Periodic Solutions for Completely Resonant Nonlinear Wave Equations with Dirichlet Boundary Conditions

We consider the nonlinear string equation with Dirichlet boundary conditions utt−uxx=ϕ(u), with ϕ(u)=Φu3+O(u5) odd and analytic, Φ≠0, and we construct small amplitude periodic solutions with frequency ω for a large Lebesgue measure set of ω close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and non-resonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect to the nonlinear wave equations utt−uxx+Mu=ϕ(u), M≠0, is that not only the P equation but also the Q equation is infinite-dimensional.

[1]  R. Kannan,et al.  Periodic solutions of nonlinear wave equations , 1983 .

[2]  Jurgen Poschel.  Quasi-periodic solutions for a nonlinear wave equation , 2007 .

[3]  Alan Weinstein,et al.  Normal modes for nonlinear hamiltonian systems , 1973 .

[4]  李幼升,et al.  Ph , 1989 .

[5]  P. Rabinowitz,et al.  Periodic Solutions of Nonlinear Hyperbolie Partial Differential Equations , 1967 .

[6]  G. Gentile,et al.  Hyperbolic Low-Dimensional Invariant Tori¶and Summations of Divergent Series , 2002 .

[7]  Dario Bambusi,et al.  Families of Periodic Solutions of Resonant PDEs , 2001, J. Nonlinear Sci..

[8]  C. D. Levermore,et al.  Singular limits of dispersive waves , 1994 .

[9]  Sergei Kuksin,et al.  Fifteen years of KAM for PDE , 2004 .

[10]  P. Rowlinson.  ALGEBRAIC GRAPH THEORY (Graduate Texts in Mathematics 207) By CHRIS GODSIL and GORDON ROYLE: 439 pp., £30.50, ISBN 0-387-95220-9 (Springer, New York, 2001). , 2002 .

[11]  D. C. Lewis,et al.  On the periodic motions near a given periodic motion of a dynamical system , 1934 .

[12]  Jean-Michel Coron,et al.  Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz , 1980 .

[13]  Jean Bourgain,et al.  Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE , 1994 .

[14]  J. Fröhlich,et al.  A rigorous approach to Anderson localization , 1984 .

[15]  Sergej B. Kuksin,et al.  Nearly Integrable Infinite-Dimensional Hamiltonian Systems , 1993 .

[16]  Massimiliano Berti,et al.  Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities , 2002, math/0211310.

[17]  C. Eugene Wayne,et al.  Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory , 1990 .

[18]  W. Craig,et al.  Nonlinear Waves and the 1:1:2 Resonance , 1994 .

[19]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[20]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[21]  J. Pöschel,et al.  Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrodinger equation , 1996 .

[22]  P. Rabinowitz.  Periodic solutions of nonlinear hyperbolic partial differential equations. II , 1969 .

[23]  Guido Gentile,et al.  Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method , 2004 .

[24]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[25]  Guido Gentile,et al.  A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems , 1999 .

[26]  Giovanni Gallavotti,et al.  Twistless KAM tori , 1993, chao-dyn/9306003.

[27]  A. Liapounoff,et al.  Problème général de la stabilité du mouvement , 1907 .

[28]  Guido Gentile,et al.  Whiskered tori with prefixed frequencies and Lyapunov spectrum , 1995 .

[29]  L. H. Eliasson,et al.  Absolutely convergent series expansions for quasi periodic motions. , 1996 .

[30]  D. Bambusi,et al.  Normal form and exponential stability for some nonlinear string equations , 2001 .

[31]  I. M. Pyshik,et al.  Table of integrals, series, and products , 1965 .

[32]  S. P. Novikov,et al.  Fifteen years of KAM for PDE , 2002 .

[33]  P. Rabinowitz,et al.  Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems , 1977 .

[34]  Walter Craig,et al.  Newton's method and periodic solutions of nonlinear wave equations , 1993 .

[35]  D. Bambusi.  Lyapunov center theorem for some nonlinear PDE's : a simple proof , 2000 .

[36]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[37]  B. Lidskii,et al.  Periodic solutions of the equation utt — uxx + u3 = 0 , 1988 .

[38]  A. M. Li︠a︡punov.  Problème général de la stabilité du mouvement , 1949 .

[39]  W. Craig.  Problèmes de petits diviseurs dans les équations aux dérivées partielles , 2000 .

[40]  H. Whitney.  Analytic Extensions of Differentiable Functions Defined in Closed Sets , 1934 .

[41]  J. Bourgain.  Harmonic Analysis and Nonlinear Partial Differential Equations , 1995 .

[42]  M. Berti,et al.  Multiplicity of periodic solutions of nonlinear wave equations , 2004 .

[43]  Jean Bourgain,et al.  QUASI-PERIODIC SOLUTIONS OF HAMILTONIAN PERTURBATIONS OF 2D LINEAR SCHRODINGER EQUATIONS , 1998 .

[44]  Frank Harary,et al.  Graphical enumeration , 1973 .

[45]  J. Bourgain.  Construction of periodic solutions of nonlinear wave equations in higher dimension , 1995 .