Nonlinear integrable equations : recursion operators, group theoretical and Hamiltonian structures of soliton equations / B.G. Konopelchenko

This work is devoted to the recursion operator method, which enables one to study and treat the recursion, group-theoretical and Hamiltonian structures of soliton equations from a common point of view. The recursion operator method gives the possibility of representing, in a compact form, the hierarchies of integrable equations connected with a given spectral problem, to construct their general Bäcklund transformations and infinite symmetry groupsand to find the hierarchies of Hamiltonian structures for these equations. The method works for a variety of one- and two-dimensional spectral problems. Following a historical introduction (1967–present) the general idea of the recursion operator method is detailed. This is followed by Chapter III on the linear matrix bundle and generalizations (Chapter IV). Chapter V discusses the Bäcklund-Calogero group and general integrable equations under reduction. The quadratic bundle with a Z2 grading is introduced in Chapter VI where we find nonlinear superposition formulas. The next generalization, polynomial and rational bundles, is given in Chapter VII. The general differential spectral problem, including the gauge group and gauge invariance, occupies Chapter VIII. These are generalized in Chapter IX, and in X the two-dimensional matrix spectral problem is examined. Chapter XI concerns the two-dimensional differential spectral problem and XII has as its goal a general theory of recursion structure for nonlinear evolution equations.
The book, reproduced from typescript, is timely and contains a worldwide bibliography of 636 papers. It is highly recommended for its bibliography and for its detailed description of the present struggles to understand a part of the nonlinear world. (MathScinet)