On the nonintegrability of equations for long- and short-wave interactions

TL;DR

This paper investigates the integrability of models describing long- and short-wave interactions in dispersive media, demonstrating nonintegrability for most cases and identifying special coefficient choices where integrability remains uncertain.

Contribution

It applies the Zakharov-Schulman method to analyze the integrability of two models, revealing nonintegrability due to resonances and highlighting specific cases needing further study.

Findings

Coupled KdV-NLS model is nonintegrable due to fourth-order resonances.

Coupled real KdV - complex KdV system is nonintegrable except for three special coefficient choices.

Higher-order calculations or alternative methods are needed for certain coefficient cases.

Abstract

We examine the integrability of two models used for the interaction of long and short waves in dispersive media. One is more classical but arguably cannot be derived from the underlying water wave equations, while the other one was recently derived. We use the method of Zakharov and Schulman to attempt to construct conserved quantities for these systems at different orders in the magnitude of the solutions. The coupled KdV-NLS model is shown to be nonintegrable, due to the presence of fourth-order resonances. A coupled real KdV - complex KdV system is shown to suffer the same fate, except for three special choices of the coefficients, where higher-order calculations or a different approach are necessary to conclude integrability or the absence thereof.