New type of solitary wave solution with coexisting crest and trough for a perturbed wave equation

2021 ◽  
Author(s):  
Lijun Zhang ◽  
Jundong Wang ◽  
Elena Shchepakina ◽  
Vladimir Sobolev
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
New Type

New Solitary Wave Solution for the Boussinesq Wave Equation Using the Semi-Inverse Method and the Exp-Function Method

2009 ◽  
Vol 64 (11) ◽  
pp. 709-712 ◽  
Author(s):  
Wenjun Liu
Keyword(s):  
Wave Equation ◽  
Variational Principle ◽  
Solitary Wave ◽  
Wave Solution ◽  
Variational Formulation ◽  
Function Method ◽  
Inverse Method ◽  
Solitary Wave Solution ◽  
Solitary Solutions

Using the semi-inverse method, a variational formulation is established for the Boussinesq wave equation. Based on the obtained variational principle, solitary solutions in the sech-function and expfunction forms are obtained

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.

scholarly journals Abundant Explicit and Exact Solutions for the Variable Coefficient mKdV Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaoxiao Zheng ◽  
Yadong Shang ◽  
Yong Huang
Keyword(s):  
Solitary Wave ◽  
Elliptic Function ◽  
Constant Coefficient ◽  
Wave Solution ◽  
Integral Method ◽  
Solitary Wave Solution ◽  
First Integral ◽  
Mkdv Equation ◽  
Variable Coefficients ◽  
First Integral Method

This paper is concerned with the variable coefficients mKdV (VC-mKdV) equation. First, through some transformation we convert VC-mKdV equation into the constant coefficient mKdV equation. Then, using the first integral method we obtain the exact solutions of VC-mKdV equation, such as rational function solutions, periodic wave solutions of triangle function, bell-shape solitary wave solution, kink-shape solitary wave solution, Jacobi elliptic function solutions, and Weierstrass elliptic function solution. Furthermore, with the aid of Mathematica, the extended hyperbolic functions method is used to establish abundant exact explicit solution of VC-mKdV equation. By the results of the equation, the first integral method and the extended hyperbolic function method are extended from the constant coefficient nonlinear evolution equations to the variable coefficients nonlinear partial differential equation.

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