Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry: 26 Softcover reprint of hardcover 1st ed. 2005 Edition

From the reviews:

“The book is concerned with mutual relations between the differential geometry of surfaces and the theory of integrable nonlinear systems of partial differential equations. It concentrates on the Darboux matrix method for constructing explicit solutions to various integrable nonlinear PDEs. … This book can be recommended for students and researchers who are interested in a differential-geometric approach to integrable nonlinear PDE’s.” (Jun-ichi Inoguchi, Mathematical Reviews, Issue 2006 i)

The Darboux transformation approach is one of the most effective methods for constructing explicit solutions of partial differential equations which are called integrable systems and play important roles in mechanics, physics and differential geometry.

This book presents the Darboux transformations in matrix form and provides purely algebraic algorithms for constructing the explicit solutions. A basis for using symbolic computations to obtain the explicit exact solutions for many integrable systems is established. Moreover, the behavior of simple and multi-solutions, even in multi-dimensional cases, can be elucidated clearly. The method covers a series of important equations such as various kinds of AKNS systems in R¹⁺ⁿ, harmonic maps from 2-dimensional manifolds, self-dual Yang-Mills fields and the generalizations to higher dimensional case, theory of line congruences in three dimensions or higher dimensional space etc. All these cases are explained in detail. This book contains many results that were obtained by the authors in the past few years.

Audience:

The book has been written for specialists, teachers and graduate students (or undergraduate students of higher grade) in mathematics and physics.