scholarly journals Symmetry Reductions of (2 + 1)-Dimensional CDGKS Equation and Its Reduced Lax Pairs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Na Lv ◽  
Xuegang Yuan ◽  
Jinzhi Wang
Keyword(s):  
Direct Method ◽  
Lax Pair ◽  
Group Method ◽  
Symmetry Groups ◽  
Invariant Solutions ◽  
Lax Pairs ◽  
Group Invariant ◽  
Lie Group Method ◽  
Group Invariant Solutions ◽  
Symmetry Transformations

With the aid of symbolic computation, we obtain the symmetry transformations of the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation by Lou’s direct method which is based on Lax pairs. Moreover, we use the classical Lie group method to seek the symmetry groups of both the CDGKS equation and its Lax pair and then reduce them by the obtained symmetries. In particular, we consider the reductions of the Lax pair completely. As a result, three reduced (1 + 1)-dimensional equations with their new Lax pairs are presented and some group-invariant solutions of the equation are given.

scholarly journals Invariant Solutions and Nonlinear Self-Adjointness of the Two-Component Chaplygin Gas Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Ben Gao ◽  
Yanxia Wang
Keyword(s):  
Chaplygin Gas ◽  
Group Method ◽  
Invariant Solutions ◽  
One Dimensional ◽  
Group Invariant ◽  
Lie Group Method ◽  
Two Component ◽  
Present Universe ◽  
Group Invariant Solutions ◽  
Similarity Reductions

In this paper, the Lie group method is performed on a special dark fluid, the Chaplygin gas, which describes both dark matter and dark energy in the present universe. Based on an optimal system of one-dimensional subalgebras, similarity reductions and group invariant solutions are given. Finally, by means of Ibragimov’s method, conservation laws are obtained.

scholarly journals Symmetries and Invariant Solutions for the Coagulation of Aerosols

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 876
Author(s):  
Mingliang Zheng
Keyword(s):  
Analytic Solution ◽  
Initial Conditions ◽  
Morphological Changes ◽  
Numerical Algorithms ◽  
Group Method ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Aerosol Particle Size ◽  
General Dynamics ◽  
Group Invariant Solutions

The coagulation of aerosol particles plays an important role in the structural morphological changes of suspended particles at any time and in any space. In this study, based on the Smoluchowski equation of population balance, a kinetic model of aerosol coalescence considering Brownian motion collision is established. By applying the developed Lie group method, we derive the allowed infinitesimal symmetries and group-invariant solutions of the integro-differential equation, as well as the exact solution under some special conditions. We also provide detailed steps and a discussion of the properties. The content and results provide an effective analytic solution for the progressive evolution of aerosol particle size considering boundary and initial conditions. This solution reveals the self-conservative phenomena in the process of aerosol coalescence and also provides validation for the numerical algorithms of general dynamics equations.

scholarly journals Some invariant solutions and conservation laws of a type of long-water wave system

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiangzhi Zhang ◽  
Yufeng Zhang
Keyword(s):  
Conservation Laws ◽  
Water Wave ◽  
Lax Pair ◽  
Wave System ◽  
Series Solutions ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Generalized System ◽  
Group Invariant Solutions ◽  
Similarity Reductions

AbstractWe propose a generalized long-water wave system that reduces to the standard water wave system. We also obtain the Lax pair and symmetries of the generalized shallow-water wave system and single out some their similarity reductions, group-invariant solutions, and series solutions. We further investigate the corresponding self-adjointness and the conservation laws of the generalized system.

New Exact Solutions to Variable Coefficient KP Equation

2012 ◽  
Vol 166-169 ◽  
pp. 3075-3078 ◽  
Author(s):  
Jun Yi Yin
Keyword(s):  
Exact Solutions ◽  
Variable Coefficient ◽  
Function Method ◽  
Group Method ◽  
Kp Equation ◽  
Group Invariant ◽  
Lie Group Method ◽  
New Exact Solutions ◽  
Solitonic Solution ◽  
Group Invariant Solutions

Two kinds of new exact solutions were offered after studying the variable coefficient KP equation, of which, the group invariant solutions of KP equation was obtained by using Lie group method, while the solitonic solution of KP equation was obtained by using hyperbola function method.

scholarly journals Exact Solutions of Newell-Whitehead-Segel Equations Using Symmetry Transformations

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Khudija Bibi ◽  
Khalil Ahmad
Keyword(s):  
Exact Solutions ◽  
Transformation Group ◽  
Symmetry Transformation ◽  
Discrete Symmetry ◽  
Transformation Groups ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Symmetry Transformations ◽  
Linear And Nonlinear

In this article, Lie and discrete symmetry transformation groups of linear and nonlinear Newell-Whitehead-Segel (NWS) equations are obtained. By using these symmetry transformation groups, several group invariant solutions of considered NWS equations have been constructed. Furthermore, some more group invariant solutions are generated by using discrete symmetry transformation group. Graphical representations of some obtained solutions are also presented.

Sign in / Sign up

Export Citation Format

Share Document