Figure - available from: Mathematical Methods in the Applied Sciences
This content is subject to copyright. Terms and conditions apply.
![The real and imaginary parts of the β‐function defined in Equation (27) for κ=1, with λ=0.1 (red), λ=0.45 (blue), and λ=0.5 (black). The functions Reβ and Imβ are, respectively, odd and even with respect to the position variable x (horizontal axis). However, they are, respectively, even and odd under the transformation λ→−λ [Colour figure can be viewed at wileyonlinelibrary.com]](publication/342836750/figure/fig1/AS:11431281176287154@1690096751413/The-real-and-imaginary-parts-of-the-b-function-defined-in-Equation27-for-k1-with.png)
The real and imaginary parts of the β‐function defined in Equation (27) for κ=1, with λ=0.1 (red), λ=0.45 (blue), and λ=0.5 (black). The functions Reβ and Imβ are, respectively, odd and even with respect to the position variable x (horizontal axis). However, they are, respectively, even and odd under the transformation λ→−λ [Colour figure can be viewed at wileyonlinelibrary.com]
Source publication
Bright and dark solitons of the cubic nonlinear Schrödinger equation are used to construct complex‐valued potentials with all‐real spectrum. The real part of these potentials is equal to the intensity of a bright soliton, whereas their imaginary part is defined by the product of such soliton with its concomitant, a dark soliton. Considering light p...
Similar publications
The main objective of this research is to investigates the intricate realm of high stochastic solitons. The underlying model is an eighth-order nonlinear Schrödinger equation, considering the effects of spatio-temporal dispersion with elevated polynomial nonlinearity and multiplicative white noise. The influence of these factors on soliton’s behavi...
The Fokas-Lenells (FL) equation, which is rich in physical property in soliton theory as well as optical fiber, is a generalization of the higher-order Schrödinger equation. We construct the periodic solutions of the FL equation based on the Jacobi elliptic function expansion method in this context. Moreover, the characteristics of the obtained sol...
Using the extended physics-informed neural network with twin subnets to study the coupled mixed derivative nonlinear Schrödinger equation (NLSE), seven types of vector solitons including vector single soliton, vector double solitons, anti-dark vector single soliton, anti-dark vector double solitons, vector rogue wave, vector bright–dark single soli...
For a one-dimensional linear lattice, earlier work has shown how to systematically construct a slowly- decaying linear potential bearing a localized eigenmode embedded in the continuous spectrum. Here, we extend this idea in two directions: The first one is in the realm of the discrete nonlinear Schrodinger equation, where the linear operator of th...
We consider one- and two-dimensional (1D and 2D) optical or matter-wave media with a maximum of the local self-repulsion strength at the center, and a minimum at periphery. If the central area is broad enough, it supports ground states in the form of flat-floor “bubbles”, and topological excitations, in the form of dark solitons in 1D and vortices...
Citations
... The SUSY approach introduces the operator intertwining [16,18,[50][51][52][53][54][55][56][57][58][59][60][61][62] two HamiltoniansĤ 1 andĤ 2 aŝ ...
The supersymmetric connection that exists between the Jaynes-Cummings (JC) and anti-Jaynes Cummings (AJC) models in quantum optics is unraveled entirely. A new method is proposed to obtain the temporal evolution of observables in the AJC model using supersymmetric techniques, providing an overview of its dynamics and extending the calculation to full photon counting statistics. The approach is general and can be applied to determine the high-order cumulants given an initial state. The analysis reveals that engineering the collapse-revival behavior and the quantum properties of the interacting field is possible by controlling the initial state of the atomic subsystem and the corresponding atomic frequency in the AJC model. The substantial potential for applications of supersymmetric techniques in the context of photonic quantum technologies is thus demonstrated.
... Besides, due to the formal equivalence between the Schrödinger equation and the paraxial Helmholtz equation [21] (also between the stationary Schrödinger equation and the Helmholtz equation), optical waveguides are suitable devices to observe, study and test quantum phenomena [22]. Therefore, either isolated waveguides or waveguide arrays (optical lattices), are susceptible to be transformed by means of Darboux or SUSY transformations [23][24][25][26][27][28][29][30][31][32][33][34], in the Hermitian [26][27][28][31][32][33][34] and non-Hermitian [9,10,[23][24][25][28][29][30] regimes. Specifically, an optical lattice associated with a Hamiltonian of the type of the quantum harmonic oscillator can be 'intertwined' with itself, giving rise to the ladder of supermodes (eigenstates) [35,36]. ...
... Besides, due to the formal equivalence between the Schrödinger equation and the paraxial Helmholtz equation [21] (also between the stationary Schrödinger equation and the Helmholtz equation), optical waveguides are suitable devices to observe, study and test quantum phenomena [22]. Therefore, either isolated waveguides or waveguide arrays (optical lattices), are susceptible to be transformed by means of Darboux or SUSY transformations [23][24][25][26][27][28][29][30][31][32][33][34], in the Hermitian [26][27][28][31][32][33][34] and non-Hermitian [9,10,[23][24][25][28][29][30] regimes. Specifically, an optical lattice associated with a Hamiltonian of the type of the quantum harmonic oscillator can be 'intertwined' with itself, giving rise to the ladder of supermodes (eigenstates) [35,36]. ...
In this study, we investigate the stationary states of the Glauber-Fock oscillator waveguide array. We begin by transforming the associated Hamiltonian into the form of a quantum harmonic oscillator Hamiltonian, allowing the implementation of a supersymmetric (SUSY) approach. By considering the simplest case for the intertwining operator, the optical ladder operators are straightforwardly constructed and shown to map eigensolutions into eigensolutions of the corresponding Hamiltonian operator, in pretty much the same manner as it is done for the quantum harmonic oscillator case. The ladder of the corresponding (eigen) supermodes is then easily established.
