Almost Periodicity in Time of Solutions of the KdV Equation

Abstract

We study the Cauchy problem for the KdV equation ${\partial }_{t}u-6u{\partial }_{x}u+{\partial }_x^{3}u=0$ with almost periodic initial data $u(x,0)=V(x)$. We consider initial data $V$, for which the associated Schrödinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify, and show the existence, uniqueness, and almost periodicity in time of solutions. This establishes a conjecture of Percy Deift for this class of initial data. The result is shown to apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.

26 pages

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Cited by in corpus (15)

• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• Asymptotics of spectral gaps of quasi-periodic Schrödinger operators
• Invariance of white noise for KdV on the line
• The nonlinear Schrodinger equation on Z and R with bounded initial data: examples and conjectures
• KdV equation beyond standard assumptions on initial data
• Global well-posedness for $H^{-1}(\mathbb{R})$ perturbations of KdV with exotic spatial asymptotics
• The inverse scattering transform for weak Wigner-von Neumann type potentials
• Almost Periodicity in Time of Solutions of the Toda Lattice
• Local Existence and Uniqueness of Spatially Quasi-Periodic Solutions to the Generalized KdV Equation
• The Quasi-Periodic Cauchy Problem for the Generalized Benjamin-Bona-Mahony Equation on the Real Line
• Stahl--Totik regularity for continuum Schrödinger operators
• Polynomial decay of the gap length for C^k quasi-periodic Schrodinger operators and spectral application
• Homogeneous Spectrum of Quasi-periodic Gevrey Schrödinger Operators with Diophantine Frequency
• The Deift Conjecture: A Program to Construct a Counterexample
• Well-posedness for good Boussinesq equations subject to quasi-periodic initial data