On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data (Q2802072)

The authors consider the KdV equation 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 \(|c(m)|\leq \exp(-\kappa_0 |m|)\) with \(\varepsilon > 0\) small, depending on \(\kappa_0 >\) and the frequency vector), global existence and uniqueness of the solution are proved.