A toy model for frequency cascade in the nonlinear Schrodinger equation

TL;DR

This paper introduces a simple model to observe frequency cascade phenomena in forced nonlinear Schrödinger equations, supported by algebraic analysis, stability results, and numerical simulations.

Contribution

It provides an explicit algebraic frequency cascade solution and analyzes its stability when derivatives are included in the model.

Findings

Explicit frequency cascade solution derived algebraically.

Stability persists when derivatives are incorporated.

Numerical simulations support the theoretical analysis.

Abstract

We present an elementary approach to observe frequency cascade on forced nonlinear Schr{\"o}dinger equations. The forcing term (which may also appear as a potential term instead) consists of a constant term, perturbed by a modulated Gaussian well. Algebraic computations provide an explicit frequency cascade when time and space derivatives are discarded from the nonlinear Schr{\"o}dinger equation. We provide stability results, showing that when derivatives are incorporated in the model, the initial algebraic solution may be little affected, possibly over long time intervals. Numerical simulations are provided, which support the analysis.