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Infinite Dimensional Lie Algebra and Symmetries of Nonlinear Equations in Three Dimension — An Overview
Author(s) -
Chowdhury A. Roy,
Sen D. C.,
Purkait S.
Publication year - 1993
Publication title -
fortschritte der physik/progress of physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.2190410308
Subject(s) - homogeneous space , dimension (graph theory) , mathematics , commutator , algebra over a field , lie algebra , nonlinear system , integrable system , pure mathematics , mathematical analysis , lie conformal algebra , physics , geometry , quantum mechanics
Symmetries of nonlinear systems integrable via inverse scattering transform in three dimension are analysed from a different point of view. Actually the classical approach of Lie Backlund transform is not so effective due to the complexity of the computations involved. So we review here the new approach for finding symmetries on the basis of Lax algebra. In the first part we mention in short the earlier techniques for the study of symmetries and in the second part the present method is reviewed in detail following the earlier work of Li Yi Shen et al., with an eye for application to nonlinear systems in three dimension. The approach is highly effective in finding symmetries of non‐local, nonlinear in equations in three dimension, which is otherwise impossible to find out. The methodology is illustrated with the help of three examples. (1) the Harry Dym equation in three dimension, (2) the non‐local KP equation and (3) dispersive water wave equation in three dimension. Lastly we demonstrate that these symmetries generate an infinite dimensional Lie algebra through a suitably defined commutator.