scholarly journals Exact Solutions and Conservation Laws of a Two-Dimensional Integrable Generalization of the Kaup-Kupershmidt Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Abdullahi Rashid Adem ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Mathematical Physics ◽  
Jacobi Elliptic Function ◽  
Two Dimensional ◽  
Lie Symmetry Method ◽  
Extended Tanh Method ◽  
Tanh Method ◽  
Integrable Generalization ◽  
Symmetry Method

We study a two-dimensional integrable generalization of the Kaup-Kupershmidt equation, which arises in various problems in mathematical physics. Exact solutions are obtained using the Lie symmetry method in conjunction with the extended tanh method and the extended Jacobi elliptic function method. In addition to exact solutions we also present conservation laws which are derived using the multiplier approach.

scholarly journals Lagrangian Formulation, Conservation Laws, Travelling Wave Solutions: A Generalized Benney-Luke Equation

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1480
Author(s):  
Sivenathi Oscar Mbusi ◽  
Ben Muatjetjeja ◽  
Abdullahi Rashid Adem
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Density ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Noether Symmetries ◽  
Extended Tanh Method ◽  
Wave Solutions ◽  
Tanh Method ◽  
Derivatives Of

The aim of this paper is to find the Noether symmetries of a generalized Benney-Luke equation. Thereafter, we construct the associated conserved vectors. In addition, we search for exact solutions for the generalized Benney-Luke equation through the extended tanh method. A brief observation on equations arising from a Lagrangian density function with high order derivatives of the field variables, is also discussed.

scholarly journals Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 170 ◽  
Author(s):  
Amjad Hussain ◽  
Shahida Bano ◽  
Ilyas Khan ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Conservation Laws ◽  
Vector Fields ◽  
Lie Symmetry ◽  
Two Dimensional ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Transport Reaction ◽  
Lie Symmetry Method ◽  
Huxley Equation ◽  
Symmetry Method

In this paper, we investigate a spatially two-dimensional Burgers–Huxley equation that depicts the interaction between convection effects, diffusion transport, reaction gadget, nerve proliferation in neurophysics, as well as motion in liquid crystals. We have used the Lie symmetry method to study the vector fields, optimal systems of first order, symmetry reductions, and exact solutions. Furthermore, using the power series method, a set of series solutions are obtained. Finally, conservation laws are derived using optimal systems.

scholarly journals Construction of invariant solutions and conservation laws to the $(2+1)$-dimensional integrable coupling of the KdV equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ben Gao ◽  
Qinglian Yin
Keyword(s):  
Conservation Laws ◽  
Kdv Equation ◽  
Lie Symmetry ◽  
Invariant Solutions ◽  
Infinitesimal Generators ◽  
Lie Symmetry Method ◽  
Tanh Method ◽  
The Kdv Equation ◽  
Integrable Coupling ◽  
Symmetry Method

AbstractUnder investigation in this paper is the $(2+1)$ ( 2 + 1 ) -dimensional integrable coupling of the KdV equation which has applications in wave propagation on the surface of shallow water. Firstly, based on the Lie symmetry method, infinitesimal generators and an optimal system of the obtained symmetries are presented. At the same time, new analytical exact solutions are computed through the tanh method. In addition, based on Ibragimov’s approach, conservation laws are established. In the end, the objective figures of the solutions of the coupling of the KdV equation are performed.

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 New Jacobi Elliptic Function Solutions for the Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Yun-Mei Zhao
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function

A generalized(G′/G)-expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics.

scholarly journals Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Maria Ihsane El Bahi ◽  
Khalid Hilal
Keyword(s):  
Differential Equation ◽  
Conservation Laws ◽  
Nonlinear Partial Differential Equation ◽  
Lie Symmetry ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Symmetry Operators ◽  
Generalized Time ◽  
Symmetry Method ◽  
Conservation Theorem

In this paper, the problem of constructing the Lie point symmetries group of the nonlinear partial differential equation appeared in mathematical physics known as the generalized KdV-Like equation is discussed. By using the Lie symmetry method for the generalized KdV-Like equation, the point symmetry operators are constructed and are used to reduce the equation to another fractional ordinary differential equation based on Erdélyi-Kober differential operator. The symmetries of this equation are also used to construct the conservation Laws by applying the new conservation theorem introduced by Ibragimov. Furthermore, another type of solutions is given by means of power series method and the convergence of the solutions is provided; also, some graphics of solutions are plotted in 3D.

