Stabilization of the Gear-Grimshaw system on a periodic domain

TL;DR

This paper proves exponential stability for a coupled Korteweg-de Vries system modeling internal gravity waves, using a Lyapunov approach and time-varying feedback in a periodic domain.

Contribution

It introduces a novel feedback law and Lyapunov method to establish exponential stability for the Gear-Grimshaw system in Sobolev spaces.

Findings

Exponential stability of solutions is achieved.

The method applies to Sobolev spaces of any positive integer order.

The system models internal gravity waves in stratified fluids.

Abstract

This paper is devoted to the study of a nonlinear coupled system of two Korteweg-de Vries equations in a periodic domain under the effect of an internal damping term. The system was introduced Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. Designing a time-varying feedback law and using a Lyapunov approach we establish the exponential stability of the solutions in Sobolev spaces of any positive integral order.