scholarly journals Exact Solutions for a Generalized KdV-MKdV Equation with Variable Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Bo Tang ◽  
Xuemin Wang ◽  
Yingzhe Fan ◽  
Junfeng Qu
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Mkdv Equation ◽  
Variable Coefficients ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Mathematical Tool ◽  
Auxiliary Equation Method

By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of variable-coefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for the KdV-MKdV equation are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving many other nonlinear partial differential equations with variable coefficients in mathematical physics.

A Generalized Auxiliary Equation Method for the Quadratic Nonlinear KG Equation

2013 ◽  
Vol 394 ◽  
pp. 571-576
Author(s):  
Sheng Zhang ◽  
Bo Xu ◽  
Ao Xue Peng
Keyword(s):  
Partial Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Mathematical Tool ◽  
Auxiliary Equation Method ◽  
Klein Gordon ◽  
Generalized Auxiliary Equation Method ◽  
Powerful Mathematical Tool

A generalized auxiliary equation method with symbolic computation is used to construct more general exact solutions of the quadratic nonlinear Klein-Gordon (KG) equation. As a result, new and more general solutions are obtained. It is shown that the generalized auxiliary equation method provides a more powerful mathematical tool for solving nonlinear partial differential equations arising in the fields of nonlinear sciences.

scholarly journals Infinitely Many Elliptic Solutions to a Simple Equation and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Long Wei ◽  
Yang Wang
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Simple Equation ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Auxiliary Equation Method ◽  
Elliptic Solutions

Based on auxiliary equation method and Bäcklund transformations, we present an idea to find infinitely many Weierstrass and Jacobi elliptic function solutions to some nonlinear problems. First, we give some nonlinear iterated formulae of solutions and some elliptic function solutions to a simple auxiliary equation, which results in infinitely many Weierstrass and Jacobi elliptic function solutions of the simple equation. Then applying auxiliary equation method to some nonlinear problems and combining the results with exact solutions of the auxiliary equation, we obtain infinitely many elliptic function solutions to the corresponding nonlinear problems. The employed approach is powerful and can be also applied to solve other nonlinear differential equations.

Quasi-Periodic, Periodic Waves, and Soliton Solutions for the Combined KdV-mKdV Equation

2009 ◽  
Vol 64 (9-10) ◽  
pp. 639-645 ◽  
Author(s):  
Emad A.-B. Abdel-Salam
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
De Vries ◽  
Jacobi Elliptic Function Method

By introducing the generalized Jacobi elliptic function, a new improved Jacobi elliptic function method is used to construct the exact travelling wave solutions of the nonlinear partial differential equations in a unified way. With the help of the improved Jacobi elliptic function method and symbolic computation, some new exact solutions of the combined Korteweg-de Vries-modified Korteweg-de Vries (KdV-mKdV) equation are obtained. Based on the derived solution, we investigate the evolution of doubly periodic and solitons in the background waves. Also, their structures are further discussed graphically.

scholarly journals Stability and Extraction of New Optical Wave Solutions of Cascaded System in (2 + 1)-dimensions

2021 ◽  
Author(s):  
Hitender Kumar ◽  
Parveen Parveen ◽  
Sunita Dahiya ◽  
Anand Kumar ◽  
Manjeet Singh Gautam
Keyword(s):  
Elliptic Function ◽  
Data Transfer ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Dynamical Behaviour ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Optical Wave ◽  
Auxiliary Equation Method ◽  
Weierstrass Elliptic Function

Abstract This paper uses the new modified sub-ODE method, the new extended auxiliary equation method, and the new Jacobi elliptic function expansion method to revisit the (2+1)-dimensional coupled nonlinear Schr¨odinger equation with cascading effect. As a consequence, dark, bright, kinkantikink, singular solitons, Weierstrass elliptic function, doubly periodic, and complex optical soliton solutions are retrieved. All solutions are described, along with the existence criterion on the parameters. As solitons are used for data transfer, the obtained results may be found usage in optical couplers, birefringed fibres, and optoelectronic devices. A comparison of the obtained results with those found in the literature is given. The dynamical behaviour of some of the obtained solutions has been explored for suitable choices of the parameters. Using the property of Hamiltonian systems, the solitons stability is determined.

Exact Solutions for N-Coupled Nonlinear Schrödinger Equations With Variable Coefficients

2016 ◽  
Vol 71 (7) ◽  
pp. 665-672 ◽  
Author(s):  
Bo Tang ◽  
Yingzhe Fan ◽  
Jixiu Wang ◽  
Shijun Chen
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Nonlinear Schrödinger Equations ◽  
Jacobi Elliptic Function ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

AbstractIn this paper, based on similarity transformation and auxiliary equation method, we construct many exact solutions ofN-coupled nonlinear Schrödinger equations with variable coefficients, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions and combined Jacobi elliptic function solutions. These solutions may give insight into many considerable physical processes.

Study on soliton solutions of the longitudinal wave equation and magneto-electro-elastic circular rod dynamical model

2021 ◽  
Vol 35 (13) ◽  
pp. 2150168
Author(s):  
Adel Darwish ◽  
Aly R. Seadawy ◽  
Hamdy M. Ahmed ◽  
A. L. Elbably ◽  
Mohammed F. Shehab ◽  
...  
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Longitudinal Wave ◽  
Physical Interpretation ◽  
Elliptic Function ◽  
Function Method ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Jacobi Elliptic Function ◽  
Two Dimensional

In this paper, we use the improved modified extended tanh-function method to obtain exact solutions for the nonlinear longitudinal wave equation in magneto-electro-elastic circular rod. With the aid of this method, we get many exact solutions like bright and singular solitons, rational, singular periodic, hyperbolic, Jacobi elliptic function and exponential solutions. Moreover, the two-dimensional and the three-dimensional graphs of some solutions are plotted for knowing the physical interpretation.

scholarly journals The Study of the Solution to a Generalized KdV-mKdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Xiumei Lv ◽  
Tengwei Shao ◽  
Jiacheng Chen
Keyword(s):  
Symbolic Computation ◽  
Elliptic Function ◽  
Mkdv Equation ◽  
High Order ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Mathematical Technique ◽  
Triangle Function

A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is employed to investigate a generalized KdV-mKdV equation which possesses high-order nonlinear terms. Some new solutions including the Jacobi elliptic function solutions, the degenerated soliton-like solutions, and the triangle function solutions to the equation are obtained.

New Jacobi elliptic solutions and other solutions with quadratic-cubic nonlinearity using two mathematical methods

2018 ◽  
Vol 13 (02) ◽  
pp. 2050043 ◽  
Author(s):  
Savaissou Nestor ◽  
Mibaile Justin ◽  
Douvagai ◽  
Gambo Betchewe ◽  
Serge Y. Doka ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Soliton Solutions ◽  
Higher Order ◽  
Auxiliary Equation ◽  
Cubic Nonlinearity ◽  
Mathematical Methods ◽  
Auxiliary Equation Method ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method ◽  
Elliptic Solutions

In this paper, we apply two powerful methods, namely, the new extended auxiliary equation method and the generalized Kudryashov method for constructing many exact solutions and other solutions for the higher order dispersive nonlinear Schrödinger’s equation to secure soliton solutions in quadratic-cubic medium. Various solutions of the resulting nonlinear ODE are obtained by using the above two methods.

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