Quasi-periodic solutions of the equation v_{tt}-v_{xx}+v^3=f(v)

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We consider 1D completely resonant nonlinear wave equations of the type vtt -vxx = -v 3 +O(v 4) with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutions turn out to be, at the first order, the superposition of a traveling wave and a modulation of long period, depending only on time.

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Discrete Contin. Dyn. Syst. 15 (2006) 883-903

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