Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2024 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data
HTML articles powered by AMS MathViewer

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

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.
References
Similar Articles
  • Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 35Q53, 35B15
  • Retrieve articles in all journals with MSC (2010): 35Q53, 35B15
Bibliographic Information
  • David Damanik
  • Affiliation: Department of Mathematics, Rice University, 6100 S. Main Street, Houston, Texas 77005-1892
  • MR Author ID: 621621
  • Email: damanik@rice.edu
  • Michael Goldstein
  • Affiliation: Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
  • MR Author ID: 674385
  • Email: gold@math.toronto.edu
  • 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.
  • © Copyright 2015 by the authors
  • Journal: J. Amer. Math. Soc. 29 (2016), 825-856
  • MSC (2010): Primary 35Q53; Secondary 35B15
  • DOI: https://doi.org/10.1090/jams/837
  • MathSciNet review: 3486173