Periodic solutions of the KdV equation

DOI10.1002/cpa.3160280105zbMath0295.35004OpenAlexW2017812896MaRDI QIDQ4047867

Peter D. Lax

Publication date: 1975

Published in: Communications on Pure and Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1002/cpa.3160280105

Mathematics Subject Classification ID

Periodic solutions to PDEs (35B10) General topics in linear spectral theory for PDEs (35P05) Schrödinger operator, Schrödinger equation (35J10) Scattering theory of linear operators (47A40)

Related Items

On polynomial symmetries of the sine-Gordon equation, Integration of the negative order Korteweg–de Vries equation with a self-consistent source in the class of periodic functions, Some inverse problems in periodic homogenization of Hamilton-Jacobi equations, Algebro-geometric solutions for the Ruijsenaars-Toda hierarchy, A New Treatment for Some Periodic Schrödinger Operators I: The Eigenvalue, The Bargmann type reduction for some Lax integrable two-dimensional generalization of the relativistic Toda lattice, Integrable three-dimensional coupled nonlinear dynamical systems related to centrally extended operator Lie algebras and their Lax type three-linearization, An Analytical Model of Periodic Waves in Shallow Water, Duals and propagators: A canonical formalism for nonlinear equations, The constraints of potentials and the finite-dimensional integrable systems, Classical Liouville completely integrable systems associated with the solutions of Boussinesq–Burgers’ hierarchy, Algebro-geometric solution of the 2+1 dimensional Burgers equation with a discrete variable, Topological phenomena in the real periodic sine-Gordon theory, A simple Lagrangian for integrable systems, Arnold--Liouville Theorem for Integrable PDEs: A Case Study of the Focusing NLS Equation, Existence of gauge field in any partially integrable systems, Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics, Periodic fixed points of Bäcklund transformations and the Korteweg–de Vries equation, Breaking Water Waves, Algebro-geometric quasi-periodic solutions to the Satsuma-Hirota hierarchy, Quasi-periodic solution of quasi-linear fifth-order KdV equation, Application of tetragonal curves to coupled Boussinesq equations, Global well-posedness for \(H^{-1}(\mathbb{R})\) perturbations of KdV with exotic spatial asymptotics, Thermal effects in short laser pulses: suppression of wave collapse, Computation of large-genus solutions of the Korteweg-de Vries equation, Integration of the negative order Korteweg-de Vries equation with a special source, Elliptic finite-band potentials of a non-self-adjoint Dirac operator, Multisolitons are the unique constrained minimizers of the KdV conserved quantities, Quasi‐periodic solutions to a hierarchy of integrable nonlinear differential–difference equations, Stability of elliptic function solutions for the focusing modified KdV equation, Conjugate points in the Bott-Virasoro group and the KdV equation, Quasi-periodic solutions to a negative-order integrable system of 2-component KdV equation, Asymptotic Analysis of a Perturbed Periodic Solution for the KdV Equation, Unnamed Item, On a class of eigenfunction identities for Hill operators, New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons, On time periodic solutions to the conformal cubic wave equation on the Einstein cylinder, RIEMANN THETA FUNCTIONS SOLUTIONS WITH RATIONAL CHARACTERISTICS FOR (2+1)-DIMENSIONAL SINH-GORDON EQUATION, Stability of the multi-solitons of the modified Korteweg–de Vries equation ^*, Derivation of Korteweg‐de Vries flow equations from fourth order nonlinear Schrödinger equation, Algebro-geometric solutions for the Gerdjikov-Ivanov hierarchy, Group theoretic aspects of conservation laws of nonlinear dispersive waves: KdV type equations and nonlinear Schrödinger equations, A Numerical Study of the 3-Periodic Wave Solutions to Toda-Type Equations, Decomposition of the Modified Kadomtsev–Petviashvili Equation and its Finite Band Solution, Boussinesq's equation on the circle, Evolution equations possessing infinitely many symmetries, Mass corrections to periodic solitons, Integration of higher Korteweg-de Vries equation with a self-consistent source in class of periodic functions, On the determination of a Hill's equation from its spectrum, Neumann type integrable reduction for nonlinear evolution equations in 1+1 and 2+1 dimensions, Integrability condition and finite-periodic Toda lattice, Freedom in the expansion and obstacles to integrability in multiple-soliton solutions of the perturbed KdV equation, Properties of the Benjamin–Ono equation, Commuting flows and conservation laws for Lax equations, GENERALIZED r-MATRIX STRUCTURE AND ALGEBRO-GEOMETRIC SOLUTION FOR INTEGRABLE SYSTEM, Hamiltonian and algebro-geometric integrals of stationary equations of KdV type, Topological charge of the real periodic finite‐gap Sine‐Gordon solutions, Lectures on the Inverse Scattering Transform, Dispersive and soliton perturbations of finite-genus solutions of the KdV equation: computational results, Hamiltonian description of the ideal fluid, An overview of periodic elliptic operators, Higher-order rational soliton solutions for the