Long-time stability of small FPU solitary waves

Amjad Khan; Dmitry E. Pelinovsky

doi:10.3934/dcds.2017088

Article Contents

2017, Volume 37, Issue 4: 2065-2075. Doi: 10.3934/dcds.2017088

This issue Previous Article Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics Next Article Almost global existence for cubic nonlinear Schrödinger equations in one space dimension

Long-time stability of small FPU solitary waves

Amjad Khan^1, and
Dmitry E. Pelinovsky^2,

1.
Department of Applied Mathematics, Western University, London, ON, N6A 3K7, Canada
2.
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

Author Bio: E-mail address: akhan659@uwo.ca; E-mail address: dmpeli@math.mcmaster.ca

Received: January 31, 2016

Revised: October 31, 2016

Published: May 2017

The work of A. Khan was performed during MSc program at McMaster University in 2013-2015. The work of D.E. Pelinovsky is supported by the NSERC grant. The authors thank E. Dumas, T. Penati, and G. Schneider for discussions and collaborations.

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Abstract

Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentialsare described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitudeapproximation is usually performed on the time scale, for which dynamics of the KdV equation is defined.We show how to extend justification analysis on longer time intervals provided dynamics of the generalized KdVequation is globally well-posed in Sobolev spaces and either the Sobolev norms are globally boundedor they grow at most polynomially. The time intervals are extended respectively by the logarithmic or double logarithmic factorsin terms of the small amplitude parameter. Controlling the approximation error on longer time intervalsallows us to deduce nonlinear metastability of small FPU solitary waves from orbital stability of the KdV solitary waves.

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Mathematics Subject Classification: Primary:37K60;Secondary:35Q53, 37K40.

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