scholarly journals Symmetry Reduction of the Two-Dimensional Ricci Flow Equation

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Mehdi Nadjafikhah ◽  
Mehdi Jafari
Keyword(s):  
Ricci Flow ◽  
Adjoint Representation ◽  
Flow Equation ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Similarity Reduction ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Group ◽  
Group Invariant Solutions

This paper is devoted to obtain the one-dimensional group invariant solutions of the two-dimensional Ricci flow ((2D) Rf) equation. By classifying the orbits of the adjoint representation of the symmetry group on its Lie algebra, the optimal system of one-dimensional subalgebras of the ((2D) Rf) equation is obtained. For each class, we will find the reduced equation by the method of similarity reduction. By solving these reduced equations, we will obtain new sets of group invariant solutions for the ((2D) Rf) equation.

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.

Symmetry Reductions and Exact Solutions of the Two-Layer Model in Atmosphere

2011 ◽  
Vol 66 (1-2) ◽  
pp. 75-86 ◽  
Author(s):  
Zhongzhou Dong ◽  
Fei Huang ◽  
Yong Chen
Keyword(s):  
Vector Fields ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Layer Model ◽  
Horizontal Structure ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Two Layer Model ◽  
Model Equations

By means of the classical symmetry method, we investigate the two-layer model in atmosphere. The symmetry group of two-layer model equations is studied and its corresponding group invariant solutions are constructed. Ignoring the discussion of the infinite-dimensional subalgebra, we construct the optimal system of one-dimensional and two-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 two-layer model equations are obtained. For some interesting solutions, the figures are given out to show their properties. Some solutions can describe the horizontal structure of tropical cyclones (TC). Especially, a new solution of double-eyewall structure of TCs is firstly found in this two-layer model.

Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System

2019 ◽  
Vol 20 (6) ◽  
pp. 713-723 ◽  
Author(s):  
Pabitra Kumar Pradhan ◽  
Manoj Pandey
Keyword(s):  
Adjoint Representation ◽  
Group Classification ◽  
Optimal System ◽  
Invariant Solutions ◽  
Horizontal Temperature Gradient ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
The One ◽  
Complete Symmetry Group

AbstractA complete symmetry group classification for the system of shallow water equations with the horizontal temperature gradient, also known as Ripa system, is presented. A rigorous and systematic procedure based on the general invariants of the adjoint representation is used to construct the one-dimensional optimal system of the Lie algebra. The complete inequivalence class of the group invariant solutions are obtained by using the one-dimensional optimal system. One such solution of the Ripa system is used to study the evolutionary behaviour of the discontinuity wave.

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 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 Group-Invariant Solutions for Two-Dimensional Free, Wall, and Liquid Jets Having Finite Fluid Velocity at Orifice

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
R. Naz
Keyword(s):  
Fluid Velocity ◽  
Invariant Solution ◽  
Jet Flows ◽  
Order Partial Differential Equation ◽  
Free Wall ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Third Order ◽  
Group Invariant ◽  
Group Invariant Solutions

The group-invariant solutions for nonlinear third-order partial differential equation (PDE) governing flow in two-dimensional jets (free, wall, and liquid) having finite fluid velocity at orifice are constructed. The symmetry associated with the conserved vector that was used to derive the conserved quantity for the jets (free, wall, and liquid) generated the group invariant solution for the nonlinear third-order PDE for the stream function. The comparison between results for two-dimensional jet flows having finite and infinite fluid velocity at orifice is presented. The general form of the group invariant solution for two-dimensional jets is given explicitly.

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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