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arXiv · 1509.07373

Almost Periodicity in Time of Solutions of the KdV Equation

Ilia Binder·David Damanik·Michael Goldstein·Milivoje Lukic
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Abstract

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 apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.

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Ilia Binder, David Damanik, Michael Goldstein, Milivoje Lukic. 2015-09-24. Almost Periodicity in Time of Solutions of the KdV Equation. https://doi.org/10.1215/00127094-2018-0015

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