Donato, Andrea and Oliveri, Francesco (2000) When Nonautonomous Equations are Equivalent to Autonomous Ones. Memorie scientifiche di Andrea Donato, LXXVII. pp. 541-551.
We consider nonlinear systems of first order partial differential equations admitting at least two one-parameter Lie groups of transformations with commuting infinitesimal operators. Under suitable conditions it is possible to introduce a variable transformation based on canonical variables which reduces the model in point to autonomous form. Remarkably, the transformed system may admit costant solutions to which three correspond non-costant solutions of the original model. The results are specialized to the case of first order quasilinear systems admitting either dilatation or spiral groups of transformations and a systematic procedure to characterize special exact solutions is given. At the end of the paper the equations of axi-symmetric gas dynamics are considered.
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