Control results for a model of resonant interaction between short and long capillary-gravity waves

TL;DR

This paper investigates the global control properties of a coupled nonlinear dispersive system combining Schrödinger and Korteweg-de Vries equations, using advanced microlocal analysis and Bourgain space estimates.

Contribution

It presents the first global control results for this coupled nonlinear dispersive system, leveraging propagation of singularities for systems with operators of different orders.

Findings

Established global control properties for the coupled system

Demonstrated propagation of singularities in the nonlinear context

Extended control analysis to systems with mixed-order differential operators

Abstract

The purpose of this article is the investigation of the global control properties of a coupled nonlinear dispersive system posed in the periodic domain $\mathbb{T}$, a system with the structure of a nonlinear Schr\"odinger equation and a nonlinear Korteweg-de Vries equation. Combining estimates derived from Bourgain spaces and using microlocal analysis we show that this system has global control properties. The main novelty of this work is twofold. One is that the global results for the nonlinear system are presented for the first time thanks to the propagation of singularities. The second one is that these propagation results are shown to a coupled dispersive system with two equations defined by differential operators with principal symbols of different orders.