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Evolution graphs of second-order lump solution: (a), (b), (c) three-dimensional plot at t=−5, t=0, and t=5, (d) density plot at t=0, and (e) the contour plot at t=−14 (black), t=−7 (green), t=0 (blue), t=7 (pink), and t=14 (red) about the moving path described by the blue line y=−x and red line y=−5/2x.
Source publication
We investigate a reduced generalized (3 + 1)-dimensional shallow water wave equation, which can be used to describe the nonlinear dynamic behavior in physics. By employing Bell’s polynomials, the bilinear form of the equation is derived in a very natural way. Based on Hirota’s bilinear method, the expression of N-soliton wave solutions is derived....
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Citations
... So, we can study the solutions of equivalent Equation (2). Researchers have studied Grammian and Pfaffian solutions [61], the lump-type solutions and their interaction solutions [62][63][64], the breather wave solutions [63], the periodic wave solutions [65], the high-order breather solutions, the high-order lump solutions and the hybrid solutions [66], the solitary wave solutions, the periodic wave solutions and the interactional solutions [67] of Equation (2). Through the dependent variable transformation ...
The superposition formulas of multi-solutions to the (3+1)-dimensional generalized shallow water wave-like Equation (GSWWLE) are proposed. There are arbitrary test functions in the superposition formulas of the mixed solutions and the interaction solutions, and we generalized to the sum of any N terms. By freely selecting the test functions and the positive integer N, we have obtained abundant solutions for the GSWWLE. First, we introduced new mixed solutions between two arbitrary functions and the multi-kink solitons, and the abundant mixed solutions were obtained through symbolic computation. Next, we constructed the multi-localized wave solutions which are the superposition of N-even power functions. Finally, the novel interaction solutions between the multi-localized wave solutions and the multi-arbitrary function solutions for the GSWWLE were obtained. The evolution behaviors of the obtained solutions are shown through 3D, contour and density plots. The received results have immensely enriched the exact solutions of the GSWWLE in the available literature.
... The interaction solutions of nonlinear partial differential equations are a topic of general interest in nonlinear systems. [1][2][3][4] Among them, shallow water wave equation has been one of the hottest issues in recent years, [5][6][7][8][9][10][11] such as marine engineering, hydrodynamics, mathematical physics in other fields. Because its exact solution is a special solution existing stably in space, [12] it has very important practical significance for many complex physical phenomena [13] and some nonlinear engineering problems. ...
... this equation has been used with tsunamis, atmospheric circulation, river transport, virtual reality, and other issues widely. Many authors obtained various forms of solutions to Eq. (2) by using the Hriota bilinear method, Darboux transformation method, etc. [6][7][8][9][10][11] Equation (1) can be transformed into the following form by using the Hriota bilinear method: ...
We investigate a (2+1)-dimensional shallow water wave equation and describe its nonlinear dynamical behaviors in physics. Based on the N -soliton solutions, the higher-order fissionable and fusionable waves, fissionable or fusionable waves mixed with soliton molecular and breather waves can be obtained by various constraints of special parameters. At the same time, by the long wave limit method, the interaction waves between fissionable or fusionable waves with higher-order lumps are acquired. Combined with the dynamic figures of the waves, the properties of the solution are deeply studied to reveal the physical significance of the waves.
... Subsequently, by virtue of various collisions between lump waves and solitary waves, Satsuma et al. established multi-lump solutions [20] through using the long wave limit method (LWLM). For recent decades, the multi-lump solutions and hybrid solutions between lumps and solitary waves of (4+1)-dimensional Fokas equation [21], (3+1)-dimensional nonlinear system [22], (2+1)dimensional HSI equation [23], the third-order evolution equation arising in the shallow water [24], a generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles [25], a reduced generalized (3+1)dimensional SWW equation [26], (3+1)-dimensional generalized YTSF equation [27] were introduced. ...
The present paper studied multi-breather, multi-lump and hybrid solutions of a novel KP-like equation by applying the Hirota bilinear method. The multi-breather solutions including one-, two- and three-breather and hybrid solutions between breathers and solitons were obtained through applying the complex conjugate method on the N-soliton solution. Via using the long wave limit method with respect to N-solution, the solution expression of multi-lump solutions was acquired; therefore, by choosing appropriate parameters on 2-, 3-, 4-, 5- and 6-soliton, one-, two-, three-lump and two types of hybrid solutions between lumps and solitons were derived. Additionally, the one-breather and one-lump solutions have been studied in-depth. In order to illustrate the wave trajectories, wave shapes and the fusion and fission process, several cases of plots with physical interpretation were given. Furthermore, the obtained solutions can be widely used to explain many interesting physical phenomena in the nature.
