Nonintegrable spatial discrete nonlocal nonlinear schrödinger equation

Abstract

Integrable and nonintegrable discrete nonlinear Schrödinger equations (NLS) are significant models to describe many phenomena in physics. Recently, Ablowitz and Musslimani introduced a class of reverse space, reverse time and reverse space-time nonlocal integrable equations, including nonlocal NLS, nonlocal sine-Gordon equation and nonlocal Davey-Stewartson equation etc. And, the integrable nonlocal discrete NLS has been exactly solved by inverse scattering transform. In this paper, we study a nonintegrable discrete nonlocal NLS which is direct discretization version of the reverse space nonlocal NLS. By applying discrete Fourier transform and modified Neumann iteration, we present its stationary solutions numerically. The linear stability of the stationary solutions is examined. Finally, we study the Cauchy problem for nonlocal NLS equation numerically and find some different and new properties on the numerical solutions comparing with the numerical solutions of the Cauchy problem for NLS equation.problem of this equation is solved under special initial values by inverse scattering transform (IST). But for general initial values, the solutions of Cauchy problem of nonlocal NLS equation can not be obtained by using IST. In our study, we investigate a nonintegrable space discrete nonlocal NLS equation. We obtain its stationary solitary wave solutions by discrete Fourier transform. The linear stability of these stationary solitary wave solutions are examined. Then, numerical simulations for Cauchy problem of nonlocal NLS equation with periodic boundary conditions are performed, in which we use the integrable scheme and the nonintegrable scheme as a spatial discrete model of nonlocal NLS equation. We find some different and new properties on the numerical solutions for Cauchy problem of nonlocal NLS equation comparing with the numerical solutions of the Cauchy problem for the NLS equation.

Nanoptera in Higher-Order Nonlinear Schrödinger Equations: Effects of Discretization

“…Discrete NLS equations also support solitary waves and GSWs (Kivshar and Campbell 1993;Pelinovsky et al 2005;Melvin et al 2006Melvin et al , 2008Melvin et al , 2009Dmitriev et al 2007;Fitrakis et al 2007;Oxtoby and Barashenkov 2007;Pelinovsky et al 2007;Pelinovsky and Kevrekidis 2008;Rothos et al 2008;Cuevas et al 2009;Syafwan et al 2012;Zhang et al 2012;Ma and Zhu 2017;Alfimov and Titov 2019;Ji et al 2019;Zhu et al 2020). See Kevrekidis et al (2001) and Kevrekidis (2009) for reviews of discrete NLS equations.…”

Nanoptera In Higher-Order Nonlinear Schrödinger Equations: Effects Of Discretization

“…Discrete NLS equations also support solitary waves [15,20,23,27,55,56,61,79,80] and GSWs [3, 4, 45, 48-51, 54, 57, 67]. See [40,42] for reviews of the theory and background of discrete NLS equations.…”

Multi-place physics and multi-place nonlocal systems

“…where l c, and l 1 are arbitrary constants. It is interesting that some known integrable nonlocal NLS (or named ABNLS) systems are just the special reductions of the nonlocal AKNS systems (34), (36) (38) and (40). For instance, taking p=q=A, s=r=B in (38), we get the known nonlocal NLS systems (29) and some others such as those in [1,25,96] and [76],…”

“…In addition to the known nonlocal NLS reductions (47), one can also obtain some types ofnovel local and nonlocal twoplace and four-place NLS type systems from the AKNS systems (34), (36), (38) and (40).…”

“…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].…”