... An important aspect of solitonic solution is the fact that one very general grounds, solitons cannot be superposed linearly. In fact, this is a characteristic property, however, there are some exceptions, that for a restricted range of frequencies, admit linear superpositions like is the case of optical solitons, [34][35][36]. We show in this section that this is also the case of the Q-ball-like solutions analyzed in section 3. We will see that in the absence of a symplectic gauge field, we are able to obtain multi-soliton solutions that naively seems to corresponds to a linear superposition but restricted to a very specific discrete set of frequencies given by the monopole contribution. ...
... In the literature it has appeared other case of optical solitons that under restricted conditions admit linear superpositions but not in the general case. See for example, [34][35][36]. ...
In this paper we obtain a family of analytic solutions to the nonlinear partial differential equations that describe the dynamics of the bosonic part of a M2-brane compactified on $M_9\times T^2$ in the LCG with worldvolume fluxes that can be induced by a constant and quantized supergravity 3-form. This sector of the theory, at supersymmetric level, has the interesting property of having a discrete spectrum. In this paper we focus in the characterization of Q-ball solitons on the M2-brane with worldvolume fluxes. We obtain two analytic families of Q-ball-like solutions that satisfy all of the constraints. We analyze two different scenarios: one in which the system is in an isotropic case and other anisotropic. In the first case we obtain two analytic families of solutions in the presence of a non-vanishing symplectic gauge field. We show that the solutions exhibit generically dispersion although they also admit solutions dispersion-free. We obtain the superposition law of the analytic solutions. They admit linear superposition but only for restricted discrete frequencies. In the non-isotropic case, we show numerically, that the system also admits Q-ball solutions, even for the simple backgrounds like the one that we are considering. The dynamics of the Q-balls are also encountered. We analyze the Lorentz boosts and Galilean transformations. Since we work in the light cone gauge, the Lorentz transformed solutions are not automatically solutions, rather some conditions must be imposed. Only a subset of the solutions remain. We discuss some examples. The Q-ball solitons of the M2-brane have a mass term that contains an interaction term between the Noether charge of the Q-ball and the topological monopole charge associated with the worldvolume flux. These solutions could be related to the new type of solitons denoted as Q-monopole-ball.
... In previous works we have studied such possibility by analyzing the resonances of the initial structure [16,47,57]. It is viable to construct the supersymmetric partners using resonances of the initial system [31], a technique implemented also in the cosh-like case [52,53] and for soliton-like models [58]. However, the transformation of resonances and/or using resonances is elaborated, so it will be analyzed elsewhere. ...
The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case.
... In previous works we have studied such possibility by analyzing the resonances of the initial structure [16,47,58]. It is viable to construct the supersymmetric partners using resonances of the initial system [31], a technique implemented also in the cosh-like case [52,53] and for soliton-like models [59]. However, the transformation of resonances and/or using resonances is elaborated, so it will be analyzed elsewhere. ...
The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux-Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case.
The construction of position-dependent mass Hamiltonian hierarchies is considered in the factorization context. It is shown that the superpotentials, the deformed potentials, and their bound state wave functions can be expressed in terms of a transformation function u that satisfy a second-order linear equation. The connection of this equation with a family of Ermakov equations allows the generation of different types of position-dependent mass models with quadratic spectra.KeywordsFactorization methodSpectrum generating algebrasPosition-dependent massMathematics Subject Classification (2010)81Q1081Q6081R15
In this work, we report the construction of different classes of complex-valued refractive index landscapes, with real spectra, in the framework of the factorization method. The particular case of guiding hyperbolic-type profiles is considered in the PT- and non-PT-symmetric configurations. In both schemes, the imaginary part of the refractive index satisfies the zero-total-area condition indicating that the total transverse optical power is preserved, allowing stable propagating modes to be obtained. The spectra and the guided modal field amplitudes are obtained and their orthogonality relations are established.
A quantum wave equation of coherence has been derived as an extension of the theory of de Broglie: particles have positions and are guided by a wavefunction. The wavefunction evolves according to the Schrödinger wave equation and can be more precisely described by a proposed quantum equation of coherence. Collective interference of Bosons (particles/waves), that are entangled, leads to a discrete coherent behaviour of waves, while their statistical character remains subordinate. As a starting point Bohm's super-implicate order is used, where inside a wave packet a super-quantum potential introduces nonlocal connectivity. A quantum equation of a coherent distribution of energy (3D semi-harmonic oscillator) is proposed: = ħ 2 + =1 3 All distributed energies can be described by ratios of 1:2, or 2:3 and close approaches thereof, is a small adaptation of Pythagorean tuning, and can be positioned as standing waves in a toroidal Riemannian Tonnetz geometry and valued weightings of 12 fundamental frequencies. There is a relation between the proposed quantum equation and lattice diagrams representing tonal spaces calculated by Fourier phase spaces by Amiot et al. The equation has been verified for many independent experiments related to energy distributions of Bose Einstein Condensates (BEC's), Bosons, Fermions, Einstein Podolsky Rosen-experiments, and superconductors. BEC's interfere like waves: thin, parallel layers of matter are separated by thin layers of empty space. The pattern forms because the waves add wherever their crests coincide and cancel where a crest meets a trough-so-called "constructive" and "destructive" interference, respectively. It is considered that the toroidal geometry is a 3D-interference pattern and can be described by two coupled equations of quantum coherence and decoherence. A challenge in making a quantum computer is entanglement and overcoming decoherence where the environment causes what is essentially noise on the quantum state of the qubit registers.
Listing the majority of soliton logic gate construction found across the literature, hopefully this proves to be useful to other people as well.
