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

Abstract

We consider the KdV equation ∂ t u + ∂ x 3 u + u ∂ x u = 0 \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 ) | ≤ ε exp ⁡ ( − κ 0 | m | ) |c(m)| \le \varepsilon \exp (-\kappa _0 |m|) with ε > 0 \varepsilon > 0 sufficiently small, depending on κ 0 > 0 \kappa _0 > 0 and the frequency vector), we prove global existence and uniqueness of the solution. The latter result relies on our recent work [Publ. Math. Inst. Hautes Études Sci. 119 (2014) 217] on the inverse spectral problem for the quasi-periodic Schrödinger equation.