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Nonclassical Symmetries and the Singular Manifold Method: Theory and Six Examples
Author(s) -
Estévez P. G.,
Gordoa P. R.
Publication year - 1995
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm199595173
Subject(s) - homogeneous space , manifold (fluid mechanics) , mathematics , center manifold , pure mathematics , mathematical analysis , algebra over a field , mathematical physics , calculus (dental) , physics , geometry , quantum mechanics , nonlinear system , mechanical engineering , hopf bifurcation , engineering , bifurcation , medicine , dentistry
In this paper we discuss a new approach to the relationship between integrability and symmetries of a nonlinear partial differential equation. The method is based heavily on ideas using both the Painlevé property and the singular manifold analysis, which is of outstanding importance in understanding the concept of integrability of a given partial differential equation. In our examples we show that the solutions of the singular manifold possess Lie point symmetries that correspond precisely to the so‐called nonclassical symmetries. We also point out the connection between the singular manifold method and the direct method of Clarkson and Kruskal. Here the singular manifold is a function of its reduced variable. Although the Painlevé property plays an essential role in our approach, our method also holds for equations exhibiting only the conditional Painlevé property. We offer six full examples of how our method works for the six equations, which we believe cover all possible cases.