On the stability of the periodic waves for the Benney system

TL;DR

This paper studies the stability of periodic traveling waves in the Benney water wave model, showing dnoidal waves are spectrally stable for certain parameters while snoidal waves are unstable, using advanced spectral analysis techniques.

Contribution

It provides the first instability results for periodic waves in the Benney system and improves stability analysis by applying the instability index theory across all parameters.

Findings

Dnoidal waves are spectrally stable for natural parameters.

Snoidal waves exhibit spectral instabilities.

The analysis advances understanding of wave stability in the Benney model.

Abstract

We analyze the Benney model for interaction of short and long waves in resonant water wave interactions. Our particular interest is in the periodic traveling waves, which we construct and study in detail. The main results are that, for all natural values of the parameters, the periodic dnoidal waves are spectrally stable with respect to perturbations of the same period. For another natural set of parameters, we construct the snoidal waves, which exhibit instabilities, in the same setup. Our results are the first instability results in this context. On the other hand, the spectral stability established herein improves significantly upon the work Angulo-Corcho-Hakkaev, which established stability of the dnoidal waves, on a subset of parameter space, by relying on the Grillakis-Shatah theory. Our approach, which turns out to give definite answer for the entire domain of parameters, relies on the instability index theory. Interestingly, end even though the linearized operators are explicit, our spectral analysis requires subtle and detailed analysis of matrix Schr\"odinger operators in the periodic context, which support some interesting features.