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Asymptotic scaling symmetries for nonlinear PDEs. (English) Zbl 1093.35004

The authors examine the concept of asymptotic scaling and translation symmetries for nonlinear partial differential equations with applications mainly to a class of equations arising in models of reaction-diffusion problems. The paper is somewhat long, but, as this is a consequence of a carefully clear development of the necessary ideas and presentation of illustrative examples, the length is probably a boon.

A formalism is developed to identify the types of asymptotic symmetries considered here. The authors do state that this formalism is somewhat specific to the class of symmetry considered in this paper, but these are the symmetries one would expect to occur most frequently. It would not surprise one if a symmetry could not be applied to the same-called generalised three scaling symmetries introduced by V. A. Galaktionov [Comput. Math. Math. Phys. 39, 1499–1505 (1999); translation from Zh. Vychisl. Mat. Mat. Fiz. 39, No. 9, 1564–1570 (1999; Zbl 0972.35120)]. The existence of numerically observed asymptotic symmetry in some practical problems is confirmed by explicit demonstration.

An obvious area of interest are the Einstein field equations of general relativity to determine which models could possess in an asymptotic sense the symmetries of the presently observed FRW universe. The idea is developed here and an earlier work by G. Gaeta [J. Math. Phys. A 27, 437–451 (1994; Zbl 0821.35127)] could be of relevance.

It is well-known that singularity analysis is related to the dominant self-similar symmetry of the system under investigation. The presence of nondominant terms is frequently incompatible with successful conclusion of the singularity analysis. However, this could be another route to find asymptotic symmetries.

The paper is marred by some obvious typographical errors (e.g. page 1089 equation (3.4), page 1103 line -10, page 1110 line 1) and the curious claim (page 1087) that u u is a symmetry only for linear equations.

MSC:
35A30Geometric theory for PDE, characteristics, transformations
58J70Invariance and symmetry properties
35K57Reaction-diffusion equations
83C05Einstein’s equations (general structure, canonical formalism, Cauchy problems)