scholarly journals A Note on the Stability and Instability of Travelling Wave of Korteweg-de Vries Type: The Periodic Case

2011 ◽  
Vol 14 ◽  
pp. 57-72
Author(s):  
José R Quintero
Keyword(s):  
Orbital Stability ◽  
Type Equation ◽  
Travelling Wave ◽  
Periodic Case ◽  
Travelling Wave Solutions ◽  
Stability And Instability ◽  
Periodic Travelling Wave ◽  
De Vries ◽  
The Mean ◽  
The Stability

In this paper we adapt the work of M. Grillakis, J. Shatah, and W. Strauss, or J. Bona, P. Souganidis and W. Strauss to the periodic case in spaces having the mean zero property in order to establish the orbital stability/instability of periodic travelling wave solutions of a generalized Korteweg-de Vries type equation.

scholarly journals Existence and Orbital Stability of Cnoidal Waves for a 1D Boussinesq Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-36 ◽  
Author(s):  
Jaime Angulo ◽  
Jose R. Quintero
Keyword(s):  
Orbital Stability ◽  
Type Equation ◽  
Elliptic Functions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Jacobian Elliptic Functions ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Complete Study ◽  
Set Up

We will study the existence and stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equationΦtt−Φxx+aΦxxxx−bΦxxtt+ΦtΦxx+2ΦxΦxt=0. Periodic travelling-wave solutions with an arbitrary fundamental periodT0will be built by using Jacobian elliptic functions. Stability (orbital) of these solutions by periodic disturbances with periodT0will be a consequence of the general stability criteria given by M. Grillakis, J. Shatah, and W. Strauss. A complete study of the periodic eigenvalue problem associated to the Lame equation is set up.

scholarly journals A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation

2018 ◽  
Vol 11 (4) ◽  
pp. 691-705 ◽  
Author(s):  
Mudassar Imran ◽  
◽  
Youssef Raffoul ◽  
Muhammad Usman ◽  
Chi Zhang ◽  
...  
Keyword(s):  
Type Equation ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
De Vries

scholarly journals Rogue waves on the general periodic travelling waves background for an extended modified Korteweg-de Vries equation

2021 ◽  
Author(s):  
Yi Zhang ◽  
Yu Lou ◽  
RS Ye
Keyword(s):  
Vries Equation ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Rogue Waves ◽  
Travelling Wave ◽  
Periodic Waves ◽  
Travelling Wave Solutions ◽  
Third Order ◽  
Periodic Travelling Wave ◽  
De Vries

Under consideration in this paper is rogue waves on the general periodic travelling waves background of an integrable extended modified Korteweg-de Vries equation. The general periodic travelling wave solutions are presented in terms of the sub-equation method. By means of the Darboux transformation and the nonlinearization of the Lax pair, we present the first-, second- and third-order rogue waves on the general periodic travelling waves background. Furthermore, the dynamic behaviors of rogue periodic waves are elucidated from the viewpoint of three-dimensional structures.

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