On the Existence and Uniqueness of Global Solutions for the Kdv Equation with Quasi-periodic Initial Data

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)| ≤ ε exp(−κ0|m|) with ε > 0 sufficiently small, depending on κ0 > 0 and the frequency vector), we prove global existence and uniqueness of the solution. The latter result relies on our recent work [DG] on the inverse spectral problem for the quasi-periodic Schrödinger equation.