Nonlocal general vector nonlinear Schrödinger equations: Integrability, PT symmetribility, and solutions

Abstract

A family of new one-parameter (ǫx = ±1) nonlinear wave models (called G (nm) ǫx model) is presented, including both the local (ǫx = 1) and new integrable nonlocal (ǫx = −1) general vector nonlinear Schrödinger (VNLS) equations with the self-phase, cross-phase, and multi-wave mixing modulations. The nonlocal G (nm) −1 model is shown to possess the Lax pair and infinite number of conservation laws for m = 1. We also establish a connection between the G (nm) ǫx model and some known models. Some symmetric reductions and exact solutions (e.g., bright, dark, and mixed brightdark solitons) of the representative nonlocal systems are also found. Moreover, we find that the new general two-parameter (ǫx, ǫt) model (called G (nm) ǫx,ǫt model) including the G (nm) ǫx model is invariant under the PT -symmetric transformation and the PT symmetribility of its self-induced potentials is discussed for the distinct two parameters (ǫx, ǫt) = (±1, ±1).

Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media

“…(18) and (19) below. While the PT symmetry was not experimentally implemented in quantum systems (and it was argued that it does not hold in the framework of the quantum field theory [11]), the possibility to realize the PT symmetry in classical photonic media with mutually balanced spatially separated gain and loss elements was elaborated both theoretically [12]- [44] and experimentally [45]- [50]. In addition to that, the same concept can be realized in optomechanics [51], acoustics [52,53], magnetism [54], and Bose-Einstein condensates [55][56][57].…”

Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions

“…Schrödinger (NLS) equations were recently introduced with the aid of two families of parameters [20][21][22]. Some other nonlocal nonlinear wave equations were presented (see, e.g., Refs.…”

N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

“…) T be four basic solutions of systems ( 9) and ( 10), then we give the following linear algebraic systems [19,20]:…”

“…The NLS equation is widely used in physics [8][9][10][11][12][13][14][15], nonlinear optics [16,17], and soft condensed matter physics [18] and there has been a vast amount of literature involving the NLS equation over the years. The applications of the DT in higher multicomponent NLS equations spatial dimensions and nonlocal equations have attracted much attention over the years [19][20][21][22]. Recently there has been an additional interest that has caused a hot topic of research, mainly due to Mark J. Ablowitz and Ziad H. Musslimani developed the nonlocal NLS equations [23,24].…”