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

摘要

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 decayexponentially and prove existence and uniqueness, in the class of functionswhich have an expansion with exponentially decaying Fourier coefficients, of asolution on a small interval of time, the length of which depends on the givendata and the frequency vector involved. For a Diophantine frequency vector andfor 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 globalexistence and uniqueness of the solution. The latter result relies on ourrecent work \cite{DG} on the inverse spectral problem for the quasi-periodicSchr\"{o}dinger equation.