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Scientific article
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English

Metastable energy strata in weakly nonlinear wave equations

Published inCommunications in partial differential equations, vol. 37, no. 8, p. 1391-1413
Publication date2012
Abstract

We consider the problem of the long-time stability of plane waves under nonlinear perturbations of linear Klein-Gordon equations. This problem reduces to studying the distribution of the mode energies along solutions of one-dimensional semilinear Klein-Gordon equations with periodic boundary conditions when the initial data are small and concentrated in one Fourier mode. It is shown that for all except finitely many values of the mass parameter, the energy remains essentially localized in the initial Fourier mode over time scales that are much longer than predicted by standard perturbation theory. The mode energies decay geometrically with the mode number with a rate that is proportional to the total energy. The result is proved using modulated Fourier expansions in time.

Keywords
  • Nonlinear wave equation
  • Modulated Fourier expansion
  • Mode energies
Citation (ISO format)
GAUCKLER, Ludwig et al. Metastable energy strata in weakly nonlinear wave equations. In: Communications in partial differential equations, 2012, vol. 37, n° 8, p. 1391–1413. doi: 10.1080/03605302.2012.683503
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Article (Accepted version)
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ISSN of the journal0360-5302
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