Periodic and Hyperbolic Soliton Solutions of a Number of Nonlocal PT-Symmetric Nonlinear Equations

TL;DR

This paper derives periodic and hyperbolic soliton solutions for various nonlocal PT-symmetric nonlinear equations, revealing that these models uniquely admit both bright and dark solitons and superpositions, unlike their local counterparts.

Contribution

The study provides explicit Jacobi elliptic and Lamé polynomial solutions for multiple nonlocal nonlinear equations, highlighting their distinct soliton properties and potential integrability.

Findings

All nonlocal models admit both bright and dark solitons.

Superpositions of elliptic functions are exact solutions.

Coupled nonlocal NLSE admits Lamé polynomial solutions of order 2.

Abstract

For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schr\"odinger equation (NLSE), nonlocal Ablowitz-Ladik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only $\dn(x,m)$ and $\cn(x,m)$ but even their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lam\'e polynomials of order 1, but it also admits solutions in terms of Lam\'e polynomials of order 2, even though they are not the solutions of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.