Renormalization-group symmetries for solutions of nonlinear boundary value problems

TL;DR

This paper introduces a new algorithm based on renormalization-group symmetries for solving nonlinear boundary value problems, providing analytical solutions in physics applications like nonlinear optics and plasma dynamics.

Contribution

It develops a modern transformation group-based algorithm for boundary value problems, expanding the application of renormalization-group symmetries beyond quantum field theory.

Findings

Describes singularity structure in laser self-focusing

Analyzes harmonic generation in plasma

Studies energy spectra in plasma expansion

Abstract

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.