Nonlinear stability of periodic-wave solutions for systems of dispersive equations

Fabrício Cristófani; Ademir Pastor

doi:10.3934/cpaa.2020225

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2020, Volume 19, Issue 10: 5015-5032. Doi: 10.3934/cpaa.2020225

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Nonlinear stability of periodic-wave solutions for systems of dispersive equations

Fabrício Cristófani^, and
Ademir Pastor

IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, Brazil

^* Corresponding author
^* Corresponding author

Received: January 31, 2020

Revised: April 30, 2020

Published: July 2020

F. C. is supported by FAPESP/Brazil grant 2017/20760-0. A. P. is partially supported by CNPq/Brazil grants 402849/2016-7 and 303098/2016-3

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Abstract

We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.

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Mathematics Subject Classification: Primary: 35B35, 35Q51; Secondary: 35Q53.

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Figure 1. Left: Phase space for $ c_0 = 1 $ and $ A_0 = 1 $. Right: Phase space for $ c_0 = 1 $ and $ A_0 = 2 $. In both cases, the orbits in black are those for which $ \varphi_{(c_0,A_0)}>1 $

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