Lie symmetry analysis, optimal system and conservation law of a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation

2021 ◽  
Author(s):  
Sheng-Nan Guan ◽  
Guang-Mei Wei ◽  
Qi Li
Keyword(s):  
Soliton Solutions ◽  
Adjoint Representation ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Optimal System ◽  
Lie Point Symmetries ◽  
Lie Symmetry Analysis ◽  
Symmetry Approach ◽  
Lie Symmetry Approach ◽  
Ito Equation

In this paper, a generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation is investigated using Lie symmetry approach. Infinitesimal generators and symmetry groups of this equation are presented, and the optimal system is given with adjoint representation. Based on the optimal system, some symmetry reductions are performed and some similarity solutions are provided, including soliton solutions and periodic solutions. With Lagrangian, it is shown that the GHSI equation is nonlinearly self-adjoint. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws are constructed. Furthermore, some physically meaningful solutions are illustrated graphically with suitable choices of parameters.

Soliton Solutions, Conservation Laws, and Reductions of Certain Classes of NonlinearWave Equations

2012 ◽  
Vol 67 (10-11) ◽  
pp. 613-620 ◽  
Author(s):  
Richard Morris ◽  
Abdul Hamid Kar ◽  
Abhinandan Chowdhury ◽  
Anjan Biswas
Keyword(s):  
Conservation Laws ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Lie Symmetry ◽  
Nonlinear Wave Equations ◽  
Conserved Quantities ◽  
Symmetry Approach ◽  
Lie Symmetry Approach

In this paper, the soliton solutions and the corresponding conservation laws of a few nonlinear wave equations will be obtained. The Hunter-Saxton equation, the improved Korteweg-de Vries equation, and other such equations will be considered. The Lie symmetry approach will be utilized to extract the conserved densities of these equations. The soliton solutions will be used to obtain the conserved quantities of these equations.

scholarly journals Application of Lie Symmetry Analysis and Simplest Equation Method for Finding Exact Solutions of Boussinesq Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Hossein Jafari ◽  
Nematollah Kadkhoda ◽  
Chaudry Massod Khalique
Keyword(s):  
Exact Solutions ◽  
Boussinesq Equations ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Equation Method ◽  
Lie Symmetry Analysis ◽  
Symmetry Approach ◽  
Simplest Equation Method ◽  
Lie Symmetry Approach

The Lie symmetry approach with simplest equation method is used to construct exact solutions of the bad Boussinesq and good Boussinesq equations. As the simplest equation, we have used the equation of Riccati.

scholarly journals Invariant Solutions and Bifurcation Analysis of the Nonlinear Transmission Line Model

2021 ◽  
Author(s):  
Sachin Kumar
Keyword(s):  
Transmission Line ◽  
Soliton Solutions ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Optimal System ◽  
Transmission Line Model ◽  
Nonlinear Transmission Line ◽  
Line Model ◽  
Similarity Reduction ◽  
Nonlinear Transmission

Abstract In this paper, the nonlinear transmission line model with the power law nonlinearity and the constant capacitance and voltage relationship is studied using Lie symmetry analysis. Corre- sponding to the infinitesimals obtained, using commutation relations, abelian and non abelian Lie subalgebras are obtained. Also, using the adjoint table, the one dimensional optimal system of subalgebra is presented. Based on the optimal system, corresponding Lie symmetry reduc- tions are obtained. Moreover, variety of new similarity solutions in the form of trigonometric functions, hyperbolic functions are obtained. Corresponding to one similarity reduction, by bifurcation of dynamical system, the stable and unstable regions are determined, which show the existence of soliton solutions from the nonlinear dynamics view point. Some of the obtained solutions are represented graphically and observations are also discussed.

Lie symmetry analysis on time scales and its application on mechanical systems

2018 ◽  
Vol 25 (3) ◽  
pp. 581-592 ◽  
Author(s):  
Xiang-Hua Zhai ◽  
Yi Zhang
Keyword(s):  
Time Scales ◽  
Mechanical Systems ◽  
Lie Symmetry ◽  
Lie Symmetries ◽  
Conserved Quantities ◽  
Lie Symmetry Analysis ◽  
Dynamical Equations ◽  
Symmetry Approach ◽  
Lie Symmetry Approach ◽  
Infinitesimal Transformations

The theory of time scales that can unify and extend continuous and discrete analysis has proved to be more accurate in modeling the dynamic process. The Lie symmetry approach, which is an effective way to deal with different kinds of dynamical equations in a variety of areas of applied science, is to be analyzed on time scales. We begin with the Lie group of point infinitesimal transformations on time scales and its corresponding extensions. And the invariance of dynamical equations on time scales under the infinitesimal transformations is discussed. More specifically, the Lie symmetries for dynamical equations of mechanical systems on time scales including Lagrangian systems on time scales, Hamiltonian systems on time scales, and Birkhoffian systems on time scales are investigated as applications. Thus, the corresponding conserved quantities for mechanical systems on time scales are derived by using the Lie symmetries. Examples are given to illustrate the application of the results.

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