEM to nonlinear long short-wave interactions systems (NLSWIS). Among our findings are sets of solutions including, but not limited to, new hyperbolic, complex, and dark soliton solutions. Not only is MEM shown to be highly adaptable for partial differential equations with strong nonlinearities, but also, it turns out to be highly efficient, despite its ease.
In this paper, a (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Konopelchenko-Schiff system in a fluid or plasma is investigated. Via the Hirota method and symbolic computation, we work out some two-resonance Y-type soliton solutions as well as some hybrid solutions composed of the two-resonance Y-type solitons and solitons/breathers. Graphically, we display some two-resonance Y-type solitons. We present the interactions between the two-resonance Y-type soliton and one soliton, among the two-resonance Y-type soliton and two solitons, between the two-resonance Y-type soliton and first-order breather as well as among the two-resonance Y-type soliton and second-order breathers.
Purpose
This manuscript addresses to investigate the optical soliton solution of the Manakov model which governs soliton transmission technology by utilizing the auxiliary equation technique.
Methodology
After obtaining the nonlinear ordinary differential equation form (NODE) of investigated nonlinear differential partial problem (NLPDE) with the complex wave transform (cWT), the serial form is suggested in accordance with the utilized auxiliary method, and after written in the NODE form, the linear algebraic system is determined by the appropriate solution of this system and the unknown parameters are calculated. Then, the proposed series form, the solution functions of the method, and the cWT are brought together to provide the main equation. By considering the functions that satisfy the main equation as solution functions, graphical presentations are made and interpreted in order to better understand the obtained results.
Findings
By applying the effective auxiliary equation method, optical soliton pulse solutions such as bright, singular, V-shaped, periodic singular, periodic bright, periodic bright-dark have been obtained, and it has been shown that the method can be applied effectively for such equations.
Originality
The auxiliary equation method has not been applied to the Manakov model and the V-shaped, periodic singular, periodic bright and periodic dark-bright optical soliton pulse solutions have not been reported before.
In this paper, we consider a generalized (2+1)-dimensional nonlinear wave equation. Based on the bilinear method, the N-soliton solutions are obtained. The resonance Y -type soliton, which is similar to the capital letter Y in the spatial structure, and the interaction solutions between different types of resonance solitons are constructed by adding some new constraints to the parameters of the N-soliton solutions. The new type of two-opening resonance Y -type soliton solutions are presented by choosing some appropriate parameters in 3-soliton solutions. The hybrid solutions consisting of resonance Y -type solitons, breathers and lumps are investigated. The trajectories of the lump waves before and after the collision with the resonance Y -type solitons are analyzed from the perspective of mathematical mechanism. Furthermore, the multi-dimensional Riemann-theta function is employed to investigate the quasi-periodic wave solutions. The one-periodic and two-periodic wave solutions are obtained. The asymptotic properties are systematically analyzed, which establish the relations between the quasi-periodic wave
solutions and the soliton solutions. The results may be helpful to provide some ef-
fective information to analyze the dynamical behaviors of solitons, fluid mechanics,
shallow water waves and optical solitons.
For a real nonlinear Klein–Gordon Lagrangian density with a special solitary wave solution (SSWS), which is essentially unstable, it is shown how adding a proper additional massless term could guarantee the energetically stability of the SSWS, without changing its dominant dynamical equation and other properties. In other words, it is a stability catalyzer. The additional term contains a parameter B, which brings about more stability for the SSWS at larger values. Hence, if one considers B to be an extremely large value, then any other solution which is not very close to the free far apart SSWSs and the trivial vacuum state, require an infinite amount of energy to be created. In other words, the possible non-trivial stable configurations of the fields with the finite total energies are any number of the far apart SSWSs, similar to any number of identical particles.
It is critical to excite pulse wave with higher amplitudes and higher frequencies in optical application. In this work, under investigation is an inhomogeneous fiber system described by a generalized variable coefficient Schrödinger equation. Analytical two-breather solution is obtained through Darboux transformation, a symbolic computation technique. The super-regular breathers can be generated via breathers colliding. The effects of amplitude and frequency amplifications can be observed as the spectral eigenvalue parameters are adjusted.
Long-wave–short-wave resonance interaction (LSRI) equations have been studied in the plasmas, gravity waves, nonlinear electron-plasma and ion-acoustic waves. By virtue of the bilinear method, two soliton solutions of the variable-coefficient LSRI equations are attained. Interaction of the solitons are studied when the coefficients are taken as the generalized Gauss functions. New types of the soliton interaction are exhibited. Position and width of the disturbances can be controlled.
