Global parametric Lie group actions, nonlinear PDEs and Hilbert's fifth problem

Since the work of Chevalley, four or five decades ago, the importance of the global action of Lie groups has been well established. In the context of PDEs however, such a globality has only been achieved in the special and rather particular case of the so called `fibre preserving' or `projectable' Lie groups of symmetries. On the other hand, even some of the most simple and basic PDEs, and even more so in the nonlinear case, have Lie symmetry groups which are not projectable, a well known example being that of the shock wave equation. In the speaker's recently published Kluwer book, global actions are constructed in the case of arbitrary Lie group symmetries which act on classical or generalized solutions of a very large class of smooth nonlinear PDEs. This construction is obtained by using parametric representations for Lie group actions, as well as for the respective solutions. One of the effects of the above is that Hilbert's fifth problem obtains a solution which is more in line with the general initial formulation of that problem, than the various earlier claimed solutions given in the 1940s and 1950s. These results extend earlier ones published by the speaker since 1992 in a joint North-Holland research monograph and in three joint papers, results which only dealt with global projectable Lie group actions on generalized solutions, and which did not use the parametric method.