##
**Symmetries and conservation laws for differential equations of mathematical physics. Transl. from the Russian by A. M. Verbovetsky and I. S. Krasil’shchik.**
*(English)*
Zbl 0911.00032

Translations of Mathematical Monographs. 182. Providence, RI: American Mathematical Society (AMS). xiv, 333 p. (1999).

Since the discovery of completely integrable systems in the 1960s the mathematical development of symmetries and conservation laws of differential equations has to a degree broken away from Sophus Lie’s heritage. Higher-order symmetries and recursion operators have been at the center of intensive research activity, and new concepts, in particular from homological algebra, have proved to provide a proper framework for the study of conservation laws. The book under review is, on the one hand, designed to provide an accessible account of these recent developments in the field and, on the other hand, meant to serve as an introductory text to the publication series by the Diffiety Institute of the Russian Academy of Natural Sciences headed by the second editor.

The book focuses on the classification and applications of higher symmetries and conservation laws of systems of differential equations. In writing the monograph the authors had in mind both readers mainly interested in the theoretical development of the subject and those interested in the computational applications. The latter audience’s needs are reflected in the large number of concrete examples one finds scattered throughout the text.

In the first two chapters ordinary differential equations and first-order scalar partial differential equations are used to introduce the reader to the basic concepts underlying the subsequent developments. Here already the authors’ preference for a coordinate-free treatment of the subject matter becomes apparent. Differential equations are regarded as submanifolds of a suitable jet space, and symmetries are introduced as transformations on these submanifolds preserving the Cartan distribution. The applicability of symmetries to the integration of differential equations is emphasized. In particular, the classical algorithm for integrating an ODE with a solvable symmetry group and the Lagrange-Charpit method for constructing complete integrals are presented.

Chapter 3 deals with classical symmetries of differential equations in general. After the basic definitions the maximal integral manifolds of the Cartan distribution on finite-order jet bundles \(J^k(\pi)\) over some fibration \(\pi: E\to M\) are found and Bäcklund’s classical theorem characterizing transformations on \(J^k(\pi)\) preserving the Cartan distribution is proved. Next, solutions to differential equations invariant under some symmetry group are discussed. Somewhat contrary to the geometric approach adopted in the text, these are found not by using Lie’s algorithm for symmetry reduction but rather by augmenting the original system by equations describing invariance. Lie symmetries for the Burgers, KdV, Khoklov-Zabolotskaya and Kadomtsev-Pogutse equations are given and examples of constructing solutions to differential equations by symmetry reduction and by factoring the solution manifold \(\mathcal E\subset J^k(\pi)\) by a symmetry group are presented. The chapter closes with an exposition of the results in [I. S. Krasil’shchik et al., Geometry of jet spaces and nonlinear partial differential equations. Transl. from the Russian by A. B. Sosinskij. New York etc.: Gordon and Breach Science Publishers (1986; Zbl 0722.35001)] describing the relationship between extrinsic and intrinsic classical symmetries.

Chapter 4 is devoted to the higher-order symmetries of differential equations. Here, unfortunately, the authors’ preference for formulating the basic definitions in a coordinate-free manner leads, perhaps inevitably, to a proliferation of notation, which tends to obscure somewhat the simplicity of the underlying ideas. This chapter also includes a number of interesting applications of higher-order symmetries. In particular, the authors touch upon the questions of classifying differential equations admitting higher-order symmetries and the relationship between higher-order intrinsic and extrinsic symmetries. Moreover, higher-order symmetries of several equations from mathematical physics are computed.

Chapters 5 and 6 on conservation laws and nonlocal symmetries form the core of the book. Chapter 5 starts with a readable account of the \(\mathcal C\)-spectral sequence associated with a system of differential equations and its relationship with conservation laws. The \(E_1\) term is first computed for the empty equation, that is, for \(\mathcal E^\infty = J^{\infty}(E)\), and the development culminates in the two-line theorem describing the \(\mathcal C\)-spectral sequence for nondegenerate systems. The computational aspects of the classification of conservation laws are illustrated with several completely integrable systems. Moreover, the correspondence between symmetries and conservation laws for locally variational and Hamiltonian systems is discussed, and the construction of conservation laws for bi-Hamiltonian systems is briefly covered. Finally, in Chapter 6, the theory of coverings of systems of differential equations is developed. These coverings are shown to provide a proper mathematical framework for potential symmetries and potential conservation laws, Bäcklund transformations and recursion operators. The classification of coverings for a system of differential equations by the cohomology classes associated with the equations is considered. Next, the problem of lifting a given symmetry to a cover is discussed and it is shown that for so-called abelian covers, the problem always admits a solution. The chapter ends with a lengthy and detailed description of symmetries of integro-differential equations.

As an appendix, the book also includes an abridged version of A. M. Vinogradov’s article [J. Geom. Phys. 14, No. 2, 146–194 (1994; Zbl 0815.58028)].

A valuable feature about the present text is that it provides an overview of a number of results that originally appeared in Russian journals and are often difficult to track down. Moreover, the text is a useful reference for the practitioner in the field not only due to the descriptions of the various relevant algorithms but also due to the numerous examples involving symmetry algebras and conservation laws of particular equations, which, because of the book’s emphasis on higher-order and potential symmetries, can often not be found in other standard references in the field. In all, this text provides a useful and readable introduction to the recent developments in the theory of symmetries and conservation laws of differential equations.

