Asymptotic stability of a nonlinear Korteweg-de Vries equation with a critical length

TL;DR

This paper proves that the nonlinear Korteweg-de Vries equation on a finite interval is locally asymptotically stable at the origin, despite the linearized system being unstable, by analyzing boundary conditions and nonlinear effects.

Contribution

It establishes the local asymptotic stability of the nonlinear Korteweg-de Vries equation with specific boundary conditions at a critical length, which was previously known to be unstable in the linear case.

Findings

The nonlinear system is asymptotically stable at the origin.

Linearized system around the origin is not asymptotically stable.

Boundary conditions influence stability properties.

Abstract

We study an initial-boundary-value problem of a nonlinear Korteweg-de Vries equation posed on a finite interval (0,2pi). The whole system has Dirichlet boundary condition at the left end-point, and both of Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the origin is not asymptotically stable for the linearized system around the origin. We prove that the origin is (locally) asymptotically stable for the nonlinear system.