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

Pietro Baldi

doi:10.3934/dcds.2006.15.883

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2006, Volume 15, Issue 3: 883-903. Doi: 10.3934/dcds.2006.15.883

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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

Received: December 31, 2004

Revised: October 31, 2005

Published: August 2006

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Abstract

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,
quasi-periodic solutions,
Lyapunov-Schmidt reduction.

Mathematics Subject Classification: 35L05, 35B15, 37K50.

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