Symmetry Approaches for Reductions of PDEs, Differential Constraints and Lagrange-Charpit Method
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Symmetry Approaches for Reductions of PDEs, Differential Constraints and Lagrange-Charpit Method Boris Kruglikov
Received: 31 October 2007 / Accepted: 17 November 2007 / Published online: 29 January 2008 © Springer Science+Business Media B.V. 2008
Abstract Many methods for reducing and simplifying differential equations are known. They provide various generalizations of the original symmetry approach of Sophus Lie. Plenty of relations between them have been noticed and in this note a unifying approach will be discussed. It is rather close to the differential constraint method, but we make this rigorous basing on recent advances in compatibility theory of non-linear overdetermined systems and homological methods for PDEs. Keywords Compatibility · Differential constraint · Generalized symmetry · Reduction · Multi-bracket · Solvability Mathematics Subject Classification (2000) 34A05 · 35N10 · 58A20 · 35A30 1 Introduction to Reduction Methods Exact solutions play an important role in investigation of equations of mathematical physics. Even though usually1 it’s only a small portion of solutions one can find explicitly, they represent significant physical phenomena and are often applied in practice. The first explicit methods have origins in the works of Euler, Lagrange, Cauchy, Hamilton, Jacobi and Monge. But the most essential contribution was made by Sophus Lie, whose continuous transformation groups changed the landscape of the theory of differential equations. These transformations are called nowadays symmetries2 and can be computed with the help of most popular symbolic software packages. If the symmetry algebra is known, is sufficiently large and has an appropriate Lie structure, then the given differential equation (or system) can be integrated completely. This
1 Except for the important class of completely integrable equations. 2 They are distinguished to be point or contact symmetries, though the latter received less circulation. Internal
symmetries are the most generic local transformations. B. Kruglikov () Institute of Mathematics and Statistics, University of Tromsø, 90-37 Tromsø, Norway e-mail: [email protected]
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works both for ODEs and PDEs. But usually the algebra is not big,3 so other methods should apply. There is a number of classical methods, like intermediate integrals, differential and nonlocal substitutions (e.g. Laplace, Backlund and Darboux transformations) and conservation laws. In more recent time a lot of other methods for obtaining exact solutions were invented. The whole list is large, but we name a few: higher and inner (= internal) symmetries [13], recursion operators, nonlocal and ghost symmetries, nonclassical symmetries and direct reduction, inverse scattering, bi-linear Hirotha method, Sato theory and so on, see e.g. [4, 7, 35]. Many interrelations have been made explicit, but the unifying theory does not exist so far. In this paper I will try to present a moderately general viewpoint for all local theories (nonlocal ones follow the same scheme, but additiona
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