PORTUGALIAE MATHEMATICA Vol. 52, No. 1, pp. 109123 (1995) 

Asymptotically Periodic Solutions for a Class of Nonlinear Coupled OscillatorsThierry Cazenave and Fred B. WeisslerAnalyse Numérique  URA CNRS 189, Univ. Pierre et Marie Curie,4, place Jussieu, F75252 Paris Cedex 05  FRANCE Laboratoire Analyse Géométrie et Applications, URA CNRS 742, Institut Galilée  Univ. Paris XIII, Avenue J.B. Clément, F93430 Villetanneuse  FRANCE Abstract: We consider the Hamiltonian system $$ \left\{\eqalign{u''+u+(u^2+v^2)^\alpha\,u&{}=0,\cr v''+k\,v+(u^2+v^2)^\alpha\,v&{}=0,\cr}\right. $$ where $k$, $\alpha$ are real numbers, $k>1$ and $\alpha >0$. This system is a special case of the nonlinear wave equation $$ u_{tt}\Delta u+\u\_{L^2}^{2\alpha}\,u=0, $$ when only two Fourier components of the solution are nonzero. We show that for sufficiently large energy, every periodic solution of the above system with $v\equiv 0$ has a nontrivial stable manifold. Thus, we obtain asymptotically periodic, and therefore nonrecurrent, solutions of this nonlinear wave equation. The same result is also true for a wider class of nonlinearities. Keywords: Conservative wave equations; Hamiltonian systems; Poincaré map; stable manifolds; nonrecurrent solutions. Classification (MSC2000): 35L70, 34D05, 34C35 Full text of the article:
Electronic version published on: 29 Mar 2001. This page was last modified: 27 Nov 2007.
© 1995 Sociedade Portuguesa de Matemática
