Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations

TL;DR

This paper analyzes the long-term behavior of solutions to the generalized Korteweg-de Vries and Benjamin-Ono equations on a one-dimensional torus, demonstrating stability and near-integrability for small initial data over extended times.

Contribution

It introduces a method to describe long-time dynamics of these equations with unbounded nonlinearities using rational normal form transformations.

Findings

Long-time Sobolev norm stability for small initial data.

Conjugation of equations to integrable systems up to high order remainders.

Applicability to equations with unbounded nonlinearities.

Abstract

We provide an accurate description of the long time dynamics of the solutions of the generalized Korteweg-De Vries (gKdV) and Benjamin-Ono (gBO) equations on the one dimension torus, without external parameters, and that are issued from almost any (in probability and in density) small and smooth initial data. We stress out that these two equations have unbounded nonlinearities. In particular, we prove a long-time stability result in Sobolev norm: given a large constant r and a sufficiently small parameter $\epsilon$, for generic initial datum u(0) of size $\epsilon$, we control the Sobolev norm of the solution u(t) for times of order $\epsilon$^{--r}. These results are obtained by putting the system in rational normal form : we conjugate, up to some high order remainder terms, the vector fields of these equations to integrable ones on large open sets surrounding the origin in high Sobolev regularity.