The Bogoliubov renormalization group and solution symmetry in mathematical physics

Evolution of the concept known in theoretical physics as the renormalization group (RG) is presented. The corresponding symmetry, that was first introduced in quantum field theory (QFT) in the mid-1950s, is a continuous symmetry of a solution with respect to transformations involving the parameters (e.g., that determine boundary condition) which specify some particular solution. After a short detour into Wilsonʹs discrete semi-group, we follow the expansion of the QFT RG and argue that the underlying transformation, being considered as a reparametrization, is closely related to the property of self-similarity. It can be treated as its generalization—Functional Self-similarity (FS). Next, we review the essential progress made in the last decade in the application of the FS concept to boundary value problems formulated in terms of differential equations. A summary of a regular approach, recently devised for discovering the RG=FS symmetries with the help of modern Lie group analysis, and some of its applications are given. As the principal physical illustration, we consider the solution of the problem of a self-focusing laser beam in a non-linear medium.