Decay for the nonlinear KdV equations at critical lengths

TL;DR

This paper studies the decay behavior of solutions to the nonlinear KdV equation at critical lengths, showing decay rates under certain conditions and extending previous results with new analytical techniques.

Contribution

It introduces a power series expansion approach and quasi-periodic function theory to analyze decay, providing new decay rate results and generalizing prior findings.

Findings

Solutions decay at least as fast as 1/√t for certain critical lengths.

Decay rate is at least logarithmic over t for all critical lengths.

New analytical methods extend previous results to broader cases.

Abstract

We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the $L^2$-norm if their initial data belong to a finite dimensional subspace $\M$. In this paper, we show that all solutions of the nonlinear KdV system decay to 0 at least with the rate $1/ t^{1/2}$ when $\dim \M = 1$ or when $\dim \M$ is even and a specific condition is satisfied, provided that their initial data is sufficiently small. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. As a consequence, we rediscover known results which were previously established for $\dim \M = 1$ or for the smallest critical length $L$ with $\dim \M = 2$ by a different approach using the center manifold theory, and obtain new results. We also show that the decay rate is not slower than $\ln (t + 2) / t$ for all critical lengths.