Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On two-sided estimates for the nonlinear Fourier transform of KdV

Pages: 3339 - 3356, Volume 36, Issue 6, June 2016      doi:10.3934/dcds.2016.36.3339

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Jan-Cornelius Molnar - Winterthrerstrasse 190, 8057 Zurich, Switzerland (email)

Abstract: The KdV-equation $u_t = -u_{xxx} + 6uu_x$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we obtain two-sided polynomial estimates of all integer Sobolev norms $||u||_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed, which are linear in the highest order.

Keywords:  Nonlinear Fourier transform, Korteweg-de Vries equation, integrable PDEs, action-angle variables, Birkhoff coordinates.
Mathematics Subject Classification:  Primary: 37K15; Secondary: 35Q53, 37K10.

Received: April 2015;      Revised: October 2015;      Available Online: December 2015.