Renormalization-group symmetries for solutions of nonlinear boundary value problems

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.

New solutions in the theory of self-focusing with saturating nonlinearity

“…(8), we apply the method of RG symmetries [30]. This method consists in finding symmetries of special kind that leave invariant the approximate solutions to Eqs.…”

Laser acceleration of ions: recent results and prospects for applications

“…The group theory approach based on the use of renormalization-group (RG) symmetries [51] is an efficient tool for the analytic solution of problems on laser±plasma acceleration of charged particles [29,52]. The kinetic equation for an electron averaged over fast laser oscillations in the model of radial ponderomotive acceleration of particles from the laser channel includes an`external' electric field specifying the radial ponderomotive force acting on plasma electrons [53] in addition to the self-consistent electric field of plasma.…”

Nonlinear acoustic waves in channels with variable cross sections

“…The first of them, X 41 , is the translation operator along the ζ axis; the second operator, X 42 , represents the dilation transformation; and X 43 corresponds to the projective transformation group. In addition to operators (16), for a channel with a constant cross section, µ ≡ ν, the MGWE also allows the infinite subgroup operator…”

“…In constructing the invariant solutions to the MGWE, we concentrate on studying solutions that are invariant with respect to the one-parameter group with the operator X 4 , because this operator represents an analog of linear combinations of operators (16), the use of which for channels with constant cross sections yields the well-known and physically meaningful particular solutions to the nonlinear Burgers equation.…”

“…To construct an approximate analytic solution for a channel with a slowly varying cross section, we use the renormalization-group symmetry algorithm [16] for boundary-value problem (10). According to perturbation theory, this algorithm allows us to extend the solutions in nonlinearity parameter a to the region of finite values of this parameter.…”