Lie Symmetries of Inviscid Burgers’ Equation

Abstract

The present paper solves completely the problem of the Lie group analysis of nonlinear equation ut(x, t) + g(u)ux(x, t) = 0, where g(u) is a smooth function of u. And apply these results on inviscid Burgers equation.

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

“…Among them, Burgers' equation may be the more intriguing one which serves as a prototype model for turbulent flows [3]. Many works in the literature are devoted to investigation of Lie symmetries and the exact solutions of Burgers' equation(s) and its generalizations [4][5][6][7][8][9][10][11][12][13][14]. Beyond its importance in mathematical physics, it appears also as a geodesic equation on diffeomorphism group of a circle with right invariant 2 metric (see [15] and references therein).…”

New conservation laws for inviscid Burgers equation

“…In addition, from Ibragimov's theorem on conservation laws [13], conservation laws for projectable Lie point symmetries (see [21]) of (1) were established. Recently Maria Luz Gandarias [8] and Nail Ibragimov [17,18,19] have generalized the previous concepts of self-adjoint equations [12,13,14].…”

Note on Lie Point Symmetries of Burgers Equations

“…This work corresponds to the talk [8] given during the XXXII CNMAC -XXXII Congresso Nacional de Matemática Aplicada e Computacional -September, [8][9][10][11] 2009, Cuiabá, Brazil.…”

“…Here the summation over the repeated indexes is understood. Applications of this method in differential equations can be found in [2,3,4,5,9,10,11,12,13,16,17]. We shall not present more preliminaries concerning the Lie point symmetries of differential equations supposing that the reader is familiar with the basic notions and methods of group analysis [2,9,12].…”