Local Wellposedness of dispersive equations with quasi-periodic initial data

Abstract

We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form $$\partial_tu + Lu + \partial_x(u^{p+1})=0,$$ where $L$ is a multiplier operator with purely imaginary symbol which grows at most exponentially. The class of equations to which our method applies includes the generalized Korteweg-de Vries equation, the generalized Benjamin-Ono equation, and the derivative nonlinear Schr\"odinger equation. We also discuss well-posedness of some dispersive models which do not have a problematic derivative in the nonlinearity, namely, the nonlinear Schr\"odinger equation and the generalized Benjamin-Bona-Mahony equation, with quasi-periodic initial data. In this way, we recover and improve upon results from arXiv:1212.2674v3 [math.AP], arXiv:2110.11263v1 [math.AP] and arXiv:2201.02920v1 [math.AP] by shorter arguments.