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Almost periodicity in time of solutions of the KdV equation

Ilia Binder, David Damanik, Michael Goldstein, Milivoje Lukic
2018 Duke mathematical journal  
We study the Cauchy problem for the KdV equation $\partial_t u - 6 u \partial_x u + \partial_x^3 u = 0$ with almost periodic initial data $u(x,0)=V(x)$. We consider initial data $V$, for which the associated Schr\"odinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify, and show the existence, uniqueness, and almost periodicity in time of solutions. This establishes a conjecture of Percy Deift for this class of initial data. The result is shown to
more » ... result is shown to apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.
doi:10.1215/00127094-2018-0015 fatcat:sybbctpwrrccpewchmlk3pjpv4