Global dynamical behaviors in a physical shallow water system

•Third order dispersive equation.•Nonlinear wave equation.•Phase portraits.•Solitary wave solution.•Breaking wave solution. The theory of bifurcations of dynamical systems is used to investigate the behavior of travelling wave solutions in an entire family of shallow water wave equations. This famil...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2016-07, Vol.36, p.285-302
Main Authors: Tchakoutio Nguetcho, Aurélien Serge, Li, Jibin, Bilbault, Jean-Marie
Format: Article
Language:English
Subjects:
Breaking wave solution
Cnoïdal waves
Compacton solution
Dark solitons
Dynamical systems
Mathematical analysis
Mathematical models
Mathematics
Mechanics
Nonlinear dynamics
Nonlinearity
Periodic wave solution
Phase portraits
Physics
Shallow water
Solitary wave solution
Third order dispersive equation
Traveling waves
Wave equations
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Summary:•Third order dispersive equation.•Nonlinear wave equation.•Phase portraits.•Solitary wave solution.•Breaking wave solution. The theory of bifurcations of dynamical systems is used to investigate the behavior of travelling wave solutions in an entire family of shallow water wave equations. This family is obtained by a perturbative asymptotic expansion for unidirectional shallow water waves. According to the parameters of the system, this family can lead to different sets of known equations such as Camassa–Holm, Korteweg–de Vries, Degasperis and Procesi and several other dispersive equations of the third order. Looking for possible travelling wave solutions, we show that different phase orbits in some regions of parametric planes are similar to those obtained with the model of the pressure waves studied by Li and Chen. Many other exact explicit travelling waves solutions are derived as well, some of them being in perfect agreement with solutions obtained in previous works by researchers using different methods. When parameters are varied, the conditions under which the above solutions appear are also shown. The dynamics of singular nonlinear travelling system is completely determined for each of the above mentioned equations. Moreover, we define sufficient conditions leading to the existence of propagating wave solutions and demonstrate how and why travelling waves lose their smoothness and develop into solutions with compact support or breaking waves.
ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2015.12.006