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

References in corpus (13)

• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• Almost reducibility and absolute continuity I
• On the spectrum and Lyapunov exponent of limit periodic Schrodinger operators
• The absolutely continuous spectrum of one-dimensional Schr"odinger operators
• Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity
• Critical density of a soliton gas
• Generic Continuous Spectrum for Ergodic Schr"odinger Operators
• Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators
• The Isospectral Torus of Quasi-Periodic Schrödinger Operators via Periodic Approximations
• Generalized reflection coefficients
• Homogeneity of the spectrum for quasi-periodic Schrödinger operators

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