Nonexistence of small, odd breathers for a class of nonlinear wave equations

Abstract

In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to $0$ in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations such as the sine Gordon equation and ${\varphi }^4$ and ${\varphi }^6$ models. It also partially answers a question of Soffer and Weinstein in \cite[p. 19]{MR1681113} about nonexistence of breathers for the cubic NLKG in dimension one.

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Cited by in corpus (9)

• Quadratic Klein-Gordon equations with a potential in one dimension
• Breathers and the dynamics of solutions to the KdV type equations
• Scattering in the energy space for Boussinesq equations
• On the asymptotic stability of the sine-Gordon kink in the energy space
• On $1$d quadratic Klein-Gordon equations with a potential and symmetries
• Variational methods for breather solutions of Nonlinear Wave Equations
• On the scattering problem for the nonlinear Schrödinger equation with a potential in 2D
• On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting
• Decay of solutions of nonlinear Dirac equations