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.

scholarly journals The First-Integral Method and Abundant Explicit Exact Solutions to the Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Yadong Shang ◽  
Xiaoxiao Zheng
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Elliptic Function ◽  
Integral Method ◽  
First Integral ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
First Integral Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion-acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell-type fornandE, the solitary wave solutions of kink-type forEand bell-type forn, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.

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.

Comparison Between Two Reliable Methods for Accurate Solution of Fractional Modified Fornberg–Whitham Equation Arising in Water Waves

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
A. K. Gupta ◽  
S. Saha Ray
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Water Waves ◽  
Integral Method ◽  
Solitary Wave Solution ◽  
First Integral ◽  
Accurate Solution ◽  
First Integral Method ◽  
Analytical Technique ◽  
Whitham Equation

In this paper, an analytical technique is proposed to determine the exact solution of fractional order modified Fornberg–Whitham equation. Since exact solution of fractional Fornberg–Whitham equation is unknown, first integral method has been applied to determine exact solutions. The solitary wave solution of fractional modified Fornberg–Whitham equation has been attained by using first integral method. The approximate solutions of fractional modified Fornberg–Whitham equation, obtained by optimal homotopy asymptotic method (OHAM), are compared with the exact solutions obtained by the first integral method. The obtained results are presented in tables to demonstrate the efficiency of these proposed methods. The proposed schemes are quite simple, effective, and expedient for obtaining solution of fractional modified Fornberg–Whitham equation.

scholarly journals Solitary and Periodic Wave Solutions of Sasa–Satsuma Equation and Their Relationship with Hamilton Energy

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17 ◽  
Author(s):  
Weiguo Zhang ◽  
Xingqian Ling ◽  
Bei-Bei Wang ◽  
Shaowei Li
Keyword(s):  
Solitary Wave ◽  
Integral Method ◽  
First Integral ◽  
Periodic Wave ◽  
Evolutionary Relationships ◽  
Solitary Wave Solutions ◽  
First Integral Method ◽  
Periodic Wave Solutions ◽  
Hamilton Energy ◽  
Wave Solutions

In this paper, we study the exact solitary wave solutions and periodic wave solutions of the S-S equation and give the relationships between solutions and the Hamilton energy of their amplitudes. First, on the basis of the theory of dynamical system, we make qualitative analysis on the amplitudes of solutions. Then, by using undetermined hypothesis method, the first integral method, and the appropriate transformation, two bell-shaped solitary wave solutions and six exact periodic wave solutions are obtained. Furthermore, we discuss the evolutionary relationships between these solutions and find that the appearance of these solutions for the S-S equation is essentially determined by the value which the Hamilton energy takes. Finally, we give some diagrams which show the changing process from the periodic wave solutions to the solitary wave solutions when the Hamilton energy changes.

Study of the Solitoff Solution and Fractal Solution for the (2+1)-Dimensional Broek-Kaup System with Variable Coefficients

2014 ◽  
Vol 532 ◽  
pp. 356-361
Author(s):  
Wei Ting Zhu
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Variable Separation ◽  
New Family ◽  
Localized Excitations ◽  
Variable Separation Method

Starting from a (G'/G)-expansion method and a variable separation method, a new family of exact solutions of the (2+1)-dimensional Broek-Kaup system with variable coefficients(VCBK) is obtained. Based on the derived solitary wave solution, we obtain some special localized excitations such as solitoff solutions and fractal solutions.

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