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

Abstract

We consider the KdV equation $${\partial }_{t}u+{\partial }_x^{3}u+u{\partial }_{x}u=0$$ 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 $|c(m)|\le \epsilon \exp (-{\kappa }_0|m|)$ with $\epsilon >0$ 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 Schrödinger equation.

arXiv comment26 pages, to appear in J. Amer. Math. Soc