Dynamics of the Drinfeld-Sokolov-Wilson system: well-posedness and (in)stability of the traveling waves

TL;DR

This paper investigates the well-posedness and stability of the Drinfeld-Sokolov-Wilson system, establishing global solutions for low regularity data and demonstrating spectral instability of explicit periodic waves.

Contribution

It proves well-posedness for low regularity initial data, extends solutions globally, and constructs explicit unstable periodic wave solutions.

Findings

Global well-posedness in $L^2$ space

Persistence of solutions in $H^1 \times L^2$

Explicit periodic waves are spectrally unstable

Abstract

We analyze the Drinfeld-Sokolob-Wilson system, which features a dispersive, KdV type evolution with a dispersionless conservation law. We establish well-posedness with low regularity initial data $L^2({\mathbb T})\times L^2({\mathbb T})$ for the Cauchy problem on periodic background, which is then extrapolated to global solutions, due to $L^2$ conservation law. We also establish a dynamically more relevant result, namely a global persistence of solutions with (large) initial data in $H^1({\mathbb T})\times L^2({\mathbb T})$. This is obtained by following a more sophisticated approach, specifically the method of normal forms. Finally, for a fixed period $L$, we construct an explicit one parameter family of periodic waves, see \eqref{2.16} below. We show that they are all spectrally unstable with respect to co-periodic perturbations. Specifically, we show that the Hamiltonian instability index is equal to one, which identifies the instability as a single positive growing mode.