Nonlocal Reductions of The Multicomponent Nonlinear Schrödinger Equation on Symmetric Spaces

Abstract

The aim of this paper is to develop the inverse scattering transform (IST) for multi-component generalisations of nonlocal reductions of the nonlinear Schrödinger (NLS) equation with PTsymmetry related to symmetric spaces. This includes: the spectral properties of the associated Lax operator, Jost function, the scattering matrix and the minimal set of scattering data, the fundamental analytic solutions. As main examples we use the Manakov vector Schrödinger equation (related to A.III-symmetric spaces) and the multi-component NLS (MNLS) equations of Kullish-Sklyanin type (related to BD.I-symmetric spaces). Furthermore, the 1and 2-soliton solutions are obtained by using an appropriate modification of the Zakharov-Shabat dressing method. It is shown, that the MNLS equations of these types allow both regular and singular soliton configurations. Finally, we present here different examples of 1-and 2-soliton solutions for both types of models, subject to different reductions.Here, S ± J and T ± J are upper-and lower-block-triangular matrices, which can be written in the form:

Inverse scattering transforms and soliton solutions of nonlocal reverse‐space nonlinear Schrödinger hierarchies

“…37,38 One can also develop the Riemann-Hilbert technique for general multicomponent NLS equations associated with simple Lie algebras 39 and their nonlocal counterparts. 7 Solution formulations, however, vary from case to case for nonlocal integrable equations, including reverse-space, reverse-time, and reverse-space-time equations (see, eg, Refs. 3, 8, 40).…”

“…Three types of nonlocal nonlinear Schrödinger (NLS) equation arises while taking group reductions. 1 The corresponding inverse scattering transforms have been recently established for the scalar case [2][3][4][5][6] and the multicomponent case, 7,8 and soliton solutions have been constructed from the Riemann-Hilbert problems whose jump is the identity, 8,9 through Darboux transformations, [10][11][12] and by the Hirota bilinear method. 13 Some other multicomponent generalizations 1,14,15 and nonlocal integrable equations 16 were also presented.…”

Employing a Modified Sumudu with a Modified Iteration Method to Solve the System of Nonlinear Partial Differential Equations

“…They can be used to model physical phenomena such as nonlinear fibre optics and wave propagation in the Kerr media [1][2][3]. Systems of NPDEs have wide-ranging engineering and science applications in many areas, such as quantum field theory and fluid mechanics [4,5]. Solutions of these equations can be used to predict the behavior of a system over time, allowing us to gain a deeper understanding of the underlying physical phenomena.…”

“…However, numerical methods can be used to approximate the solutions, allowing us to gain insight into the behavior of the system being modeled. This is because the fact that these equations can represent intricate systems with a large number of interacting components provides a more sophisticated description of physical systems than linear equations [4,5].…”

Multi-place physics and multi-place nonlocal systems

“…In addition to the nonlocal NLS system (1), there are many other types of two-place nonlocal models, such as the NLS equations with different non-localities [27], the nonlocal KdV systems [25,26,28,29], nonlocal modified KdV (MKdV) systems [25,[30][31][32][33][34][35], nonlocal discrete NLS systems [36][37][38], nonlocal coupled NLS systems [39][40][41][42][43][44][45], nonlocal derivative NLS equation [46], nonlocal Davey-Stewartson systems [47][48][49][50], generalized nonlocal NLS equation [51], nonlocal nonautonomous KdV equation [52], nonlocal peakon systems [53], nonlocal KP systems, nonlocal sine Gordon systems, nonlocal Toda systems [25,26], nonlocal Sawada-Kortera equations [54], nonlocal Kaup-Kupershmidt equations [54] and many others [55][56][57][58][59][60][61][62][63][64][65].…”