scholarly journals Exact solutions and conservation laws for the modified equal width-Burgers equation

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 795-800 ◽  
Author(s):  
Chaudry Masood Khalique ◽  
Innocent Simbanefayi
Keyword(s):  
Conservation Laws ◽  
Burgers Equation ◽  
Expansion Method ◽  
Momentum Conservation ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Lie Symmetry Method ◽  
Long Wave ◽  
Symmetry Method ◽  
Conservation Theorem

AbstractIn this paper we study the modified equal width-Burgers equation, which describes long wave propagation in nonlinear media with dispersion and dissipation. Using the Lie symmetry method in conjunction with the (G'/G)− expansion method we construct its travelling wave solutions. Also, we determine the conservation laws by invoking the new conservation theorem due to Ibragimov. As a result we obtain energy and linear momentum conservation laws.

scholarly journals Analysis of the Symmetries and Conservation Laws of the Nonlinear Jaulent-Miodek Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Mehdi Nadjafikhah ◽  
Mostafa Hesamiarshad
Keyword(s):  
Conservation Laws ◽  
Symmetry Group ◽  
Symbolic Computation ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Reduced Equations ◽  
Symmetry Generators ◽  
Symmetry Method

Lie symmetry method is performed for the nonlinear Jaulent-Miodek equation. We will find the symmetry group and optimal systems of Lie subalgebras. The Lie invariants associated with the symmetry generators as well as the corresponding similarity reduced equations are also pointed out. And conservation laws of the J-M equation are presented with two steps: firstly, finding multipliers for computation of conservation laws and, secondly, symbolic computation of conservation laws will be applied.

scholarly journals Comparison between the (G’/G) - expansion method and the modified extended tanh method

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 88-94 ◽  
Author(s):  
Şamil Akçaği ◽  
Tuğba Aydemir
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
The Other ◽  
Extended Tanh Method ◽  
New Exact Solutions ◽  
Tanh Method

AbstractIn this paper, firstly, we give a connection between well known and commonly used methods called the $\left( {{{G'} \over G}} \right)$ -expansion method and the modified extended tanh method which are often used for finding exact solutions of nonlinear partial differential equations (NPDEs). We demonstrate that giving a convenient transformation and formula, all of the solutions obtained by using the $\left( {{{G'} \over G}} \right)$ - expansion method can be converted the solutions obtained by using the modified extended tanh method. Secondly, contrary to the assertion in some papers, the $\left( {{{G'} \over G}} \right)$-expansion method gives neither all of the solutions obtained by using the other method nor new solutions for NPDEs. Namely, while the modified extended tanh method gives more solutions in a straightforward, concise and elegant manner without reproducing a lot of different forms of the same solution. On the other hand, the $\left( {{{G'} \over G}} \right)$-expansion method provides less solutions in a rather cumbersome form. Lastly, we obtain new exact solutions for the Lonngren wave equation as an illustrative example by using these methods.

scholarly journals Exact Explicit Traveling Wave Solution for the Generalized (2+1)-Dimensional Boussinesq Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Libing Zeng ◽  
Keding Qin ◽  
Shengqiang Tang
Keyword(s):  
Traveling Wave ◽  
Boussinesq Equation ◽  
Nonlinear Partial Differential Equations ◽  
Mathematical Physics ◽  
Traveling Wave Solution ◽  
Mathematical Tool ◽  
Extended Tanh Method ◽  
Physical Problems ◽  
Tanh Method ◽  
Powerful Mathematical Tool

The sine-cosine method and the extended tanh method are used to construct exact solitary patterns solution and compactons solutions of the generalized (2+1)-dimensional Boussinesq equation. The compactons solutions and solitary patterns solutions of the generalized (2+1)-dimensional Boussinesq equation are successfully obtained. These solutions may be important and of significance for the explanation of some practical physical problems. It is shown that the sine-cosine and the extended tanh methods provide a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.

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