Almost periodicity in time of solutions of the KdV equation

DOI10.1215/00127094-2018-0015zbMath1406.35325arXiv1509.07373OpenAlexW2215671126WikidataQ129376569 ScholiaQ129376569MaRDI QIDQ1626578

David Damanik, Milivoje Lukic, Michael Goldstein, Ilia A. Binder

Publication date: 21 November 2018

Published in: Duke Mathematical Journal (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1509.07373

zbMATH Keywords

almost periodic solutionKorteweg-de Vries equation

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) KdV equations (Korteweg-de Vries equations) (35Q53) Almost and pseudo-almost periodic solutions to PDEs (35B15)

Related Items

Absence of absolutely continuous spectrum for generic quasi-periodic Schrödinger operators on the real line ⋮ The inverse scattering transform for weak Wigner–von Neumann type potentials ^* ⋮ Homogeneous spectrum of quasi-periodic Gevrey Schrödinger operators with Diophantine frequency ⋮ Exponential upper bounds on the spectral gaps and homogeneous spectrum for the non-critical extended Harper's model ⋮ The nonlinear Schrödinger equation on Z and R with bounded initial data: examples and conjectures ⋮ Global well-posedness for \(H^{-1}(\mathbb{R})\) perturbations of KdV with exotic spatial asymptotics ⋮ The quasi-periodic Cauchy problem for the generalized Benjamin-Bona-Mahony equation on the real line ⋮ Uniform Estimate of Potentials by Reflection Coefficients and its Application to KdV Flow ⋮ KdV equation beyond standard assumptions on initial data ⋮ Mini-workshop: Reflectionless operators: the Deift and Simon conjectures. Abstracts from the mini-workshop held October 22--28, 2017 ⋮ KdV hierarchy via Abelian coverings and operator identities ⋮ The isospectral torus of quasi-periodic Schrödinger operators via periodic approximations ⋮ Some open problems in random matrix theory and the theory of integrable systems. II ⋮ Invariance of white noise for KdV on the line ⋮ Polynomial decay of the gap length for \(C^k\) quasi-periodic Schrödinger operators and spectral application ⋮ Periodic and almost periodic solutions for a coupled system of two Korteweg-de Vries equations with boundary forces ⋮ Uniqueness of solutions of the KdV-hierarchy via Dubrovin-type flows ⋮ Limit-periodic Dirac operators with thin spectra

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous
• The isospectral torus of quasi-periodic Schrödinger operators via periodic approximations
• The absolutely continuous spectrum of Jacobi matrices
• Limit-periodic continuum Schrödinger operators with zero measure Cantor spectrum
• Generic singular spectrum for ergodic Schrödinger operators
• On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators
• Global theory of one-frequency Schrödinger operators
• Generic continuous spectrum for ergodic Schrödinger operators
• Critical points of Green's function, harmonic measure, and the corona problem
• The rotation number for almost periodic potentials
• Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation
• The periodic problem for the Korteweg-de Vries equation
• The spectrum of Hill's equation
• Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials
• Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II: The KdV-equation
• Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions
• On solutions of nonlinear Schrödinger equations with Cantor-type spectrum
• Asymptotic behavior of polynomials orthonormal on a homogeneous set
• The trace formula for Schrödinger operators on the line
• Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum
• Primitive potentials and bounded solutions of the KdV equation
• On the inverse spectral problem for the quasi-periodic Schrödinger equation
• On the Cauchy problem for the Korteweg-de Vries equation with steplike finite-gap initial data. II: Perturbations with finite moments
• Generalized reflection coefficients
• The absolutely continuous spectrum of one-dimensional Schrödinger operators
• Singular density of states measure for subshift and quasi-periodic Schrödinger operators
• Korteweg-de Vries equation: a completely integrable Hamiltonian system
• Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions
• On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data
• Nonuniqueness for Solutions of the Korteweg-DeVries Equation
• Theta functions and non-linear equations
• Method for Solving the Korteweg-deVries Equation
• Some open problems in random matrix theory and the theory of integrable systems
• On perturbations of a Schrödinger operator with periodic potential
• Well-Posedness of the Initial Value Problem for the Korteweg-de Vries Equation
• Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points
• Critical density of a soliton gas
• Schrödinger operators with dynamically defined potentials
• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Korteweg-de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion
• Korteweg-de Vries Equation and Generalizations. IV. The Korteweg-de Vries Equation as a Hamiltonian System
• Zeros of the Wronskian and Renormalized Oscillation Theory
• Local Well-Posedness of the KdV Equation with Quasi-Periodic Initial Data
• Soliton Theory and Hankel Operators
• On the evolution of a reflection coefficient under the Korteweg–de Vries flow
• A multi-scale analysis scheme on Abelian groups with an application to operators dual to Hill’s equation
• Integrals of nonlinear equations of evolution and solitary waves