Travelling wave solutions for some two-component shallow water models

Denys Dutykh; Delia Ionescu-Kruse

doi:10.1016/j.jde.2016.03.035

Travelling wave solutions for some two-component shallow water models

Denys Dutykh
, Delia Ionescu-Kruse

Mathematics

Simion Stoilow Institute of Mathematics of the Romanian Academy Research Unit

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

In the present study we perform a unified analysis of travelling wave solutions to three different two-component systems which appear in shallow water theory. Namely, we analyze the celebrated Green-Naghdi equations, the integrable two-component Camassa-Holm equations and a new two-component system of Green-Naghdi type. In particular, we are interested in solitary and cnoidal-type solutions, as two most important classes of travelling waves that we encounter in applications. We provide a complete phase-plane analysis of all possible travelling wave solutions which may arise in these models. In particular, we show the existence of new type of solutions.

Original language	British English
Pages (from-to)	1099-1114
Number of pages	16
Journal	Journal of Differential Equations
Volume	261
Issue number	2
DOIs	https://doi.org/10.1016/j.jde.2016.03.035
State	Published - 15 Jul 2016

Keywords

Camassa-Holm equations
Cnoidal waves
Green-Naghdi model
Phase-plane analysis
Serre equations
Solitary waves
Travelling waves

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10.1016/j.jde.2016.03.035

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title = "Travelling wave solutions for some two-component shallow water models",

abstract = "In the present study we perform a unified analysis of travelling wave solutions to three different two-component systems which appear in shallow water theory. Namely, we analyze the celebrated Green-Naghdi equations, the integrable two-component Camassa-Holm equations and a new two-component system of Green-Naghdi type. In particular, we are interested in solitary and cnoidal-type solutions, as two most important classes of travelling waves that we encounter in applications. We provide a complete phase-plane analysis of all possible travelling wave solutions which may arise in these models. In particular, we show the existence of new type of solutions.",

keywords = "Camassa-Holm equations, Cnoidal waves, Green-Naghdi model, Phase-plane analysis, Serre equations, Solitary waves, Travelling waves",

author = "Denys Dutykh and Delia Ionescu-Kruse",

note = "Funding Information: The authors acknowledge the support of this work by “ Laboratoire Europ{\'e}en Associ{\'e} CNRS Franco-Roumain de Math{\'e}matiques et Mod{\'e}lisation ” LEA Math-Mode. D. Dutykh would like to thank Professors D. Clamond (Universit{\'e} de Nice Sophia-Antipolis, France) and D. Mitsotakis (Victoria University of Wellington, New Zealand) for helpful discussions on shallow water models, and M. Raibaut (Universit{\'e} Savoie Mont Blanc, France) for helpful discussions on implicitly defined curves. Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

year = "2016",

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doi = "10.1016/j.jde.2016.03.035",

language = "British English",

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T1 - Travelling wave solutions for some two-component shallow water models

AU - Dutykh, Denys

AU - Ionescu-Kruse, Delia

N1 - Funding Information: The authors acknowledge the support of this work by “ Laboratoire Européen Associé CNRS Franco-Roumain de Mathématiques et Modélisation ” LEA Math-Mode. D. Dutykh would like to thank Professors D. Clamond (Université de Nice Sophia-Antipolis, France) and D. Mitsotakis (Victoria University of Wellington, New Zealand) for helpful discussions on shallow water models, and M. Raibaut (Université Savoie Mont Blanc, France) for helpful discussions on implicitly defined curves. Publisher Copyright: © 2016 Elsevier Inc.

PY - 2016/7/15

Y1 - 2016/7/15

N2 - In the present study we perform a unified analysis of travelling wave solutions to three different two-component systems which appear in shallow water theory. Namely, we analyze the celebrated Green-Naghdi equations, the integrable two-component Camassa-Holm equations and a new two-component system of Green-Naghdi type. In particular, we are interested in solitary and cnoidal-type solutions, as two most important classes of travelling waves that we encounter in applications. We provide a complete phase-plane analysis of all possible travelling wave solutions which may arise in these models. In particular, we show the existence of new type of solutions.

AB - In the present study we perform a unified analysis of travelling wave solutions to three different two-component systems which appear in shallow water theory. Namely, we analyze the celebrated Green-Naghdi equations, the integrable two-component Camassa-Holm equations and a new two-component system of Green-Naghdi type. In particular, we are interested in solitary and cnoidal-type solutions, as two most important classes of travelling waves that we encounter in applications. We provide a complete phase-plane analysis of all possible travelling wave solutions which may arise in these models. In particular, we show the existence of new type of solutions.

KW - Camassa-Holm equations

KW - Cnoidal waves

KW - Green-Naghdi model

KW - Phase-plane analysis

KW - Serre equations

KW - Solitary waves

KW - Travelling waves

UR - https://www.scopus.com/pages/publications/84962028740

U2 - 10.1016/j.jde.2016.03.035

DO - 10.1016/j.jde.2016.03.035

M3 - Article

AN - SCOPUS:84962028740

SN - 0022-0396

VL - 261

SP - 1099

EP - 1114

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -

Travelling wave solutions for some two-component shallow water models

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Keywords

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