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

Damanik, David¹ ; Goldstein, Michael²

¹ Department of Mathematics, Rice University, 6100 S. Main Street, Houston, Texas 77005-1892
² Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4

Journal of the American Mathematical Society, Tome 29 (2016) no. 3, pp. 825-856

Résumé

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.

Détail

DOI : 10.1090/jams/837

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