Thursday, December 13, 2012

1212.2674 (David Damanik et al.)

On the Existence of Global Solutions for the KdV Equation with
Quasi-Periodic Initial Data

David Damanik, Michael Goldstein
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. For any such data and with no limitations on the frequency vector involved (in particular for periodic data), we prove well-posedness of the equation, that is, the existence of a solution on a small interval of time, depending 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 $|c(m)| \le \ve \exp(-\kappa_0 |m|)$ with $\ve$ being small enough, depending on $\kappa_0 > 0$ and the frequency vector), we prove global existence of the solution. The latter result relies on our recent work on the inverse spectral problem for the quasi-periodic Schr\"{o}dinger equation.
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