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 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 Some General New Einstein Walker Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Mehdi Nadjafikhah ◽  
Mehdi Jafari
Keyword(s):  
Partial Differential Equations ◽  
Symmetry Group ◽  
Lie Symmetry ◽  
Group Method ◽  
Invariant Solutions ◽  
Point Symmetry Group ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Walker Manifold

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

Analytic study of solutions for a two-component Novikov system

2020 ◽  
pp. 2150025
Author(s):  
Hui Gao ◽  
Gangwei Wang
Keyword(s):  
Exact Solutions ◽  
Stationary Solutions ◽  
Invariant Solutions ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Two Component ◽  
Group Invariant Solutions ◽  
Symmetry Transformations ◽  
Symmetry Method ◽  
Similarity Reductions

Under investigation in this paper is a two-component Novikov system (also called Geng-Xue equation), which was proposed by Geng and Xue in 2009. Firstly, via the Lie symmetry method, infinitesimal generators, commutator table of Lie algebra and symmetry groups of the two-component Novikov system are presented. At the same time, some group invariant solutions are computed through similarity reductions. In particular, we construct peakon solution by applying the distribution theory. In addition, based on obtained group invariant solutions and symmetry transformations, we derive some new exact solutions, which include stationary solutions, smooth solutions, and a weak solution. The analytical properties to some of group invariant solutions and new exact solutions are discussed, such as decay, asymptotic behavior, and boundedness.

Optimal systems and their group-invariant solutions to geodesic equations

2019 ◽  
Vol 16 (09) ◽  
pp. 1950135
Author(s):  
Bismah Jamil ◽  
Tooba Feroze ◽  
Muhammad Safdar
Keyword(s):  
Nonlinear Systems ◽  
Separation Of Variables ◽  
Noether Symmetries ◽  
Invariant Solutions ◽  
One Dimensional ◽  
Group Invariant ◽  
First Order ◽  
Geodesic Equations ◽  
Integrating Factor ◽  
Group Invariant Solutions

We find one-dimensional optimal systems of the Lie subalgebras of Noether symmetries associated with systems of geodesic equations. Further, we find invariants corresponding to each element of the derived optimal system. The derived invariants are shown to reduce systems of geodesic equations (nonlinear systems of quadratically semi-linear second-order ordinary differential equations (ODEs)) to nonlinear systems of first-order ODEs. The resulting systems are solved via known methods (e.g. separation of variables, integrating factor, etc.). In some cases, we provide exact solutions of these systems of geodesic equations.

scholarly journals New Nonlinear Systems Admitting Virasoro-Type Symmetry Algebra and Group-Invariant Solutions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Lizhen Wang ◽  
Qing Huang ◽  
Yanmei Di
Keyword(s):  
Nonlinear Systems ◽  
Symmetry Algebra ◽  
Coupled Systems ◽  
Model Systems ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Two Component ◽  
Type Symmetry ◽  
Group Invariant Solutions ◽  
Virasoro Type

With the aid of symbolic computation by Maple, we extend the application of Virasoro-type symmetry prolongation method to coupled systems with two-component nonlinear equations. New nonlinear systems admitting infinitely dimensional centerless Virasoro-type symmetry algebra are constructed. Taking one of them as an example, we present some group-invariant solutions to one of the new model systems.

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.

Symmetry Reductions and Exact Solutions of the (2+1)-Dimensional Navier-Stokes Equations

2010 ◽  
Vol 65 (6-7) ◽  
pp. 504-510 ◽  
Author(s):  
Xiaorui Hua ◽  
Zhongzhou Dongb ◽  
Fei Huangc ◽  
Yong Chena
Keyword(s):  
Vector Fields ◽  
Stokes Equations ◽  
Stationary Wave ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Group ◽  
Group Invariant Solutions

By means of the classical symmetry method, we investigate the (2+1)-dimensional Navier-Stokes equations. The symmetry group of Navier-Stokes equations is studied and its corresponding group invariant solutions are constructed. Ignoring the discussion of the infinite-dimensional subalgebra, we construct an optimal system of one-dimensional group invariant solutions. Furthermore, using the associated vector fields of the obtained symmetry, we give out the reductions by one-dimensional and two-dimensional subalgebras, and some explicit solutions of Navier-Stokes equations are obtained. For three interesting solutions, the figures are given out to show their properties: the solution of stationary wave of fluid (real part) appears as a balance between fluid advection (nonlinear term) and friction parameterized as a horizontal harmonic diffusion of momentum.

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.

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.

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