Some case example exact solutions for quadratically nonlinear optical media with $\mathcal{PT}$-symmetric potentials

TL;DR

This paper derives exact solutions for a nonlinear optical system with $ ext{PT}$-symmetric potentials, analyzing their stability and nonlinear dynamics to understand localized and extended waveforms.

Contribution

It introduces a method to find exact $ ext{PT}$-symmetric potentials and solutions in a $ ext{chi}^{(2)}$ nonlinear optical system, including stability and dynamical analysis.

Findings

Exact cnoidal solutions for the system

Stability regions identified for soliton and periodic solutions

Numerical simulations reveal dynamical behaviors and instabilities

Abstract

In the present paper we consider an optical system with a $\chi^{(2)}$-type nonlinearity and unspecified $\mathcal{PT}$-symmetric potential functions. Considering this as an inverse problem and positing a family of exact solutions in terms of cnoidal functions, we solve for the resulting potential functions in a way that ensures the potentials obey the requirements of $\mathcal{PT}$-symmetry. We then focus on case examples of soliton and periodic solutions for which we present a stability analysis as a function of their amplitude parameters. Finally, we numerically explore the nonlinear dynamics of the associated waveforms to identify the outcome of the relevant dynamical instabilities of localized and extended states.