scholarly journals Approximate Symmetry Analysis and Approximate Conservation Laws of Perturbed KdV Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yu-Shan Bai ◽  
Qi Zhang
Keyword(s):  
Conservation Laws ◽  
Kdv Equation ◽  
Symmetry Algebra ◽  
Optimal System ◽  
Approximate Symmetry ◽  
Invariant Solutions ◽  
Approximate Symmetries ◽  
One Dimensional ◽  
Perturbed Kdv Equation ◽  
Partial Lagrangian

Approximate symmetries, which are admitted by the perturbed KdV equation, are obtained. The optimal system of one-dimensional subalgebra of symmetry algebra is obtained. The approximate invariants of the presented approximate symmetries and some new approximately invariant solutions to the equation are constructed. Moreover, the conservation laws have been constructed by using partial Lagrangian method.

scholarly journals Perturbed Korteweg-de Vries equations symmetry analysis and conservation laws

2019 ◽  
Vol 23 (4) ◽  
pp. 2281-2289
Author(s):  
Yu-Shan Bai ◽  
Qi Zhang
Keyword(s):  
Conservation Laws ◽  
Coupled System ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Transformation Groups ◽  
Invariant Solutions ◽  
Approximate Symmetries ◽  
De Vries ◽  
Small Parameters ◽  
Partial Lagrangian

Approximate symmetries for a coupled system of perturbed Korteweg-de Vries equations with small parameters are constructed by applying the method of approximate transformation groups. The optimal system of the presented approximate symmetries and a few approximate invariant solutions to the coupled system are obtained. Moreover, approximate conservation laws are constructed by using the partial Lagrangian method.

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.

scholarly journals Optimal System and New Approximate Solutions of a Generalized Ames’s Equation

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1230
Author(s):  
Marianna Ruggieri ◽  
Maria Paola Speciale
Keyword(s):  
Lie Algebra ◽  
Algebraic Structure ◽  
Approximate Solutions ◽  
Optimal System ◽  
Approximate Symmetry ◽  
Invariant Solutions ◽  
One Dimensional

In this paper, by applying Valenti’s theory for the approximate symmetry, we introduce and define the concept of a one-dimensional optimal system of approximate subalgebras for a generalized Ames’s equation; furthermore, the algebraic structure of the approximate Lie algebra is discussed. New approximately invariant solutions to the equation are found.

scholarly journals Exact solutions of equal-width equation and its conservation laws

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 505-511 ◽  
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie ◽  
Innocent Simbanefayi
Keyword(s):  
Wave Propagation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Optimal System ◽  
Invariant Solutions ◽  
Cnoidal Waves ◽  
Dispersion Process ◽  
One Dimensional ◽  
Linear Medium ◽  
D Wave

Abstract In this work we investigate the equal-width equation, which is used for simulation of (1-D) wave propagation in non-linear medium with dispersion process. Firstly, Lie symmetries are determined and then used to establish an optimal system of one-dimensional subalgebras. Thereafter with its aid we perform symmetry reductions and compute new invariant solutions, which are snoidal and cnoidal waves. Additionally, the conservation laws for the aforementioned equation are established by invoking multiplier method and Noether’s theorem.

scholarly journals Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Tahir Ayaz ◽  
Farhad Ali ◽  
Wali Khan Mashwani ◽  
Israr Ali Khan ◽  
Zabidin Salleh ◽  
...  
Keyword(s):  
Conservation Laws ◽  
Order Differential Equation ◽  
Kdv Equation ◽  
Third Order ◽  
Lagrange Method ◽  
Approximate Symmetries ◽  
Weakly Nonlinear ◽  
The Kdv Equation ◽  
De Vries ◽  
Perturbed Kdv Equation

The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. This paper presents the analysis of the approximate symmetries along with conservation laws corresponding to the perturbed KdV equation for different classes of the perturbed function. Partial Lagrange method is used to obtain the approximate symmetries and their corresponding conservation laws of the KdV equation. The purpose of this study is to find particular perturbation (function) for which the number of approximate symmetries of perturbed KdV equation is greater than the number of symmetries of KdV equation so that explore something hidden in the system.

scholarly journals Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
M. N. Qureshi ◽  
A. Q. Khan ◽  
M. Ayub ◽  
Q. Din
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Symmetry Algebra ◽  
Invariant Solutions ◽  
Source Terms ◽  
One Dimensional

One-dimensional optimal systems for nonhomogeneous discrete heat equation with different source terms are calculated. By utilizing these optimal systems invariant solutions are found. Also generating solutions are calculated, using the elements of the symmetry algebra.

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.

Generalized symmetries and higher-order Conservation laws of the Camassa–Holm equation

2019 ◽  
Vol 9 (2) ◽  
pp. 20-25
Author(s):  
Parastoo Kabi-Nejad ◽  
Keyword(s):  
Vector Field ◽  
Conservation Laws ◽  
Adjoint Representation ◽  
Higher Order ◽  
Optimal System ◽  
Homotopy Formula ◽  
One Dimensional ◽  
Holm Equation ◽  
Generalized Symmetries ◽  
Symmetry Criterion

In the present paper, we derive generalized symmetries of order three of the Camassa–Holm equation by infinite prolongation of a generalized vector field and applying infinitesimal symmetry criterion. In addition, one-dimensional optimal system of Lie subalgebras investigated by applying the adjoint representation. Furthermore, determining equation for multipliers and the 2- dimensional homotopy formula employed to construct higher–order conservation laws for the Camassa–Holm equation.

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