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August  2006, 15(3): 883-903. doi: 10.3934/dcds.2006.15.883

Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$

Pietro Baldi 1,

1. 

Sissa, via Beirut 2-4, 34014 Trieste, Italy

Received  January 2005 Revised  November 2005 Published  April 2006

We consider 1D completely resonant nonlinear wave equations of the type $v_{t t}$ - $v_{x x}$$ = -v^3 + \mathcal{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.
Keywords: Nonlinear wave equation, infinite dimensional Hamiltonian systems, Lyapunov-Schmidt reduction., quasi-periodic solutions.
Mathematics Subject Classification: 35L05, 35B15, 37K5.
Citation: Pietro Baldi. Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 883-903. doi: 10.3934/dcds.2006.15.883
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