Asymptotic stability of a Korteweg-de Vries equation with a two-dimensional center manifold

TL;DR

This paper proves local asymptotic stability of a nonlinear Korteweg-de Vries equation with specific boundary conditions on a finite interval, using a two-dimensional center manifold to analyze decay rates.

Contribution

It introduces a novel application of the center manifold method to establish stability and decay rates for a Korteweg-de Vries equation with boundary conditions.

Findings

Existence of a two-dimensional local center manifold.

Polynomial decay rate of solutions.

Local exponential attractivity of the center manifold.

Abstract

Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg-de Vries equation posed on a finite interval $\left[0, 2\pi \sqrt{7/3}\right]$. The equation comes with a Dirichlet boundary condition at the left end-point and both of the Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the associated linearized equation around the origin is not asymptotically stable. In this paper, the nonlinear Korteweg-de Vries equation is proved to be locally asymptotically stable around the origin through the center manifold method. In particular, the existence of a two-dimensional local center manifold is presented, which is locally exponentially attractive. By analyzing the Korteweg-de Vries equation restricted on the local center manifold, a polynomial decay rate of the solution is obtained.