Symmetry approach to the integrability problem.

*(English. Russian original)*Zbl 1029.37041
Theor. Math. Phys. 125, No. 3, 1603-1661 (2000); translation from Teor. Mat. Fiz. 125, No. 3, 355-424 (2000).

The paper is a survey article, in which the authors summarize their twenty years of development in classifying a few classes of integrable equations and integrable lattice equations. The basis for the classification is the existence of Lie-Bäcklund symmetries including master-symmetries. The generalized Toda chains and their related equations of the nonlinear Schrödinger type, discrete transformations, and hyperbolic systems are main topics in the whole theory of so-called symmetry approach. The lists of integrable equations among the KdV, nonlinear Schrödinger, Boussinesq, Toda and Volterra type equations are given and a general problem of classifying scalar evolution equations of arbitrary order is put in one appendix. The paper also considers the equations of the Painlevé type, the local property of master symmetries, and the problem of integrability criteria for \(2+1\)-dimensional equations. In particular, the authors introduce a new notion of \(B\)-integrable equations, for which there exist changes of variables transforming the equations to their maser symmetries. The effective tests for integrability and the algorithms for reduction to the canonical form are also elaborated.

Reviewer: Ma Wen-Xiu (Tampa)

##### MSC:

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

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35Q53 | KdV equations (Korteweg-de Vries equations) |

35Q58 | Other completely integrable PDE (MSC2000) |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37K60 | Lattice dynamics; integrable lattice equations |