Transverse instability of periodic standing waves for the generalized nonlinear Schrodinger equation

TL;DR

This paper investigates the transverse stability of periodic standing waves in a generalized nonlinear Schrödinger equation with fractional power nonlinearity, identifying conditions for stability and instability.

Contribution

It extends the analysis of transverse instability to fractional nonlinear Schrödinger equations and characterizes the sign-changing behavior of periodic solutions.

Findings

Periodic waves exist via constrained minimization.

The stability depends on the nonlinearity power.

Transverse instability is established using existing theoretical results.

Abstract

In this paper, we determine the transverse instability of periodic standing wave solutions for the generalized Schr\"odinger equation with fractional power nonlinearity. The existence of periodic waves is determined by using a constrained minimization problem in the complex setting, and it is shown that the corresponding real solution, depending on the power nonlinearity, is always positive or changes its sign. The transverse instability results are then determined by applying the main result given in \cite{RoussetTzvetkov} for the periodic case.