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Existence and linearized stability of solitary waves for a quasilinear Benney system

Part of: General higher-order equations and systems Representations of solutions

Published online by Cambridge University Press:  21 March 2016

João-Paulo Dias ,
Mário Figueira  and
Filipe Oliveira
João-Paulo Dias
Affiliation:
CMAF-UL, Av. Prof. Gama Pinto, 1649-003 Lisboa, Portugal and DM-FCUL, Campo Grande, 1749-016 Lisboa, Portugal (dias@ptmat.fc.ul.pt; figueira@ptmat.fc.ul.pt)
Mário Figueira
Affiliation:
CMAF-UL, Av. Prof. Gama Pinto, 1649-003 Lisboa, Portugal and DM-FCUL, Campo Grande, 1749-016 Lisboa, Portugal (dias@ptmat.fc.ul.pt; figueira@ptmat.fc.ul.pt)
Filipe Oliveira
Affiliation:
CMAF-UL, Av. Prof. Gama Pinto, 1649-003 Lisboa, Portugal and FCT-UNL, Monte da Caparica, 2829-516 Caparica, Portugal (fso@fct.unl.pt)
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Extract

We prove the existence of solitary wave solutions to the quasilinear Benney system

where , –1 < p < +∞ and a, γ > 0. We establish, in particular, the existence of travelling waves with speed arbitrarily large if p < 0 and arbitrarily close to 0 if . We also show the existence of standing waves in the case , with compact support if – 1 < p < 0. Finally, we obtain, under certain conditions, the linearized stability of such solutions.

Keywords

long wave–short wave interactionssolitary wavesdispersive equationshyperbolic systemslinearized stability

MSC classification

Secondary: 35G55: Initial value problems for nonlinear higher-order systems 35C07: Traveling wave solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 3 , June 2016 , pp. 547 - 564
Copyright
Copyright © Royal Society of Edinburgh 2016 

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