Nonlinear Schr\"odinger equations with amplitude-dependent Wadati potentials

TL;DR

This paper introduces a nonlinear Schr"odinger equation with amplitude-dependent Wadati potentials, revealing that such models retain key features like soliton families, symmetry-breaking bifurcations, and spectral properties, thus broadening understanding of non-Hermitian nonlinear systems.

Contribution

It generalizes Wadati potentials by allowing the base function to depend on amplitude, demonstrating that key properties are preserved in this nonlinear extension.

Findings

Existence of continuous soliton families

Symmetry-breaking bifurcations in PT-symmetric cases

Presence of eigenvalue quartets in spectra

Abstract

Complex Wadati-type potentials of the form $V(x)=-w^2(x) + iw_x(x)$, where $w(x)$ is a real-valued function, are known to possess a number of intriguing features, unusual for generic non-Hermitian potentials. In the present work, we introduce a class of nonlinear Schr\"odinger-type problems which generalize the Wadati potentials by assuming that the base function $w(x)$ depends not only on the transverse spatial coordinate but also on the amplitude of the field. Several examples of prospective physical relevance are discussed, including models with the nonlinear dispersion or with the derivative nonlinearity. The numerical study indicates that the generalized model inherits the remarkable features of standard Wadati potentials, such as the existence of continuous soliton families, the possibility of symmetry-breaking bifurcations when the model obeys the parity-time symmetry, the existence of constant-amplitude waves, and the eigenvalue quartets in the linear-instability spectra. Our results deepen the current understanding of the interplay between nonlinearity and non-Hermiticity and expand the class of systems which enjoy the exceptional combination of properties unusual for generic dissipative nonlinear models.