Genuine Lie semigroups and semi-symmetries of PDEs

TL;DR

This paper explores the concept of genuine Lie semigroups, which are broader than traditional Lie groups, and their role in transforming solutions of PDEs through semi-symmetries, expanding the scope of symmetry analysis.

Contribution

It introduces and studies genuine Lie semigroups (GLS), which are not necessarily invertible, and examines their application to semi-symmetries of PDEs, extending classical symmetry methods.

Findings

Genuine Lie semigroups can transform PDE solutions without invertibility.

GLS extend the class of semi-symmetries beyond classical Lie group symmetries.

Potential for broader solution transformations in PDE analysis.

Abstract

Any Lie group G acting on a Euclidean nonvoid open subset M can be seen as a subgroup of the smooth diffeomorphisms Diff^\infty(M,M) of M into itself. Thus actions by such Lie groups G correspond to smooth coordinate transforms on M which, in particular, have smooth inverses. In Rosinger [1, chap. 13], the study of Lie semigroups G in the vastly larger semigroup C^\infty(M,M) of smooth maps of M into itself was initiated. Such semigroups were named "genuine Lie semigroups", or in short, GLS, since they are no longer contained in Diff^\infty(M,M), thus they correspond to smooth coordinate transforms which need not have smooth inverses. Genuine Lie semigroups, or GLS, have a major interest since they still can transform solutions of linear or nonlinear PDEs into other solutions of the respective equations, thus leading to the vastly larger class of "semi-symmetries" of such equations. Certain Lie semigroups have earlier been studied in the literature, Hilgert, et.al. However, such semigroups have always been contained in suitable Lie groups, thus they have been contained in Diff^\infty(M,M) as sub-semigroups. In particular, the coordinate transforms defined by them were always invertible, unlike in the case of "genuine Lie semigroups", or GLS, studied here.