Asymptotically Periodic Solutions for a Class of Nonlinear Coupled Oscillators

Thierry Cazenave and Fred B. Weissler

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

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