Symmetry Reduction of the Two-Dimensional Ricci Flow Equation

Abstract

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, the optimal system of one-dimensional subalgebras of the ((2D) Rf) equation is obtained. For each class, we will find the reduced equation by method of similarity reduction. By solving these reduced equations we will obtain new sets of group invariant solutions for the ((2D) Rf) equation.In this paper we want to obtain new solutions of the ((2D) Rf) equation by method of Lie symmetry group. As it is well known, the Lie symmetry group method has an important role in the analysis of differential equations. The theory of Lie symmetry groups of differential equations was developed by Sophus Lie [8]. By this method we can reduce the order of ODEs and investigate the invariant solutions. Also we can construct new solutions from known ones (for more details about the applications of Lie symmetries, see [10,1,3]). Lei's method led to an algorithmic approach to find special solution of differential equation by its symmetry group. These solutions are called group invariant solutions and obtain by solving the reduced system of differential equation having fewer independent variables than the original system. Bluman and Cole generalized the Lie's symmetry method for finding the group-invariant solutions [2]. In this paper we apply this method to obtain the invariant solutions of ((2D) Rf) equation and classify them.This paper is organized as follows. In section 2, by using the mechanical model of Ricci flow, Lie symmetries of ((2D) Rf) equation will be state and some results yield from the structure of the Lie algebra of the Lie symmetry group. In section 3, we will construct an optimal system of one-dimensional subalgebras of the ((2D) Rf) equation which is useful for classifying of group invariant solutions. In section 4, the reduced equation for each element of optimal system is obtained. In section 5, we will solve the reduced equations by method of Lie symmetry group and obtain the group invariant solutions of ((2D) Rf) equation.

Invariant Subspaces of the Two-Dimensional Nonlinear Evolution Equations

“…Lie's classical symmetries of Equation (24) were computed in [40][41][42][43][44][45]. Indeed, Equation (24) admits the Lie group of symmetry with infinitesimal generator:…”

Analytic solutions for the one-dimensional compressible Euler equation with heat conduction and with different kind of equations of state

“…Additional solutions are available for the 2+1 dimensional NS also via symmetry reduction techniques by [16]. There is a full three dimensional Lie group analysis is available for the three dimensional Euler equation of gas dynamics, with polytropic EOS unfortunately without any heat conduction mechanism [17]. Of course one may find numerical methods for solving equations of fluid dynamics [18].…”

“…(We still use h as shape function). However, if we consider (17), it can be rewritten in the following form…”