The Galilean relativistic principle and nonlinear partial differential equations

Abstract

The second-order partial differential equations invariant under transformations of Galilei, rotation, scale and projection are described.

Lie symmetries and exact solutions of nonlinear equations of heat conductivity with convection term

“…Thus, operators (8) belong to the algebra of invariance of all equations of the form (7). Theorem 1.…”

Nonlinear Partial Differential Equations

“…Recall that the invariants u 11 u 22 −u 2 12 and W II are nothing else but the right-hand-side (RHS) of the well-known Monge-Ampère equation in two-and three-dimensional space, respectively [25], while the invariant determinant W I was derived for the first time in [26].…”

Galilean-invariant Nonlinear PDEs and their Exact Solutions

“…(1). The Galilei operators are also known [3] to be closely related with the fundamental solution of the diffusion equation. We recall that if some system of PDEs is invariant with respect to the Galilei algebra or its extention, then it gives a wide range of possibilities for construction of multiparametric families of exact solutions [1,9,11].…”

“…The proof of this and the following theorems is based on the classical Lie scheme, which is realized in [2,3] for obtaining the Galilei invariant equations. The detailed cumbersome calculations are omitted.…”

“…Moreover, in paper [3] we have constructed all quasilinear generalizations of the heat equation that are invariant with respect to subalgebras of the generalized Galilei algebra AG 2 (1.n) . In particular we have found the following nonlinear equation…”