Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry-猫眼课题宝

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2005-06

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作者：

C. Gu;He-sheng Hu;Zixiang Zhou

前言。-1.1+1维可积系统。-1.1KdV方程，MKdV方程及其达布变换。1.1.1原始达布变换。1.1.2 KdV方程的达布变换。1.1.3 MKdV方程的达布变换。1.1.4例：单孤子解和双孤子解。1.1.5 KdV方程和MKdV方程的达布变换之间的关系。1.2 AKNS系统。1.2.1 2 x 2 AKNS系统。1.2.2 N x N AKNS系统。1.3达布变换。1.3.1 AKNS系统的达布变换。1.3.2达布变换下方程的不变性。1.3.3高次Darboux变换与可置换性定理1.3.4关于一次达布矩阵的更多结果。1.4KdV系统、MKdV-SG系统、NLS系统和具有u(N)约简的AKNS系统。1.4.1 KDV层次结构。1.4.2 MKdV-SG层次结构。1.4.3 NLS层次结构。1.4.4具有u(N)约简的AKNS系统。1.5达布变换与散射，逆散射理论。1.5.1 2×2 AKNS系统的散射和逆散射理论概述。1.5.2 SU(2)AKNS体系达布变换下散射数据的变化。2.2+1维可积系统.-2.1KP方程及其达布变换.2.2 2+1维AKNS系统和DS方程。2.3达布变换。2.3.1一般宽松配对。2.3.2一次达布变换。2.3.3高次Darboux变换和置换性定理。2.4 DS方程的达布变换和二元达布变换。2.4.1 DS II方程的达布变换。2.4.2 DSI方程的达布变换和二元达布变换。2.5应用于1+1维Gelfand-Dickey系统。2.6 2+1维非线性约束和达布变换。3.n+1维可积系统-3.1n+1维AKNS系统。3.1.1 n+1维AKNS系统。3.1.2示例。3.2达布变换和孤子解。3.2.1达布变换。3.2.2u(N)箱。3.2.3孤子解。3.3 Rn上的精简系统。4.常曲率的曲面，Backlund同余。-4.1欧氏空间R3中的曲面理论。4.2常负高斯曲率曲面、Sine-Gordon方程和Backlund变换。4.2.1 Sine-Gordon方程与R3中常负高斯曲率曲面的关系4.2.2伪球同余。4.2.3 Backlund变换。4.2.4达布变换。4.2.5示例。4.3 Minkowski空间R2，1中的常高斯曲率曲面与伪球同余。4.3.1 Minkowski空间的曲面理论R2，1.4.3.2常高斯曲率曲面的切比雪夫坐标。4.3.3 R2中的伪球同余1.4.3.4 R2中常高斯曲率曲面的Backlund变换和Darboux变换。4.5常平均曲率的曲面。4.5.1欧氏空间中的平行曲面。4.5.2表面的构造。4.5.3 Minkowski空间的情况。5.达布变换和调和映射。5.1调和映射和基本方程的定义。5.2从R2或R1到S2、H2或S1的调和映射。5.3从R1、1到U(N)的调和映射。5.3.1 U(N)上的黎曼度量。5.3.2从R1，1到U(N)的调和映射。5.3.3单孤子解。5.3.4多孤子解。5.4从R2到U(N)的调和映射。5.4.1从R2到U(N)的调和映射及其达布变换。5.4.2孤子解。5.4.3 Uniton。5.4.4单位元达布变换和奇异达布变换。6.广义自对偶Yang-Mills和Yang-Mills-Higgs方程-6.1广义自对偶杨-Mills流。6.1.1广义自对偶杨-米尔流。6.1.2达布变换。6.1.3示例。6.1.4与AKNS系统的关系。6.2杨-米尔斯-希格斯

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 x 2 AKNS system. 1.2.2 N x 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 x 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, Backlund Congruences.- 4.1 Theory of surfaces in the Euclidean space R3. 4.2 Surfaces of constant negative Gauss curvature, sine-Gordon equation and Backlund 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 Backlund 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 Backlund 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

Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry

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