Applications of Symmetry Methods to Partial Differential Equations

Description

sequel to Bluman & Anco (2002)

MSC classifications
Willy Hereman:

4) GeM (Maple) by A. F. Cheviakov (a.k.a. Shevyakov), project started in 2004, mentioned in his book with Bluman and Anco (PDF pp. 376-86):

• Applications of Symmetry Methods to Partial Differential Equations (2010) by G.W. Bluman, A. F. Cheviakov, and S. C. Anco, Springer: Applied Mathematical Sciences, Vol. 168, ISBN 978-0-387-98612-8.

This is an acessible book on the advanced symmetry methods for differential equations, including such subjects as conservation laws, Lie-Bäcklund symmetries, contact transformations, adjoint symmetries, Nöther's Theorem, mappings with some modification, potential symmetries, nonlocal symmetries, nonlocal mappings, and non-classical method. Of use to graduate students and researchers in mathematics and physics.

"The book contains a wealth of practically relevant examples as well as numerous exercises to allow the reader to gain a working knowledge of advanced symmetry methods. Each chapter concludes with a discussion that provides helpful connections to the journal literature (collected in an extensive list of references). This book is carefully written and provides an excellent overview of this highly active branch of applied mathematics. Like its predecessors, it will be a standard reference in the field for years to come." Michael Kunzinger, Mathematical Reviews, Issue 2011 d

Symmetry methods for differential equations constitute a vast subject with numerous applications in mathematics, mathematical physics and engineering sciences. One of the standard references in the field is [G. W. Bluman and S. C. Anco, Symmetry and integration methods for differential equations, Appl. Math. Sci., 154, Springer, New York, 2002; MR1914342], based in turn on [G. W. Bluman and S. Kumei, Symmetries and differential equations, Appl. Math. Sci., 81, Springer, New York, 1989; MR1006433]. The book under review is a sequel to these works, with an emphasis on direct methods for the determination of symmetries of (systems of) differential equations. In some more detail, the contents of this volume are as follows.
Chapter 1 starts out with a review of some fundamental notions of symmetry analysis, e.g., point and contact transformations, higher-order symmetries and equivalence transformations. This is followed by a section on direct methods for constructing conservation laws for systems of PDEs. Also, Noether's theorem, its limitations and certain extensions are thoroughly analyzed. The chapter concludes with a study of the connections between symmetries and conservation laws.
The topic of chapter 2 is the construction of mappings relating differential equations. For a given PDE system and a target system (or class of systems), the authors systematically show how to determine whether the system can be mapped invertibly into the target class, using its symmetries or conservation laws. Target systems of interest here are linear systems or linear systems with constant coefficients.
As a main application of the conservation law techniques developed in chapter 1, chapter 3 studies nonlocally related PDE systems. The basic method is the embedding of a given system of PDEs in a larger (augmented) system which is nonlocally related to the original system. Here, local symmetries or conservation laws of the augmented system may give rise to nonlocal symmetries or conservation laws of the given system and vice versa. The construction of such augmented systems is usually based on conservation laws of the original system giving rise to corresponding potential variables. Using this and related methods the authors construct an extended tree of nonlocally related PDE systems for a given system.
Chapter 4 deals with applications of nonlocally related systems of PDEs. In particular, a number of concrete examples illustrate the methods for finding nonlocal symmetries introduced in chapter 3. Moreover, the main emphasis lies on the construction of non-invertible mappings relating systems of PDEs.
The final chapter, chapter 5, collects miscellaneous extensions of symmetry methods. Here, systematic methods for constructing particular solutions of a given system of PDEs are presented. In particular, this comprises the classical Lie theoretic method, the nonclassical method introduced by Bluman as well as methods based on the theory developed in the previous chapters. Finally, the use of symbolic software for determining local symmetries and conservation laws is illustrated.
The book contains a wealth of practically relevant examples as well as numerous exercises to allow the reader to gain a working knowledge of advanced symmetry methods. Each chapter concludes with a discussion that provides helpful connections to the journal literature (collected in an extensive list of references). This book is carefully written and provides an excellent overview of this highly active branch of applied mathematics. Like its predecessors, it will be a standard reference in the field for years to come. Reviewed by Michael Kunzinger