Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions

DOI10.1007/s00220-004-1255-8zbMath1094.35021arXivmath/0402262OpenAlexW2154789637MaRDI QIDQ1781841

Guido Gentile, Michela Procesi, Vieri Mastropietro

Publication date: 8 June 2005

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0402262

zbMATH Keywords

periodic solutionsLyapunov-Schmidt methodstring equationsmall divisors

Mathematics Subject Classification ID

Initial-boundary value problems for second-order hyperbolic equations (35L20) Periodic solutions to PDEs (35B10) Second-order nonlinear hyperbolic equations (35L70) Applications of operator theory to differential and integral equations (47N20)

Related Items

An abstract Nash-Moser theorem with parameters and applications to PDEs ⋮ A KAM theorem with applications to partial differential equations of higher dimensions ⋮ Conservation of resonant periodic solutions for the one-dimensional nonlinear Schrödinger equation ⋮ Quasi-periodic solutions of completely resonant nonlinear wave equations ⋮ Periodic solutions for a class of nonlinear partial differential equations in higher dimension ⋮ Small amplitude periodic solutions in time for one-dimensional nonlinear wave equations ⋮ Small amplitude periodic solutions of Klein-Gordon equations ⋮ Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing ⋮ Families of Periodic Solutions for Some Hamiltonian PDEs ⋮ Invariant tori for the Hamiltonian derivative wave equation with higher order nonlinearity ⋮ Quasiperiodic solutions of completely resonant wave equations with quasiperiodic forced terms ⋮ Boundary-value problem with mixed conditions for typeless linear partial differential equations ⋮ A modified nonlinear POD method for order reduction based on transient time series ⋮ Deterministic learning of completely resonant nonlinear wave systems with Dirichlet boundary conditions ⋮ Cantor families of periodic solutions for completely resonant wave equations ⋮ Cantor families of periodic solutions for wave equations via a variational principle ⋮ Periodic solutions for the Schrödinger equation with nonlocal smoothing nonlinearities in higher dimension ⋮ Torsional instability in suspension bridges: the Tacoma Narrows Bridge case ⋮ Quasiperiodic motions in dynamical systems: Review of a renormalization group approach ⋮ Periodic solutions of Birkhoff-Lewis type for the nonlinear wave equation ⋮ A Birkhoff-Lewis type theorem for the nonlinear wave equation ⋮ Cantor families of periodic solutions for completely resonant nonlinear wave equations ⋮ Forced vibrations of wave equations with non-monotone nonlinearities ⋮ Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies ⋮ Quasi-Periodic Solutions of Completely Resonant Forced Wave Equations ⋮ Degenerate elliptic resonances

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory
• Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems
• Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations
• Nearly integrable infinite-dimensional Hamiltonian systems
• Twistless KAM tori
• Periodic solutions of nonlinear wave equations with general nonlinearities
• Multiplicity of periodic solutions of nonlinear wave equations.
• Periodic solutions of the equation \(u_{tt}-u_{xx}+u^ 3=0\)
• On the periodic motions near a given periodic motion of a dynamical system
• Hyperbolic low-dimensional invariant tori and summations of divergent series
• Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method
• Construction of periodic solutions of nonlinear wave equations in higher dimension
• Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation
• Quasi-periodic solutions for a nonlinear wave equation
• Normal modes for nonlinear Hamiltonian systems
• Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz
• Periodic solutions of hamiltonian systems
• Forced vibrations for a nonlinear wave equation
• Newton's method and periodic solutions of nonlinear wave equations
• A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems
• Analytic Extensions of Differentiable Functions Defined in Closed Sets
• Whiskered tori with prefixed frequencies and Lyapunov spectrum
• Periodic Solutions of Nonlinear Hyperbolie Partial Differential Equations
• Families of periodic solutions of resonant PDEs
• Normal form and exponential stability for some nonlinear string equations