Based on the KP hierarchy reduction method, we construct the general bright-dark mixed N-soliton solution of the two-dimensional (2D) (M+1)-component Maccari system comprised of M-component short waves (SWs) and one-component long wave (LW) with all possible combinations of nonlinearities. We firstly consider two types of mixed N-soliton solutions (two-bright-one-dark and one-bright-two-dark solitons in SW components) to the (3+1)-component Maccari system in detail. Then by extending our analysis to the (M+1)-component Maccari system, its general m-bright-(M–m)-dark mixed N-soliton solution is obtained. The formula obtained also contains the general all-bright and all-dark N-soliton solutions as special cases. For the two-bright-one-dark mixed soliton solution of the (3+1)-component Maccari system, it can be shown that solioff excitation and solioff interaction take place in the two SW components supporting bright solitons, whereas the SW component supporting dark solitons and the LW component possess V-type solitary and interaction.
Dilute-gas Bose-Einstein condensates are an exceptionally versatile testbed for the investigation of novel solitonic structures. While matter-wave solitons in one- and two-component systems have been the focus of intense research efforts, an extension to three components has never been attempted in experiments, to the best of our knowledge. Here, we experimentally demonstrate the existence of robust dark-bright-bright (DBB) and dark-dark-bright (DDB) solitons in a spinor $F=1$ condensate. We observe lifetimes on the order of hundreds of milliseconds for these structures. Our theoretical analysis, based on a multiscale expansion method, shows that small-amplitude solitons of these types obey universal long-short wave resonant interaction models, namely Yajima-Oikawa systems. Our experimental and analytical findings are corroborated by direct numerical simulations highlighting the persistence of, e.g., the DBB states, as well as their robust oscillations in the trap.
In this paper, we develop a new analytical method called as the modified exp (−Ω(ξ))-expansion function method giving more analytical solutions for partial differential equations with have powerful nonlinearity especially, and based on the exp(−Ω(ξ))-expansion method. We have applied this new approach to the nonlinear long shortwave interaction system being stand for relationships among waves such as water, gravity, high cosmic and so on. Afterwards, we have found some new hyperbolic function solution, trigonometric function solution and complex function solution for the nonlinear long shortwave interaction system by using this new the modified exp(−Ω(ξ))-expansion function method.
We present new version of previously published Fortran and C programs for solving the Gross-Pitaevskii equation for a Bose-Einstein condensate with contact interaction in one, two and three spatial dimensions in imaginary and real time, yielding both stationary and non-stationary solutions. To reduce the execution time on multicore processors, new versions of parallelized programs are developed using Open Multi-Processing (OpenMP) interface. The input in the previous versions of programs was the mathematical quantity nonlinearity for dimensionless form of Gross-Pitaevskii equation, whereas in the present programs the inputs are quantities of experimental interest, such as, number of atoms, scattering length, oscillator length for the trap, etc. New output files for some integrated one- and two-dimensional densities of experimental interest are given. We also present speedup test results for the new programs.
The long wave--short wave resonance model arises physically when the
phase velocity of a long wave matches the group velocity of a short
wave. It is a system of nonlinear evolution equations solvable by the
Hirota bilinear method and also possesses a Lax pair formulation.
``Rogue wave'' modes, algebraically localized entities in both space and
time, are constructed from the breathers by a singular limit involving a
``coalescence'' of wavenumbers in the long wave regime. In contrast with
the extensively studied nonlinear Schrödinger case, the frequency
of the breather cannot be real and must satisfy a cubic equation with
complex coefficients. The same limiting procedure applied to the finite
wavenumber regime will yield mixed exponential-algebraic solitary waves,
similar to the classical ``double pole'' solutions of other evolution
systems.
Exact explicit rogue-wave solutions of intricate structures are presented for the long-wave-short-wave resonance equation. These vector parametric solutions feature coupled dark- and bright-field counterparts of the Peregrine soliton. Numerical simulations show the robustness of dark and bright rogue waves in spite of the onset of modulational instability. Dark fields originate from the complex interplay between anomalous dispersion and the nonlinearity driven by the coupled long wave. This unusual mechanism, not available in scalar nonlinear wave equation models, can provide a route to the experimental realization of dark rogue waves in, for instance, negative index media or with capillary-gravity waves.
