Local Well-Posedness and Orbital Stability of Solitary Wave Solutions for the Generalized Camassa–Holm Equation

2005 ◽  
Vol 30 (5-6) ◽  
pp. 761-781 ◽  
Author(s):  
Sevdzhan Hakkaev ◽  
Kiril Kirchev
Keyword(s):  
Solitary Wave ◽  
Orbital Stability ◽  
Well Posedness ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Holm Equation

scholarly journals A Note on Solitary Wave Solutions of the Nonlinear Generalized Camassa-Holm Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Lei Zhang ◽  
Xing Tao Wang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Solitary Wave ◽  
Numerical Simulations ◽  
Solitary Wave Solutions ◽  
Simple Method ◽  
Wave Solutions ◽  
Holm Equation

We give a simple method for applying ordinary differential equation to solve the nonlinear generalized Camassa-Holm equation ut+2kux−uxxt+aumux−2uxuxx+uuxxx=0. Furthermore we give a new ansätz. In the cases where m=1,2,3, the numerical simulations demonstrate the results.

scholarly journals On Solitary Wave Solutions for the Camassa-Holm and the Rosenau-RLW-Kawahara Equations with the Dual-Power Law Nonlinearities

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nattakorn Sukantamala ◽  
Supawan Nanta
Keyword(s):  
Solitary Wave ◽  
Nonlinear Wave ◽  
Traveling Wave ◽  
Analytic Solution ◽  
Traveling Wave Solutions ◽  
Well Posedness ◽  
Solitary Wave Solutions ◽  
Kawahara Equation ◽  
Ansatz Method ◽  
Wave Solutions

The nonlinear wave equation is a significant concern to describe wave behavior and structures. Various mathematical models related to the wave phenomenon have been introduced and extensively being studied due to the complexity of wave behaviors. In the present work, a mathematical model to obtain the solution of the nonlinear wave by coupling the classical Camassa-Holm equation and the Rosenau-RLW-Kawahara equation with the dual term of nonlinearities is proposed. The solution properties are analytically derived. The new model still satisfies the fundamental energy conservative property as the original models. We then apply the energy method to prove the well-posedness of the model under the solitary wave hypothesis. Some categories of exact solitary wave solutions of the model are described by using the Ansatz method. In addition, we found that the dual term of nonlinearity is essential to obtain the class of analytic solution. Besides, we provide some graphical representations to illustrate the behavior of the traveling wave solutions.

scholarly journals Diverse approaches to search for solitary wave solutions of the fractional modified Camassa–Holm equation

2021 ◽  
pp. 104882
Author(s):  
Asim Zafar ◽  
M. Raheel ◽  
Kamyar Hosseini ◽  
Mohammad Mirzazadeh ◽  
Soheil Salahshour ◽  
...  
Keyword(s):  
Solitary Wave ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Holm Equation
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