The Bogoliubov renormalization group and solution symmetry in mathematical physics.

*(English)* Zbl 0988.81082
Summary: 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., those, determining boundary conditions) which specify some particular solution. After a short detour into Wilsonâ€™s 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 nonlinear medium.

##### MSC:

81T17 | Renormalization group methods (quantum theory) |