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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.
The book has been written for specialists, teachers and graduate students (or undergraduate students of higher grade) in mathematics and physics.
Content Level »Research
Keywords »Darboux transformations - Minkowski space - differential equations - differential geometry - integrable systems - inverse scattering theory - scattering theory - two-dimensional manifolds
Preface.- 1. 1+1 Dimensional Integrable Systems.- 1.1 KdV equation, MKdV equation and their Darboux transformations. 1.1.1 Original Darboux transformation. 1.1.2 Darboux transformation for KdV equation. 1.1.3 Darboux transformation for MKdV equation. 1.1.4 Examples: single and double soliton solutions. 1.1.5 Relation between Darboux transformations for KdV equation and MKdV equation. 1.2 AKNS system. 1.2.1 2 × 2 AKNS system. 1.2.2 N × N AKNS system. 1.3 Darboux transformation. 1.3.1 Darboux transformation for AKNS system. 1.3.2 Invariance of equations under Darboux transformations. 1.3.3 Darboux transformations of higher degree and the theorem of permutability. 1.3.4 More results on the Darboux matrices of degree one. 1.4 KdV hierarchy, MKdV-SG hierarchy, NLS hierarchy and AKNS system with u(N) reduction. 1.4.1 KdV hierarchy. 1.4.2 MKdV-SG hierarchy. 1.4.3 NLS hierarchy. 1.4.4 AKNS system with u(N) reduction. 1.5 Darboux transformation and scattering, inverse scattering theory. 1.5.1 Outline of the scattering and inverse scattering theory for the 2 × 2 AKNS system . 1.5.2 Change of scattering data under Darboux transformations for su(2) AKNS system. 2. 2+1 Dimensional Integrable Systems.- 2.1 KP equation and its Darboux transformation. 2.2 2+1 dimensional AKNS system and DS equation. 2.3 Darboux transformation. 2.3.1 General Lax pair. 2.3.2 Darboux transformation of degree one. 2.3.3 Darboux transformation of higher degree and the theorem of permutability. 2.4 Darboux transformation and binary Darboux transformation for DS equation. 2.4.1 Darboux transformation for DSII equation. 2.4.2 Darboux transformation and binary Darboux transformation for DSI equation. 2.5 Application to 1+1 dimensional Gelfand-Dickey system. 2.6 Nonlinear constraints and Darboux transformation in 2+1 dimensions. 3. N + 1 Dimensional Integrable Systems.- 3.1 n + 1 dimensional AKNS system. 3.1.1 n + 1 dimensional AKNS system. 3.1.2Examples. 3.2 Darboux transformation and soliton solutions. 3.2.1 Darboux transformation. 3.2.2 u(N) case. 3.2.3 Soliton solutions. 3.3 A reduced system on Rn. 4. Surfaces of Constant Curvature, Bäcklund Congruences.- 4.1 Theory of surfaces in the Euclidean space R3. 4.2 Surfaces of constant negative Gauss curvature, sine-Gordon equation and Bäcklund transformations. 4.2.1 Relation between sine-Gordon equation and surface of constant negative Gauss curvature in R3. 4.2.2 Pseudo-spherical congruence. 4.2.3 Bäcklund transformation. 4.2.4 Darboux transformation. 4.2.5 Example. 4.3 Surface of constant Gauss curvature in the Minkowski space R2,1 and pseudo-spherical congruence. 4.3.1 Theory of surfaces in the Minkowski space R2,1. 4.3.2 Chebyshev coordinates for surfaces of constant Gauss curvature. 4.3.3 Pseudo-spherical congruence in R2,1. 4.3.4 Bäcklund transformation and Darboux transformation for surfaces of constant Gauss curvature in R2,1. 4.4 Orthogonal frame and Lax pair. 4.5 Surface of constant mean curvature. 4.5.1 Parallel surface in Euclidean space. 4.5.2 Construction of surfaces. 4.5.3 The case in Minkowski space. 5. Darboux Transformation and Harmonic Map.- 5.1 Definition of harmonic map and basic equations. 5.2 Harmonic maps from R2 or R1,1 to S2, H2 or S1,1. 5.3 Harmonic maps from R1,1 to U(N). 5.3.1 Riemannian metric on U(N). 5.3.2 Harmonic maps from R1,1 to U(N). 5.3.3 Single soliton solutions. 5.3.4 Multi-soliton solutions. 5.4 Harmonic maps from R2 to U(N). 5.4.1 Harmonic maps from R2 to U(N) and their Darboux transformations. 5.4.2 Soliton solutions. 5.4.3 Uniton. 5.4.4 Darboux transformation and singular Darboux transformation for unitons. 6. Generalized Self-Dual Yang-Mills and Yang-Mills-Higgs Equations.- 6.1 Generalized self-dual Yang-Mills flow. 6.1.1 Generalized self-dual Yang-Mills flow. 6.1.2 Darboux transformation. 6.1.3 Example. 6.1.4 Relation with AKNS system. 6.2 Yang-Mills-Higgs