Stationary modes and integrals of motion in nonlinear lattices with PT-symmetric linear part

TL;DR

This paper studies nonlinear lattices with PT-symmetric linear parts, analyzing bifurcations of stationary modes, especially near degeneracies and exceptional points, and identifies conditions for integrals of motion based on pseudo-Hermiticity.

Contribution

It introduces a framework for understanding bifurcations of nonlinear modes in PT-symmetric systems with degeneracies and exceptional points, and links integrals of motion to pseudo-Hermiticity.

Findings

Bifurcation analysis near degenerate eigenvalues and exceptional points.

Construction of formal expansions for small-amplitude nonlinear modes.

Identification of nonlinearities that admit integrals of motion.

Abstract

We consider finite-dimensional nonlinear systems with linear part described by a parity-time (PT-) symmetric operator. We investigate bifurcations of stationary nonlinear modes from the eigenstates of the linear operator and consider a class of PT-symmetric nonlinearities allowing for existence of the families of nonlinear modes. We pay particular attention to the situations when the underlying linear PT-symmetric operator is characterized by the presence of degenerate eigenvalues or exceptional-point singularity. In each of the cases we construct formal expansions for small-amplitude nonlinear modes. We also report a class of nonlinearities allowing for the system to admit one or several integrals of motion, which turn out to be determined by the pseudo-Hermiticity of the nonlinear operator.