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

Damanik, David; Goldstein, Michael

doi:10.1090/jams/837

On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data
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by David Damanik and Michael Goldstein

J. Amer. Math. Soc. 29 (2016), 825-856

DOI: https://doi.org/10.1090/jams/837

Published electronically: June 29, 2015

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Abstract:

We consider the KdV equation $\partial _t u +\partial ^3_x u +u\partial _x u=0$ 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 $|c(m)| \le \varepsilon \exp (-\kappa _0 |m|)$ with $\varepsilon > 0$ sufficiently small, depending on $\kappa _0 > 0$ 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.

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