scholarly journals Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 341 ◽  
Author(s):  
Aliyu Aliyu ◽  
Mustafa Inc ◽  
Abdullahi Yusuf ◽  
Dumitru Baleanu
Keyword(s):  
Conservation Laws ◽  
Travelling Wave ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Sixth Order ◽  
Lie Point Symmetries ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Analysis Technique ◽  
Conservation Theorem

In this work, we study the completely integrable sixth-order nonlinear Ramani equation. By applying the Lie symmetry analysis technique, the Lie point symmetries and the optimal system of one-dimensional sub-algebras of the equation are derived. The optimal system is further used to derive the symmetry reductions and exact solutions. In conjunction with the Riccati Bernoulli sub-ODE (RBSO), we construct the travelling wave solutions of the equation by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction. We show that the equation is nonlinearly self-adjoint and construct the conservation laws (CL) associated with the Lie symmetries by invoking the conservation theorem due to Ibragimov. Some figures are shown to show the physical interpretations of the acquired results.

scholarly journals Symmetry Analysis and Conservation Laws of a Generalized Two-Dimensional Nonlinear KP-MEW Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Khadijo Rashid Adem ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Expansion Method ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Two Dimensional ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Similarity Reductions

Lie symmetry analysis is performed on a generalized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation. The symmetries and adjoint representations for this equation are given and an optimal system of one-dimensional subalgebras is derived. The similarity reductions and exact solutions with the aid ofG′/G-expansion method are obtained based on the optimal systems of one-dimensional subalgebras. Finally conservation laws are constructed by using the multiplier method.

On Symmetry Analysis and Conservation Laws of the AKNS System

2016 ◽  
Vol 71 (8) ◽  
pp. 741-750 ◽  
Author(s):  
Zhonglong Zhao ◽  
Bo Han
Keyword(s):  
Conservation Laws ◽  
Wave Model ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Wave System ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Point Transformations ◽  
Akns System ◽  
Similarity Reductions

AbstractThe Lie symmetry analysis is applied to study the Ablowitz–Kaup–Newell–Segur (AKNS) system of water wave model. The AKNS system can be obtained from a dispersive-wave system via a variable transformation. Lie point symmetries and corresponding point transformations are determined. The optimal system of one-dimensional subalgebras is presented. On the basis of the optimal system, the similarity reductions and the invariant solutions are obtained. Some conservation laws are derived using the multipliers. In addition, the AKNS system is quasi self-adjoint. The conservation laws associated with the symmetries are also constructed.

Lie Symmetry Analysis, Analytical Solutions, and Conservation Laws of the Generalised Whitham–Broer–Kaup–Like Equations

2017 ◽  
Vol 72 (3) ◽  
pp. 269-279 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Shou-Fu Tian ◽  
Chun-Yan Qin ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Vector Fields ◽  
Wave Equations ◽  
Boussinesq Equations ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Lie Symmetry Analysis ◽  
Bidirectional Propagation

AbstractIn this article, a generalised Whitham–Broer–Kaup–Like (WBKL) equations is investigated, which can describe the bidirectional propagation of long waves in shallow water. The equations can be reduced to the dispersive long wave equations, variant Boussinesq equations, Whitham–Broer–Kaup–Like equations, etc. The Lie symmetry analysis method is used to consider the vector fields and optimal system of the equations. The similarity reductions are given on the basic of the optimal system. Furthermore, the power series solutions are derived by using the power series theory. Finally, based on a new theorem of conservation laws, the conservation laws associated with symmetries of this equations are constructed with a detailed derivation.

scholarly journals Exact Solutions and Conservation Laws of the (3 + 1)-Dimensional B-Type Kadomstev–Petviashvili (BKP)-Boussinesq Equation

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 97 ◽  
Author(s):  
Ben Gao ◽  
Yao Zhang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Boussinesq Equation ◽  
Lie Symmetry ◽  
Optimal System ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Reduced Equations ◽  
Tanh Method ◽  
Similarity Reductions

In this paper, Lie symmetry analysis is presented for the (3 + 1)-dimensional BKP-Boussinesq equation, which seriously affects the dispersion relation and the phase shift. To start with, we derive the Lie point symmetry and construct the optimal system of one-dimensional subalgebras. Moreover, according to the optimal system, similarity reductions are investigated and we obtain exact solutions of reduced equations by means of the Tanh method. In the end, we establish conservation laws using Ibragimov’s approach.

scholarly journals Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 522
Author(s):  
Na Liu
Keyword(s):  
Traveling Wave Solutions ◽  
Optimal System ◽  
Time Dependent ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Lie Symmetry Analysis ◽  
Similarity Reduction ◽  
One Dimensional ◽  
Power Series Solutions ◽  
Akns Equation

The Lie point symmetries are reported by performing the Lie symmetry analysis to the Ablowitz-Kaup-Newell-Suger (AKNS) equation with time-dependent coefficients. In addition, the optimal system of one-dimensional subalgebras is constructed. Based on this optimal system, several categories of similarity reduction and some new invariant solutions for the equation are obtained, which include power series solutions and travelling and non-traveling wave solutions.

scholarly journals On optimal system, exact solutions and conservation laws of the modified equal-width equation

2018 ◽  
Vol 3 (2) ◽  
pp. 409-418 ◽  
Author(s):  
Chaudry Masood Khalique ◽  
Oke Davies Adeyemo ◽  
Innocent Simbanefayi
Keyword(s):  
Wave Propagation ◽  
Conservation Laws ◽  
Optimal System ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Dimensional Wave

AbstractIn this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

Lie symmetry analysis, explicit solutions and conservation laws of the time fractional Clannish Random Walker’s Parabolic equation

2020 ◽  
pp. 2150074
Author(s):  
Panpan Wang ◽  
Wenrui Shan ◽  
Ying Wang ◽  
Qianqian Li
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Vector Fields ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Series Solutions ◽  
Lie Symmetry Analysis ◽  
Similarity Reduction ◽  
Power Series Solutions ◽  
Conservation Theorem

In this paper, we mainly study the symmetry analysis and conservation laws of the time fractional Clannish Random Walker’s Parabolic (CRWP) equation. The vector fields and similarity reduction of the time fractional CRWP equation are obtained. In addition, based on the power series theory, a simple and effective approach for constructing explicit power series solutions is proposed. Finally, by use of the new conservation theorem, the conservation laws of the time fractional CRWP equation are constructed.

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