Abstract (Invited)

Lie Group Analysis of Nonlinear Problems

N.H.Ibragimov (Research Centre ALGA: Advances in Lie Group Analysis, Department of Health, Science and Mathematics, Blekinge Institute of Technology, Karlskrona, Sweden)

e-mail: nib@bth.se

Lie group analysis provides a rigorous mathematical formulation of intuitive ideas of symmetry and invariance. These ideas permeate all mathematical models, in particular those formulated in terms of differential or integro-differential equations. Theoretically, Lie group analysis combines, via Lie's infinitesimal method, all three principal branches of mathematics: algebra, analysis and geometry. It augments intuition in understanding and using symmetry properties in mathematical models. Practically, Lie group analysis discloses possible approaches to solving complex problems. It furnishes a universal and effective method for solving nonlinear equations analytically. It is particularly useful when other means of integration fail. Numerous physical phenomena can be investigated using Lie symmetries to unearth various group invariant solutions and conservation laws that provide significant physical insights into the problem. The present talk is a general survey of the basic methods from classical Lie group theory and its developments in modern group analysis. The methods are illustrated by means of various nonlinear models typical for problems of nonlinear acoustics.

Section : 1