fifth-order modified KdV and KdV equations, STRINGS, WAVES, DRUMS: SPECTRA AND INVERSE PROBLEMS, Quasi-periodic solutions of the negative-order Jaulent–Miodek hierarchy, A symmetry approach to exactly solvable evolution equations, [https://portal.mardi4nfdi.de/wiki/Publication:5230893 Hyperelliptic function solutions with finite genus ������ of coupled nonlinear differential equations*], The Application of Neumann Type Systems for Solving Integrable Nonlinear Evolution Equations, An explicit formula for the cubic Szegő equation, Green's Function Method for Periodic Traveling Wave Solutions of the KdV Equation, On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KdV poten\-tials, Finite-gap solutions of 2+1 dimensional integrable nonlinear evolution equations generated by the Neumann systems, Soliton gas in integrable dispersive hydrodynamics, Compactness of Floquet isospectral sets for the matrix Hill's equation, Some algebro-geometric solutions for the coupled modified Kadomtsev-Petviashvili equations arising from the Neumann type systems, Numerical evaluations of periodic wave solutions, integrable time discretization and their applications to the mKdV–sine-Gordon equation, Isoinertial operators around the KdV multi-solitons, A numerical study on the \(N\)-periodic wave solutions of two coupled bilinear equations, On the spectra of real and complex Lamé operators, Algebro-geometric solutions of the generalized Burgers hierarchy associated with a \(3\times 3\) matrix spectral problem based on Riemann surface, \(N=2\) supersymmetric QCD and elliptic potentials, Hamiltonian formulation of energy conservative variational equations by wavelet expansion, Lie-algebraic structure of the Lax-integrable \((2|1 + 1)\)-dimensional supersymmetric matrix dynamical systems, Integration of a loaded Korteweg-de Vries equation in a class of periodic functions, Two-component description of dynamical systems that can be approximated by solitons: the case of the ion acoustic wave equations of plasma physics, Study of stationary Schrödinger equations and integrals of Gelfand-Levitan and Korteweg-de-Vries equations. Solitons and the inverse scattering method., Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations, Bifurcation structure and stability of steady gravity water waves with constant vorticity, Orbital stability of double solitons for the Benjamin-Ono equation, Global wellposedness of KdV in \(H^{-1}(\mathbb T,\mathbb R)\), Almost periodicity for the nonlinear cubic Schrödinger equation, On the modified Korteweg-de-Vries equation with loaded term, Solitary wave solutions for the KdV and mKdV equations by differential transform method, On the hierarchies of higher order mKdV and KdV equations, A Riemann-Hilbert problem for the finite-genus solutions of the KdV equation and its numerical solution, Algebraic aspects of nonlinear differential equations, Orbital bifurcations and shoaling of cnoidal waves, Analyticity and smoothing effect for the fifth order KdV type equation, Quelques généralisations de l'équation de Korteweg-de Vries. II, Generalized symmetries and recursive operators of some diffusive equations, On solutions of Liouville's equation, A simple and direct method for generating travelling wave solutions for nonlinear equations, A numerical scheme for periodic travelling-wave simulations in some nonlinear dispersive wave models, A numerical study of the 3-periodic wave solutions to KdV-type equations, Generalized conditional symmetries of evolution equations, Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains, An integrable couplings of G-WKI equations hierarchy with self-consistent sources, On the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy, Solitons and periodic solutions for the fifth-order KdV equation with the Exp-function method, Tangential polynomials and elliptic solitons, On the complete integrability of nonlinear dynamical systems on functional manifolds within the gradient-holonomic approach, A kind of integrable couplings of soliton equations hierarchy with self-consistent sources associated with \(\tilde{sl}(4)\), Non-isospectral integrable couplings of Ablowitz-Ladik hierarchy with self-consistent sources, Numerical solutions of the Korteweg-de Vries equation using the periodic scattering transform \(\mu\)-representation, Central affine curve flow on the plane, Computational synergetics and mathematical innovation, The Korteweg-de Vries equation with a self-consistent source in the class of periodic functions, Integrable couplings of C-KdV equations hierarchy with self-consistent sources associated with \(\tilde{sl}(4)\), The Camassa Holm equation: Conserved quantities and the initial value problem, Finite gap integration of the derivative nonlinear Schrödinger equation: a Riemann-Hilbert method, Degenerate Riemann theta functions, Fredholm and Wronskian representations of the solutions to the KdV equation and the degenerate rational case, Breather-like director reorientations in a nematic liquid crystal with nonlocal nonlinearity, Hyperelliptic Riemann surfaces of infinite genus and solutions of the KdV equation, Commuting flows and invariant tori: Korteweg-de Vries, Quasi-periodic