We consider the multicomponent Yajima-Oikawa (YO) system and show that the two-component YO system can be derived in a physical setting of a three-coupled nonlinear Schrödinger (3-CNLS) type system by the asymptotic reduction method. The derivation is further generalized to the multicomponent case. This set of equations describes the dynamics of nonlinear resonant interaction between a one-dimensional long wave and multiple short waves. The Painlevé analysis of the general multicomponent YO system shows that the underlying set of evolution equations is integrable for arbitrary nonlinearity coefficients which will result in three different sets of equations corresponding to positive, negative, and mixed nonlinearity coefficients. We obtain the general bright N-soliton solution of the multicomponent YO system in the Gram determinant form by using Hirota's bilinearization method and explicitly analyze the one- and two-soliton solutions of the multicomponent YO system for the above mentioned three choices of nonlinearity coefficients. We also point out that the 3-CNLS system admits special asymptotic solitons of bright, dark, anti-dark, and gray types, when the long-wave-short-wave resonance takes place. The short-wave component solitons undergo two types of energy-sharing collisions. Specifically, in the two-component YO system, we demonstrate that two types of energy-sharing collisions-(i) energy switching with opposite nature for a particular soliton in two components and (ii) similar kind of energy switching for a given soliton in both components-result for two different choices of nonlinearity coefficients. The solitons appearing in the long-wave component always exhibit elastic collision whereas those of short-wave components exhibit standard elastic collisions only for a specific choice of parameters. We have also investigated the collision dynamics of asymptotic solitons in the original 3-CNLS system. For completeness, we explore the three-soliton interaction and demonstrate the pairwise nature of collisions and unravel the fascinating state restoration property.
In this communication, we investigate the two-component long-wave-short-wave resonance interaction equation and show that it admits the Painlevé property. We then suitably exploit the recently developed truncated Painlevé approach to generate exponentially localized solutions for the short-wave components S(1) and S(2) while the long wave L admits a line soliton only. The exponentially localized solutions driving the short waves S(1) and S(2) in the y-direction are endowed with different energies (intensities) and are called 'multimode dromions'. We also observe that the multimode dromions suffer from intramodal inelastic collision while the existence of a firewall across the modes prevents the switching of energy between the modes.
Two-dimensional resonant interaction between an internal gravity wave and a surface gravity wave packet in a two-layer fluid is investigated. The equations describing this interaction are derived. Modulational instability of a plane wave solution of the quations is discussed. Some interesting solutions which follow from a two-soliton solution to these equations are given.
We construct and discuss a semi-rational, multi-parametric vector solution of
coupled nonlinear Schr\"odinger equations (Manakov system). This family of
solutions includes known vector Peregrine solutions, bright-dark-rogue
solutions, and novel vector unusual freak waves. The vector freak (or rogue)
waves could be of great interest in a variety of complex systems, from optics
to Bose-Einstein condensates and finance.
We demonstrate experimentally multi-bound-soliton solutions of the Nonlinear
Schr\"odinger equation (NLS) in the context of surface gravity waves. In
particular, the Satsuma-Yajima N-soliton solution with N=2,3,4 is investigated
in detail. Such solutions, also known as breathers on zero background, lead to
periodic self-focussing in the wave group dynamics, and the consequent
generation of a steep localized carrier wave underneath the group envelope. Our
experimental results are compared with predictions from the NLS for low
steepness initial conditions where wave-breaking does not occur, with very good
agreement. We also show the first detailed experimental study of irreversible
massive spectral broadening of the water wave spectrum, which we refer to by
analogy with optics as the first controlled observation of hydrodynamic
supercontinuum a process which is shown to be associated with the fission of
the initial multi-soliton bound state into individual fundamental solitons
similar to what has been observe in optics.
Using explicit, bright-soliton solutions for the coupled Manakov system recently described by Radhakrishnan, Lakshmanan, and Hietarinta, we show that collisions of these solitons can be completely described by explicit linear fractional transformations of a complex-valued polarization state. We design sequences of solitons operating on other sequences of solitons that effect logic operations, including controlled NOT gates. Both data and logic operators have the self-restoring and reusability features of digital logic circuits. This suggests a method for implementing computation in a bulk nonlinear medium without interconnecting discrete components.
In this paper, we construct the bright-soliton bound states of an integrable (2 + 1)-dimensional multicomponent long wave-short wave resonance interaction (LSRI) system by using the exact bright-soliton solutions obtained in Ref. [24] and analyze their interesting collision dynamics. We show that the beating and breathing oscillations of the bound solitons can be controlled by tuning the polarization parameters. Also, we explore the interaction between the bound-soliton and a standard soliton. We also point out that the two bound-soliton state seems to be robust against collision with a standard soliton and remain to be bounded even after collision.
