  Approximately 10 years ago, the method of renormalization-group symmetries
entered the field of boundary value problems of classical mathematical physics,
stemming from the concepts of functional self-similarity and of the Bogoliubov
renormalization group treated as a Lie group of continuous transformations.
Overwhelmingly dominating practical quantum field theory calculations, the
renormalization-group method formed the basis for the discovery of the
asymptotic freedom of strong nuclear interactions and underlies the Grand
Unification scenario. This paper describes the logical framework of a new
algorithm based on the modern theory of transformation groups and presents the
most interesting results of application of the method to differential and/or
integral equation problems and to problems that involve linear functionals of
solutions. Examples from nonlinear optics, kinetic theory, and plasma dynamics
are given, where new analytical solutions obtained with this algorithm have
allowed describing the singularity structure for self-focusing of a laser beam
in a nonlinear medium, studying generation of harmonics in weakly inhomogeneous
plasma, and investigating the energy spectra of accelerated ions in expanding
plasma bunches.