solutions of \((3+1)\) generalized BKP equation by using Riemann theta functions, Coupled discrete Sawada-Kotera equations and their explicit quasi-periodic solutions, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Algebro-geometric solutions of the coupled Chaffee-Infante reaction diffusion hierarchy, Exact solutions for the fourth order nonlinear Schrödinger equations with cubic and power law nonlinearities, Bargmann-type finite-dimensional reductions of the Lax-integrable supersymmetric Boussinesq hierarchy and their integrability, Time behaviour of the error when simulating finite-band periodic waves. The case of the KdV equation, Nonlinear evolution equations and hyperelliptic covers of elliptic curves, Small dispersion limit of the Korteweg-de Vries equation with periodic initial conditions and analytical description of the Zabusky-Kruskal experiment, Semiclassical limit of the focusing NLS: Whitham equations and the Riemann-Hilbert problem approach, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Lump, mixed lump-stripe and rogue wave-stripe solutions of a \((3+1)\)-dimensional nonlinear wave equation for a liquid with gas bubbles, Local analyticity in the time and space variables and the smoothing effect for the fifth-order KdV-type equation, Reliable analysis for nonlinear Schrödinger equations with a cubic nonlinearity and a power law nonlinearity, Almost periodic behavior of nonlinear waves, Matrix finite-zone operators, PSEUDO: applications of streams and lazy evaluation to integrable models, Periodic finite-genus solutions of the KdV equation are orbitally stable, Remarks on the Korteweg-de Vries equation, A reliable treatment for nonlinear Schrödinger equations, Anomalous waves triggered by abrupt depth changes: laboratory experiments and truncated KdV statistical mechanics, A kind of explicit Riemann theta functions periodic waves solutions for discrete soliton equations, Theta function solutions for two discrete equations, New soliton and periodic solutions of \((1+2)\)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity, Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations, A discrete nonlinear Schrödinger-type hierarchy, its finite-dimensional reduction analysis, and the numerical integration scheme, Quasiperiodic solutions of the negative-order Korteweg-de Vries hierarchy, An integrable couplings of Dirac soliton hierarchy with self-consistent sources, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II: The KdV-equation, The stationary equations of a coupled nonlinear Schrödinger system, The KP equation with quasiperiodic initial data, Involutive functionals, infinite dimensional tori, and neighboring tori, The expansions of spectral function and the corresponding finite dimensional integrable systems, Solitons and periodic solutions for the fifth-order KdV equation, On the instability tongues of the Hill equation coupled with a conservative nonlinear oscillator, Low regularity conservation laws for integrable PDE, Periodic solutions and Whitham modulation equations for the Lakshmanan-Porsezian-Daniel equation, The inverse periodic problem for Hill's equation with a finite-gap potential, Invariant measures for some partial differential equations, On normal and non-normal wave statistics implied by a canonical-microcanonical Gibbs ensemble of the truncated KdV system, Conservation laws of evolution equations: Generic non-existence, Algebro-geometric quasi-periodic solutions to the Bogoyavlensky lattice 2(3) equations, Nonlinear stability of multi-solitons for the Hirota equation, Two-dimensional Schrödinger operators in periodic fields, Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method, Derivation of the Korteweg-de Vries equation for an operator with a mixed spectrum, Spectral stability of multi-solitons for generalized Hamiltonian system. I: The Caudrey-Dodd-Gibbon-Sawada-Kotera equation, Double cnoidal waves of the Korteweg-de Vries equation: A boundary value approach, Stability of smooth multi-solitons for the Camassa-Holm equation, Orbital stability of KdV multisolitons in \(H^{-1}\), Algebro-geometric finite gap solutions to the Korteweg-de Vries equation as primitive solutions, Periodic problem for nonlinear Ablowitz model

Uses Software

• GEAR

Cites Work

• Unnamed Item
• Unnamed Item
• The periodic problem for the Korteweg-de Vries equation
• The spectrum of Hill's equation
• Korteweg-de Vries equation; asymptotic behavior of solutions
• On finite-difference methods for the Korteweg-de Vries equation
• Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States
• Korteweg‐devries equation and generalizations. VI. methods for exact solution
• Method for Solving the Korteweg-deVries Equation
• On the Toda Lattice. II: Inverse-Scattering Solution
• 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
• A Rapidly Convergent Descent Method for Minimization
• Integrals of nonlinear equations of evolution and solitary waves
• The decay of the continuous spectrum for solutions of the Korteweg-deVries equation
• Stable hill equations
• On the determination of a Hill's equation from its spectrum
• On the determination of a Hill's equation from its spectrum