The exact bright one- and two-soliton solutions of a particular type of coherently coupled nonlinear Schrödinger equations, with alternate signs of nonlinearities among the two components, are obtained using the non-standard Hirota's bilinearization method. We find that in contrary to the coherently coupled nonlinear Schrödinger equations with same signs of nonlinearities the present system supports only coherently coupled solitons arising due to an interplay between dispersion and the nonlinear effects, namely, self-phase modulation, cross-phase modulation, and four-wave mixing process, thereby depend on the phases of the two co-propagating fields. The other type of soliton, namely, incoherently coupled solitons which are insensitive to the phases of the co-propagating fields and arise in a similar kind of coherently coupled nonlinear Schrödinger equations but with same signs of nonlinearities are not at all possible in the present system. The present system can support regular solution for the choice of soliton parameters for which mixed coupled nonlinear Schrödinger equations admit only singular solution. Our analysis on the collision dynamics of the bright solitons reveals the important fact that in contrary to the other types of coupled nonlinear Schrödinger systems the bright solitons of the present system can undergo only elastic collision in spite of their multicomponent nature. We also show that regular two-soliton bound states can exist even for the choice for which the same system admits singular one-soliton solution. Another important effect identified regarding the bound solitons is that the breathing effects of these bound solitons can be controlled by tuning the additional soliton parameters resulting due to the multicomponent nature of the system which do not have any significant effects on bright one soliton propagation and also in soliton collision dynamics.
In this paper we define the Painlevé property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Bäcklund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painlevé property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.
It is shown that a generalized nonlinear Schrödinger equation proposed by Malomed and Stenflo admits, for a specific range of parameters, resonant soliton interaction. The equation is transformed to the 'resonant' nonlinear Schrödinger equation, as originally introduced to describe black holes in a Madelung fluid and recently derived in the context of uniaxial wave propagation in a cold collisionless plasma. A Hirota bilinear representation is obtained and soliton solutions are thereby derived. The one-soliton solution interpretation in terms of a black hole in two-dimensional spacetime is given. For the two-soliton solution, resonant interactions of several kinds are found. The addition of a quantum potential term is considered and the reduction is obtained to the resonant NLS equation.
We consider the integrable multicomponent coherently coupled nonlinear
Schr\"odinger (CCNLS) equations describing simultaneous propagation of multiple
fields in Kerr type nonlinear media. The correct bilinear equations of
$m$-CCNLS equations are obtained by using a non-standard type of Hirota's
bilinearization method and the more general bright one solitons with single
hump and double hump profiles including special flat-top profiles are obtained.
The solitons are classified as coherently coupled solitons and incoherently
coupled solitons depending upon the presence and absence of coherent
nonlinearity arising due to the existence of the co-propagating
modes/components. Further, by obtaining the more general two-soliton solutions
using this non-standard bilinearization approach we demonstrate that the
collision among coherently coupled soliton and incoherently coupled soliton
displays a non-trivial collision behaviour in which the former always undergoes
energy switching accompanied by an amplitude dependent phase-shift and change
in the relative separation distance, leaving the latter unaltered. But the
collision between coherently coupled solitons alone is found to be standard
elastic collision. Our study also reveals the important fact that the collision
between incoherently coupled solitons arising in the $m$-CCNLS system with
$m=2$ is always elastic, whereas for $m>2$ the collision becomes intricate and
for this case the $m$-CCNLS system exhibits interesting energy sharing
collision of solitons characterized by intensity redistribution, amplitude
dependent phase-shift and change in relative separation distance which is
similar to that of the multicomponent Manakov soliton collisions. This suggests
that the $m$-CCNLS system can also be a suitable candidate for soliton
collision based optical computing in addition to the Manakov system.
Interaction of two long-crested shallow water waves is analysed in the framework of the two-soliton solution of the Kadomtsev-Petviashvili equation. The wave system is decomposed into the incoming waves and the interaction soliton that represents the particularly high wave hump in the crossing area of the waves. Shown is that extreme surface elevations up to four times exceeding the amplitude of the incoming waves typically cover a very small area but in the near-resonance case they may have considerable extension. An application of the proposed mechanism to fast ferries wash is discussed.
We characterize dark-type vector optical solitons of arbitrary polarization in isotropic, Kerr-type media by applying Hirota's method to the integrable Manakov model with a defocusing nonlinearity. We find that nonuniformly polarized solitons comprise a rich solution family that can be divided into two categories: dark-dark and dark-bright vector solitons. We consider the propagation dynamics and the interactions of these vector solitons by deriving multisoliton solutions, and show the existence of stationary bound states, a phenomenon not observed for scalar dark solitons.
Under certain conditions it is found that the interactions between short and long waves are especially important. The type of analysis presented has applications to many nonlinear dispersive wave systems, but the detailed discussion is restricted to gravity-capillary wave interactions.
We derive a (2+1)-dimensional multicomponent long-wave-short-wave resonance interaction (LSRI) system as the evolution equation for propagation of N-dispersive waves in weak Kerr-type nonlinear medium in the small-amplitude limit. The mixed- (bright-dark) type soliton solutions of a particular (2+1)-dimensional multicomponent LSRI system, deduced from the general multicomponent higher-dimensional LSRI system, are obtained by applying the Hirota's bilinearization method. Particularly, we show that the solitons in the LSRI system with two short-wave components behave like scalar solitons. We point out that for an N-component LSRI system with N>3, if the bright solitons appear in at least two components, interesting collision behavior takes place, resulting in energy exchange among the bright solitons. However, the dark solitons undergo standard elastic collision accompanied by a position shift and a phase shift. Our analysis on the mixed bound solitons shows that the additional degree of freedom which arises due to the higher-dimensional nature of the system results in a wide range of parameters for which the soliton collision can take place.
A complete analysis is given of the long-wave - short-wave resonance equations which appear in fluid mechanics as well as plasma physics. Using the inverse-scattering technique, these equations can be reduced to a pair of linear integral equations (Marchenko equations), with the N-soliton solutions intimately related to the asymptotic state of evolution equations. The interaction of solitons and the conserved quantities are discussed.
We investigate the propagation of a longitudinal-transverse elastic pulse in a statically deformed crystal containing paramagnetic impurities and placed in an external magnetic field. We derive a system of three nonlinear wave equations describing the interaction of the pulse with the paramagnetic impurities in the quasiresonance approximation in the Faraday geometry. We assume that the transverse components of the pulse, which cause quantum transitions, have carrier frequencies and are short-wave (acoustic), while the longitudinal component has no carrier frequency and is long-wave. We show that in the case of an equilibrium initial distribution of populations of quantum levels of paramagnetic impurities, the coupling between the longitudinal and transverse components is weak, the pulse is therefore strictly transverse, and its dynamics are described by the Manakov system. With a nonequilibrium initial distribution of populations, conditions of effective interaction between all components of the elastic pulse can be reached, and their nonlinear dynamics are described by a vector generalization of the Zakharov equations. In the case of a unidirectional propagation of the pulse, these equations reduce to the Yajima-Oikawa vector system. We show that the obtained system of equations and its version with an arbitrary number of short-wave components can be integrated using the inverse scattering transform. We construct infinite hierarchies of solutions of the Yajima-Oikawa vector system (including a solution on a nontrivial background). We consider stationary (complex-valued Garnier system) and self-similar reductions of that system, also admitting a representation in the form of compatibility conditions.
N-dark–dark solitons in the integrable coupled NLS equations are derived by the KP-hierarchy reduction method. These solitons exist when nonlinearities are all defocusing, or both focusing and defocusing nonlinearities are mixed. When these solitons collide with each other, energies in both components of the solitons completely transmit through. This behavior contrasts collisions of bright–bright solitons in similar systems, where polarization rotation and soliton reflection can take place. It is also shown that in the mixed-nonlinearity case, two dark–dark solitons can form a stationary bound state.
A vector generalization of the Yajima-Oikawa equations for the interaction between a quasi-resonance circularly polarized optical pulse and a long-wave electromagnetic spike has been derived. This new system has been shown to be integrable using the method of the inverse scattering transformation. The soliton solutions have been found and analyzed.
Coupled nonlinear partial differential equations, which describe a nonlinear interaction between short and long capillary-gravity waves on a liquid layer of uniform depth, are derived by the derivative expansion method. The short and the long waves can exchange energy in a resonant manner, if the group velocity of the short wave is close to the phase velocity of the long wave. It is found that the long wave can take a form of rarefactive (convex downwards) solitary wave due to the resonant interaction. This should be compared with the well-known gravity wave soliton which is compressive (convex upwards) in the absence of the short wave.
We consider behaviour of sonic-Langmuir solitons which are Langmuir oscillations trapped in regions of reduced plasma density caused by the ponderomotive force due to a high-frequency field. In particular, the formation and the interaction of solitons are studied by the inverse scattering technique in the case of the Langmuir waves coupled with ionacoustic waves propagating in one-direction.
Nonlinear partial differential equations which permit both short and long wave solutions are studied. The theory reveals a variety of phenomena relevant to a broad class of physical problems. Among the properties investigated are resonances, instabilities and steady state solutions.
We obtain solutions describing stationary one-dimensional propagation of a coupled nonlinear electron-plasma wave and a nonlinear ion-acoustic wave. These waves have amplitudes linearly proportional to one another, and propagate with approximately the ion-acoustic velocity in the form of periodic wave trains, including solitary waves as a special case.
The two-dimensional nonlinear interaction of two planar ion-acoustic solitons has been studied experimentally. When the angle between the wave vectors of the two interacting solitons is small and the soliton amplitudes approach a critical value, a resonant three-soliton interaction occurs.
The Zakharov-Shabat theory of integrable systems with more than one spatial dimension breaks down when certain resonances between solitons occur. Remarkably, the resonance condition is precisely the same as that for the strong interaction of three weakly coupled linear dispersive waves. This provides a mechanism for the creation of new particles by collision.
The long-short wave interactions between an internal gravity wave and a surface gravity wave packet in a two-layer fluid are investigated. When the phase speed of the internal wave coincides with the group velocity of the surface wave packet, a strong interaction occurs. The equations describing this interaction are derived both for a shallow (in comparison with a length scale of the long wave) fluid layer and for a deep one, and are studied numerically.
The evolution equations describing the interaction of long and short waves when they are resonantly coupled are discussed. Envelope-pulse soliton solutions of these equations are constructed using the method of Hirota and the existence of a breather state is shown. The wavetrain modulational characteristics and envelope-hole solitary wave solutions are also presented.
The three dimensionality of drift vortex solitons in a convective motion is investigated. The propagation of vortex solitons is described by the Kadomtsev-Petviashvili equation with negative dispersion. It is pointed out that under a certain condition the vortex soliton resonance is possible.
We characterize dark-type vector optical solitons of arbitrary polarization in isotropic, Kerr-type media by applying Hirota's method to the integrable Manakov model with a defocusing nonlinearity. We find that nonuniformly polarized solitons comprise a rich solution family that can be divided into two categories: dark-dark and dark-bright vector solitons. We consider the propagation dynamics and the interactions of these vector solitons by deriving multisoliton solutions, and show the existence of stationary bound states, a phenomenon not observed for scalar dark solitons.
The derivative expansion method is applied to an investigation of the
weakly nonlinear self-interactions of capillary-gravity waves on a
liquid layer of uniform depth. The stability characteristics of a wave
train are examined on the basis of the nonlinear Schrödinger
equation. It is shown that the effect of capillarity is of critical
importance to the modulational instability.
An inviscid, incompressible, stably stratified fluid occupies a horizontal channel, along which an internal gravity-wave packet is propagating. The wave induced mean motions are calculated, and the equations describing the evolution of the wave amplitude derived. When the group velocity of the wave packet coincides with a long-wave speed there is a resonance, and the equations describing this resonance are derived.
An analysis is presented which describes the slow-time evolution of an internal gravity wave in an arbitrarily specified stratification. The weakly nonlinear description of a single-wave mode, governed by the nonlinear Schrödinger equation, breaks down when certain resonant conditions are satisfied. One such condition occurs when the group velocity of the wavetrain is equal to the phase velocity of a higher-mode long wave of the system. The resonant interaction occurs on a faster time scale and is described by a coupled pair of nonlinear partial differential equations governing the evolution of both the short-wave and the long-wave modes. This long-wave/short-wave interaction is pursued further in an experimental investigation by measuring the modal interchange of energy between two internal waves of disparate length and time scales. The resulting data are compared with numerical solutions of the long-wave/short-wave resonant interaction equations. In general, the agreement between the theory and the experiment is reasonably good in the range of operating conditions for which the theory is valid.
Resonant (phase-locked) interactions among three obliquely oriented solitary waves are studied. It is shown that such interactions are associated with the parametric end points of the singular regime for interactions between two solitary waves. The latter include regular reflexion at a rigid wall, which is impossible for φi < (3α)½ (φ = angle of incidence, α = amplitude/depth [double less-than sign] 1), and it is shown that the observed phenomenon of ‘Mach reflexion’ can be described as a resonant interaction in this regime. The run-up at the wall is calculated as a function of φi/(3α)½ and is found to have a maximum value of 4αd for φi = (3α)½. This same resonant interaction also describes diffraction of a solitary wave at a corner of internal angle π − ψi, −(3α)½, and suggests that a solitary wave cannot turn through an angle in excess of (3α)½ at a convex corner without separating or otherwise losing its identity.
Nonlinear oblique interactions between two slightly dispersive gravity waves (in particular, solitary waves) of dimensionless amplitudes α 1 and α 2 (relative to depth) and relative inclination 2ϕ (between wave normals) are classified as weak if sin ² ϕ α 1,2 or strong if ϕ ² = O (α 1,2 ). Weak interactions permit superposition of the individual solutions of the Korteweg-de Vries equation in first approximation; the interaction term, which is O (α 1 α 2 ), then is determined from these basic solutions.
Strong interactions are intrinsically nonlinear. It is shown that these interactions are phase-conserving (the sum of the phases of the incoming waves is equal to the sum of the phases of the outgoing waves) if |α 2 -α 1 > (2ϕ) ² but not if |α 2 -α 1 | (2ϕ) ² (e.g. the reflexion problem, for which the interacting waves are images and α 2 = α 1 ). It also is shown that the interactions are singular, in the sense that regular incoming waves with sech ² profiles yield singular outgoing waves with - csch ² profiles, if \[ \psi_{-}< |\psi| < \psi_{+},\quad{\rm where}\quad\psi_{\pm}={\textstyle\frac{1}{2}}\left|(3\alpha_2)^{\frac{1}{2}}\pm(3\alpha_1)^{\frac{1}{2}}\right|. \]
Regular interactions appear to be impossible within this singular regime, and its end points, |ϕ| = ϕ±, are associated with resonant interactions.
The two-soliton solution of the KdV equation is known not to degenerate into a solution describing resonant triads of solitons in 1 + 1 dimensions. However, we show that two integrable coupled-KdV systems of the Drinfeld–Sokolov class possess solutions which may degenerate into resonant triads. These solutions are associated with a resonance relation which generalizes the usual one previously considered by Hirota and Ito.
Resonant multisoliton interactions in one space dimension, involving a resonance triad and an arbitrary number N of solitary waves are discussed for the good Boussinesq (GB) equation. It is shown that any of the special (regular) N-soliton solutions, which are obtained at the boundary of the regularity domain, describe one of the four basic processes (related by time reversal and/or space reversal) in the presence of (N-2) (or (N-3)) 'spectator' solitons. In contrast with the two-dimensional case (KP equation), the GB resonant vertices cannot include more than three solitary waves.
A periodic soliton is turned into a line soliton accordingly as a parameter point approaches to the boundary of the existing domain in the parameter space for a nonsingular periodic-soliton solution. We will call the periodic soliton with parameters of the neighborhood of the boundary a quasi-line soliton in this paper, which seems to be the line soliton. The interaction between two quasi-line solitons is the same as the interaction between two line solitons, except for very small parameter-sensitive regions. However, in such parameter regions, there are new long-range interactions between two quasi-line solitons through the periodic soliton as the messenger under some conditions, which cannot be described by the two-line-soliton solution.
It is shown that the resonant Davey–Stewartson (RDS) system can pass the Painlev test. By truncating the Laurent series to a constant level term, a dependent variable transformation is naturally derived, which leads to the bilinear forms of the RDS system. From the bilinear equations, through making suitable assumptions, some new soliton solutions are obtained. Some representative profiles of the solitary waves are graphically displayed including the two-line soliton solution, "Y" soliton solution, "V" soliton solution, solitoff, etc. The solutions might be useful to describe the nonlinear phenomena in Madelung fluids, capillarity fluids, and so on.
We present C programming language versions of earlier published Fortran
programs (Muruganandam and Adhikari, Comput. Phys. Commun. 180 (2009) 1888) for
calculating both stationary and non-stationary solutions of the time-dependent
Gross-Pitaevskii (GP) equation. The GP equation describes the properties of
dilute Bose-Einstein condensates at ultra-cold temperatures. C versions of
programs use the same algorithms as the Fortran ones, involving real- and
imaginary-time propagation based on a split-step Crank-Nicolson method. In a
one-space-variable form of the GP equation, we consider the one-dimensional,
two-dimensional, circularly-symmetric, and the three-dimensional
spherically-symmetric harmonic-oscillator traps. In the two-space-variable
form, we consider the GP equation in two-dimensional anisotropic and
three-dimensional axially-symmetric traps. The fully-anisotropic
three-dimensional GP equation is also considered. In addition to these twelve
programs, for six algorithms that involve two and three space variables, we
have also developed threaded (OpenMP parallelized) programs, which allow
numerical simulations to use all available CPU cores on a computer. All 18
programs are optimized and accompanied by makefiles for several popular C
compilers. We present typical results for scalability of threaded codes and
demonstrate almost linear speedup obtained with the new programs, allowing a
decrease in execution times by an order of magnitude on modern multi-core
computers.
Bright and bright-dark type multisoliton solutions of the integrable N-coupled nonlinear Schrödinger (CNLS) equations with
focusing, defocusing and mixed type nonlinearities are obtained by using Hirota’s bilinearization method. Particularly, for
the bright soliton case, we present the Gram type determinant form of the n-soliton solution (n:arbitrary) for both focusing
and mixed type nonlinearities and explicitly prove that the determinant form indeed satisfies the corresponding bilinear equations.
