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Darboux transformations and solitons. (English) Zbl 0744.35045
Springer Series in Nonlinear Dynamics. Berlin etc.: Springer-Verlag. viii, 120 p. (1991).
The discovery of a solution technique for a special nonlinear partial differential equation in 1967 was the foundation for a novel field in mathematical physics: the theory of solitons. For the past 25 years it has been developed to become one of the major theories in nonlinear sciences. Meanwhile a huge number of nonlinear equations are identified to reside in the class of “integrable systems” for which exact solutions with the spectacular soliton properties can be found. These nonlinear equations share the common property that they are given by the compatibility condition of linear equations.
It was discovered at an early stage that the classical results of Darboux on the transformation properties of the 1-dimensional Schrödinger equation gives rise to a simple construction of the solitons for one of these systems, the Korteweg-de Vries (KdV) equation. The basic idea of this technique is to identify an invariance of the linear scattering problems describing the nonlinear soliton system under consideration. Hence, transformations of the linear systems will give rise to solutions of the nonlinear equations in terms of the solution of the linear equations. Such invariances have been found for a variety of linear scattering problems. The corresponding generalizations of the Darboux transformation for the Schrödinger operator were shown to be an effective tool to obtain nontrivial solutions of the linear equations. This, in turn, provides classes of exact solutions for a large number of nonlinear systems associated with these scattering equations.
The authors start by establishing the generalized Darboux transformations for some classes of scalar linear problems. These include ordinary differential operators, partial differential operators and differential- difference operators. General “Wronskian-type” solution formulas are obtained from iterated applications of the general Darboux theorems. Relations to factorization methods and supersymmetric quantum mechanics are pointed out. These results are then applied to the Kadomtsev- Petviashvili hierarchy and its reductions such as the KdV equation, the Boussinesq equation, the cylindrical KdV equation etc. Solitons as well as lump solutions are constructed and discussed. It is shown how statements about phase shifts in the soliton interactions or stability properties may be obtained from the Darboux approach. Then matrix-valued linear scattering problems are considered, the corresponding Darboux theorems are established for the generalized Zakharov-Shabat linear problems. Soliton solutions are constructed for a variety of nonlinear soliton systems which are described in terms of such scattering equations. Using the Darboux results for linear differential-difference equations, a number of integrable lattice equations is discussed. Among other systems the abelian and non-abelian Toda lattice equation, discretized KdV and Burgers equations are considered, solution formulas involving Cazorati determinants are derived. The connection to the sine- Gordon equation and its multi-solitons is pointed out. A detailed discussion is devoted to localized soliton solutions (“ dromions”) of 2+1-dimensional nonlinear equations. In particular, the Davey-Stewartson equation and the Veselov-Novikov equation are discussed. Finally, some remarks on various further aspects of the Darboux transformation are given. Its Hamiltonian interpretation as well as its relationship to the Zakharov-Shabat Dressing Method is pointed out. A brief discussion of the long-time asymptotic behavior of multisoliton solutions finishes the last chapter.
Concentrating on the Darboux approach and essentially ignoring other techniques in the field of solitons, the book provides a remarkably simple and direct approach to solitons and other exact solutions for a variety of nonlinear system. Thus, Darboux transformations are shown to be a simple, but yet effective and powerful, tool for studying solitons and their interactions. Other methods such as the much more involved inverse scattering transforms are only briefly commented on, no structural results obtained from these very technical methods are needed to understand the Darboux approach. Thus, the book provides a selfcontained and easily digestible introduction to solitons, also suitable for advanced students or readers with more experimental interests. The book represents a concise account of what Darboux theorems can do for investigating integrable equations, a variety of new results is included. Hence, this book will also be of interest to specialists in soliton theory. A short section with comments on the literature is included to make the connection with related developments and results in other areas of soliton theory.

35Q51 Soliton equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35Q53 KdV equations (Korteweg-de Vries equations)
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds