Donato, Andrea and Oliveri, Francesco (2000) Reduction to Autonomous Form by Group Analysis and Exact Solutions os Axisymmetric MHD Equations. Memorie scientifiche di Andrea Donato, LXXVIII. pp. 449-456.
Motivated by many physical applications, we consider a general first order system of nonlinear partial differential equations involving two independent variables x, t and a vector field u (x,t). We suppose that the system admits two one parameter Lie groups of transformations with commuting infinitesimal operators. Then, by introducing canonical variables, it is possible to show, under suitable conditions, that the original governing system may be written in autonomous form. Of course, constant solutions of the transformed system are nonconstant solutions of the original system. By using the above mentioned result, we are able to point out a systematic procedure that allows us to build up special nonconstant solutions provided that the governing system admits two commuting infinitesimal operators. The results obtained in , in the case when the system under consideration is invariant with respect to the stretching group of transformation can be recovered as a special case of our procedure. In dealing with waves propagating in media having cylindrical or spherical symmetry, as well as in regions with inhomogeneities, we are led, in the one-dimensional case, to consider hyperbolic systems where the independent variables appear explicity in the coefficients of the system. Consequently, it becomes a suitable procedure to transform the system in autonomous form characterizing, in the meantime, exact solutions. Moreover, it is possible to study discontinuity and shock waves propagating in such special nonconstant states as propagation problems in constant states of the transformed system
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