On the Existence and Uniqueness of Global Solutions for the KdV Equation with Quasi-Periodic Initial Data

TL;DR

This paper proves the existence and uniqueness of solutions to the KdV equation with quasi-periodic initial data, establishing local results generally and global results under smallness and Diophantine conditions, using inverse spectral methods.

Contribution

It demonstrates global existence and uniqueness for the KdV equation with small quasi-periodic data and Diophantine frequencies, extending previous inverse spectral techniques.

Findings

Existence and uniqueness of solutions for quasi-periodic initial data.

Global solutions for small data with Diophantine frequencies.

Dependence of solution interval on initial data and frequency vector.

Abstract

We consider the KdV equation $$ \partial_t u +\partial^3_x u +u\partial_x u=0 $$ with quasi-periodic initial data whose Fourier coefficients decay exponentially and prove existence and uniqueness, in the class of functions which have an expansion with exponentially decaying Fourier coefficients, of a solution on a small interval of time, the length of which depends on the given data and the frequency vector involved. For a Diophantine frequency vector and for small quasi-periodic data (i.e., when the Fourier coefficients obey $|c(m)| \le \varepsilon \exp(-\kappa_0 |m|)$ with $\varepsilon > 0$ sufficiently small, depending on $\kappa_0 > 0$ and the frequency vector), we prove global existence and uniqueness of the solution. The latter result relies on our recent work \cite{DG} on the inverse spectral problem for the quasi-periodic Schr\"{o}dinger equation.