Nonlinear wave dynamics near phase transition in $\mathcal{PT}$-symmetric localized potentials

TL;DR

This paper analytically investigates nonlinear wave behavior near phase transition in $\ ext{PT}$-symmetric localized potentials, revealing stable solitons, oscillating modes, and growth dynamics through a reduced ODE model.

Contribution

It derives a general condition for phase transition and develops a reduced nonlinear ODE model to predict complex wave phenomena near the transition point.

Findings

Stable solitons exist above phase transition for certain nonlinearities.

Periodically oscillating nonlinear modes are predicted away from solitons.

Solutions can grow unboundedly depending on nonlinearity sign.

Abstract

Nonlinear wave propagation in parity-time ($\mathcal{PT}$) symmetric localized potentials is investigated analytically near a phase-transition point where a pair of real eigenvalues of the potential coalesce and bifurcate into the complex plane. Necessary conditions for phase transition to occur are derived based on a generalization of the Krein signature. Using multi-scale perturbation analysis, a reduced nonlinear ODE model is derived for the amplitude of localized solutions near phase transition. Above phase transition, this ODE model predicts a family of stable solitons not bifurcating from linear (infinitesimal) modes under a certain sign of nonlinearity. In addition, it predicts periodically-oscillating nonlinear modes away from solitons. Under the opposite sign of nonlinearity, it predicts unbounded growth of solutions. Below phase transition, solution dynamics is predicted as well. All analytical results are compared to direct computations of the full system and good agreement is observed.