The book focuses on the classification and applications of higher symmetries and conservation laws of systems of differential equations. In writing the monograph the authors had in mind both readers mainly interested in the theoretical development of the subject and those interested in the computational applications. The latter audience’s needs are reflected in the large number of concrete examples one finds scattered throughout the text.

In the first two chapters ordinary differential equations and first-order scalar partial differential equations are used to introduce the reader to the basic concepts underlying the subsequent developments. Here already the authors’ preference for a coordinate-free treatment of the subject matter becomes apparent. Differential equations are regarded as submanifolds of a suitable jet space, and symmetries are introduced as transformations on these submanifolds preserving the Cartan distribution. The applicability of symmetries to the integration of differential equations is emphasized. In particular, the classical algorithm for integrating an ODE with a solvable symmetry group and the Lagrange-Charpit method for constructing complete integrals are presented.

Chapter 3 deals with classical symmetries of differential equations in general. After the basic definitions the maximal integral manifolds of the Cartan distribution on finite-order jet bundles \(J^k(\pi)\) over some fibration \(\pi: E\to M\) are found and Bäcklund’s classical theorem characterizing transformations on \(J^k(\pi)\) preserving the Cartan distribution is proved. Next, solutions to differential equations invariant under some symmetry group are discussed. Somewhat contrary to the geometric approach adopted in the text, these are found not by using Lie’s algorithm for symmetry reduction but rather by augmenting the original system by equations describing invariance. Lie symmetries for the Burgers, KdV, Khoklov-Zabolotskaya and Kadomtsev-Pogutse equations are given and examples of constructing solutions to differential equations by symmetry reduction and by factoring the solution manifold \(\mathcal E\subset J^k(\pi)\) by a symmetry group are presented. The chapter closes with an exposition of the results in [I. S. Krasil’shchik et al., Geometry of jet spaces and nonlinear partial differential equations. Transl. from the Russian by A. B. Sosinskij. New York etc.: Gordon and Breach Science Publishers (1986; Zbl 0722.35001)] describing the relationship between extrinsic and intrinsic classical symmetries.

Chapter 4 is devoted to the higher-order symmetries of differential equations. Here, unfortunately, the authors’ preference for formulating the basic definitions in a coordinate-free manner leads, perhaps inevitably, to a proliferation of notation, which tends to obscure somewhat the simplicity of the underlying ideas. This chapter also includes a number of interesting applications of higher-order symmetries. In particular, the authors touch upon the questions of classifying differential equations admitting higher-order symmetries and the relationship between higher-order intrinsic and extrinsic symmetries. Moreover, higher-order symmetries of several equations from mathematical physics are computed.

Chapters 5 and 6 on conservation laws and nonlocal symmetries form the core of the book. Chapter 5 starts with a readable account of the \(\mathcal C\)-spectral sequence associated with a system of differential equations and its relationship with conservation laws. The \(E_1\) term is first computed for the empty equation, that is, for \(\mathcal E^\infty = J^{\infty}(E)\), and the development culminates in the two-line theorem describing the \(\mathcal C\)-spectral sequence for nondegenerate systems. The computational aspects of the classification of conservation laws are illustrated with several completely integrable systems. Moreover, the correspondence between symmetries and conservation laws for locally variational and Hamiltonian systems is discussed, and the construction of conservation laws for bi-Hamiltonian systems is briefly covered. Finally, in Chapter 6, the theory of coverings of systems of differential equations is developed. These coverings are shown to provide a proper mathematical framework for potential symmetries and potential conservation laws, Bäcklund transformations and recursion operators. The classification of coverings for a system of differential equations by the cohomology classes associated with the equations is considered. Next, the problem of lifting a given symmetry to a cover is discussed and it is shown that for so-called abelian covers, the problem always admits a solution. The chapter ends with a lengthy and detailed description of symmetries of integro-differential equations.

As an appendix, the book also includes an abridged version of A. M. Vinogradov’s article [J. Geom. Phys. 14, No. 2, 146–194 (1994; Zbl 0815.58028)].

A valuable feature about the present text is that it provides an overview of a number of results that originally appeared in Russian journals and are often difficult to track down. Moreover, the text is a useful reference for the practitioner in the field not only due to the descriptions of the various relevant algorithms but also due to the numerous examples involving symmetry algebras and conservation laws of particular equations, which, because of the book’s emphasis on higher-order and potential symmetries, can often not be found in other standard references in the field. In all, this text provides a useful and readable introduction to the recent developments in the theory of symmetries and conservation laws of differential equations.

Reviewer: Juha Pohjanpelto (MR 2000f:58076)

### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

58-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

34C14 | Symmetries, invariants of ordinary differential equations |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

35L65 | Hyperbolic conservation laws |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |

58A20 | Jets in global analysis |

58J72 | Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds |

00B25 | Proceedings of conferences of miscellaneous specific interest |

35-06 | Proceedings, conferences, collections, etc. pertaining to partial differential equations |

58-06 | Proceedings, conferences, collections, etc. pertaining to global analysis |