Lie symmetries of differential equations:direct and inverse problems

This paper reviews some relevant problems arising within the context of Lie group analysis of dieren tial equations either in the direct approach or in the inverse one. For what concerns the direct approach, there are considered two results, the rst related to the reduction through an invertible point transformation of a system of PDE's to an equivalent autonomous form, and the second related to the reduction of a nonlinear rst order system of PDE's to linear form. Two applications of the results are given. The Navier{Stokes{Fourier model equations for a viscous and heat conducting monatomic gas in a rotating frame are mapped in two dieren t autonomous forms, and some explicit exact solutions are determined. Moreover, the rst order system corresponding to the most general second order completely exceptional equation in (1 + 1) dimensions (which is a Monge{Amp ere equation) is reduced to linear form. Finally, within the context of the inverse approach of Lie group analysis, there is introduced the concept of Lie remarkable systems and it is shown that second order Monge{ Amp ere equations and the third order Monge{Amp ere equation in (1 + 1) dimensions are Lie remarkable.

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