Based on this, we also write down the multisoliton form for the mixed (bright-dark) type solitons. For the focusing and mixed
type nonlinearities with vanishing boundary conditions the pure bright solitons exhibit different kinds of nontrivial shape
changing/energy sharing collisions characterized by intensity redistribution, amplitude dependent phase-shift and change in
relative separation distances. Due to nonvanishing boundary conditions the mixed N-CNLS system can admit coupled bright-dark
solitons. Here we show that the bright solitons exhibit nontrivial energy sharing collision only if they are spread up in
two or more components, while the dark solitons appearing in the remaining components undergo mere standard elastic collisions.
Energy sharing collisions lead to exciting applications such as collision based optical computing and soliton amplification.
Finally, we briefly discuss the energy sharing collision properties of the solitons of the (2+1) dimensional long wave-short
wave resonance interaction (LSRI) system.
We study soliton solutions of the Kadomtsev-Petviashvili II equation (-4u(t)+6uu(x)+3u(xxx))(x)+u(yy)=0 in terms of the amplitudes and directions of the interacting solitons. In particular, we classify elastic N-soliton solutions, namely, solutions for which the number, directions, and amplitudes of the N asymptotic line solitons as y-->infinity coincide with those of the N asymptotic line solitons as y-->-infinity. We also show that the (2N-1)!! types of solutions are uniquely characterized in terms of the individual soliton parameters, and we calculate the soliton position shifts arising from the interactions.
As is well known, Korteweg–de Vries equation is a typical one which has planar solitary waves. By considering the higher-dimensional nonlinear waves, we studied a Kadomtsev–Petviashvili (KP) equation and found some interesting results which explain experimental results well enough. Two same amplitude soliton solution of KP equation explain resonance phenomena reported by some experiments. Two arbitrary amplitude soliton solution of KP equation is also obtained in this Letter, which can also results in resonance phenomena. The phase shift after interaction between two soliton are obtained theoretically in this Letter. It is in agreement with experimental results.
We present a review of results of the so-called Painlevé singularity approach to the investigation of the integrability of dynamical systems with finite and infinite number of degrees of freedom. Rigorous results based on the theorems of Yoshida and Ziglin concerning proofs of non-integrability are also presented, as well as an application of the new “poly-Painlevé” method due to Kruskal. Finally a section is devoted to the singularity analysis of the solutions of non-integrable dynamical systems.
In recent investigations on nonlinear dynamics, the singularity structure analysis pioneered by Kovalevskaya, Painlevé and contempories, which stresses the meromorphic nature of the solutions of the equations of motion in the complex-time plane, is found to play an increasingly important role. Particularly, soliton equations have been found to be associated with the so-called Painlevé property, which implies that the solutions are free from movable critical points/manifolds. Finite-dimensional integrable dynamical systems have also been found to possess such a property. In this review, after briefly presenting the historical developments and various features of the Painlevé (P) method, we demonstrate how it provides an effective tool in the analysis of nonlinear dynamical systems, starting from simple examples. We apply this method to several important coupled nonlinear oscillators governed by generic Hamiltonians of polynomial type with two, three and arbitrary (N) degrees of freedom and classify all the P-cases. Sufficient numbers of involutive integrals of motion for each of the P-cases are constructed by employing other direct methods. In particular, we examine the question of integrability from the viewpoint of symmetries, explicitly demonstrate the existence of nontrivial extended Lie symmetries for the P-cases, and obtain the required integrals of motion by direct integration of symmetries. Furthermore, we briefly explain how the singularity structure analysis can be used to understand some of the intrinsic properties of nonintegrability and chaos with special reference to the two-coupled quartic anharmonic oscillators and Henon-Heiles systems.
We show that long wave\char21{}short wave resonance can be achieved in a second-order nonlinear negative refractive index medium when the short wave lies on the negative index branch. With the medium exhibiting a second-order nonlinear susceptibility, a number of nonlinear phenomena such as solitary waves, paired solitons, and periodic wave trains are possible or enhanced through the cascaded second-order effect. Potential applications include the generation of terahertz waves from optical pulses.
We obtain explicit bright one- and two-soliton solutions of the integrable
case of the coherently coupled nonlinear Schr{\"o}dinger equations by applying
a non-standard form of the Hirota's direct method. We find that the system
admits both degenerate and non-degenerate solitons in which the latter can take
single hump, double hump, and flat-top profiles. Our study on the collision
dynamics of solitons in the integrable case shows that the collision among
degenerate solitons and also the collision of non-degenerate solitons are
always standard elastic collisions. But the collision of a degenerate soliton
with a non-degenerate soliton induces switching in the latter leaving the
former unaffected after collision, thereby showing a different mechanism from
that of the Manakov system.