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


Authors: David Damanik and Michael Goldstein
Journal: J. Amer. Math. Soc. 29 (2016), 825-856
MSC (2010): Primary 35Q53; Secondary 35B15
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 $ \vert c(m)\vert \le \varepsilon \exp (-\kappa _0 \vert m\vert)$ 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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Additional Information

David Damanik
Affiliation: Department of Mathematics, Rice University, 6100 S. Main Street, Houston, Texas 77005-1892
Email: damanik@rice.edu

Michael Goldstein
Affiliation: Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
Email: gold@math.toronto.edu

DOI: https://doi.org/10.1090/jams/837
Received by editor(s): January 15, 2013
Received by editor(s) in revised form: February 12, 2015, and May 15, 2015
Published electronically: June 29, 2015
Additional Notes: The first author was partially supported by a Simons Fellowship and NSF grants DMS-0800100, DMS-1067988, and DMS-1361625.
The second author was partially supported by a Guggenheim Fellowship and an NSERC grant.
Article copyright: © Copyright 2015 by the authors