Solitary wave solutions and cnoidal wave solutions to the combined KdV and mKdV equation

1994 ◽  
Vol 17 (5) ◽  
pp. 339-347 ◽  
Author(s):  
Sen-Yue Lou ◽  
Li-Li Chen
Keyword(s):  
Solitary Wave ◽  
Mkdv Equation ◽  
Cnoidal Wave ◽  
Solitary Wave Solutions ◽  
Wave Solutions

A New Type of Solitary Wave Solution of the mKdV Equation Under Singular Perturbations

2020 ◽  
Vol 30 (11) ◽  
pp. 2050162
Author(s):  
Lijun Zhang ◽  
Maoan Han ◽  
Mingji Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Solitary Wave ◽  
Singular Perturbations ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Mkdv Equation ◽  
Geometric Singular Perturbation Theory ◽  
Solitary Wave Solutions ◽  
Wave Speeds ◽  
Wave Solutions ◽  
New Type

In this work, we examine the solitary wave solutions of the mKdV equation with small singular perturbations. Our analysis is a combination of geometric singular perturbation theory and Melnikov’s method. Our result shows that two families of solitary wave solutions of mKdV equation, having arbitrary positive wave speeds and infinite boundary limits, persist for selected wave speeds after small singular perturbations. More importantly, a new type of solitary wave solution possessing both valley and peak, named as breather in physics, which corresponds to a big homoclinic loop of the associated dynamical system is observed. It reveals an exotic phenomenon and exhibits rich dynamics of the perturbed nonlinear wave equation. Numerical simulations are performed to further detect the wave speeds of the persistent solitary waves and the nontrivial one with both valley and peak.

Solitary wave solutions of the MKdV− equation

1995 ◽  
Vol 124 (4) ◽  
pp. 321-333 ◽  
Author(s):  
L.R.T. Gardner ◽  
G.A. Gardner ◽  
T. Geyikli
Keyword(s):  
Solitary Wave ◽  
Mkdv Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions
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