Homoclinic and heteroclinic solutions of the nonlinear Schr\"odinger equation with a complex Wadati potential

TL;DR

This paper characterizes stationary solutions of the nonlinear Schrödinger equation with a PT-symmetric complex potential, focusing on homoclinic and heteroclinic solutions that connect nonlinear plane waves, using asymptotic analysis and numerical simulations.

Contribution

It introduces a detailed analysis of homoclinic and heteroclinic solutions supported by a Wadati potential in a PT-symmetric nonlinear Schrödinger equation, highlighting their bifurcations and structure.

Findings

Existence of homoclinic and heteroclinic solutions supported by Wadati potentials.

Bifurcation analysis reveals solution structure and stability.

Numerical simulations confirm analytical predictions and illustrate wave dynamics.

Abstract

Stationary solutions asymptoting to nonlinear plane waves of the nonlinear Schr\"odinger equation with a PT-symmetric, complex linear potential are characterized. The potential includes both a spatially varying gain-loss profile and a repulsive real part, generated by a Wadati potential function,that support the existence of homoclinic and heteroclinic solutions that asymptote to the same or different, respectively, nonlinear plane waves in the far field. Asymptotic analysis and numerical simulations are used to examine solution existence, bifurcations, and structure. Such solutions play an important role in resonant nonlinear wave generation of dispersive media with localized gain and loss.