李-巴克兰-达布变换

Chapter 1 Introduction

Chapter 2 A Brief Account on Backlund rlyansformations

2.1 A Warm-Up Approach

2.2 Chen's Method

2.3 Clairin's Method

2.4 Hirota's Bilinear Operator Method

2.5 Wahlquist-Estabrook Procedure

Chapter 3 Nonlinear Schrodinger Equation

3.1 Physical Background

3.2 Lax Pair and Floquet Theory

3.3 Darboux rlyansformations and Homoclinic Orbit

3.4 Linear Instability

3.5 Quadratic Products of Eigenfunctions

3.6 Melnikov Vectors

3.7 Melnikov Integrals

Chapter 4 Sine-Gordon Equation

4.1 Background

4.2 Lax Pair

4.3 Darboux Transformations

4.4 Melnikov Vectors

4.5 Heteroclinic Cycle

4.6 Melnikov Vectors Along the Heteroclinic Cycle

Chapter 5 Heisenberg Ferromagnet Equation

5.1 Background

5.2 Lax Pair

5.3 Darboux Transformations

5.4 Figure Eight Structures Connecting to the Domain Wall

5.5 Floquet Theory

5.6 Melnikov Vectors

5.7 Melnikov Vectors Along the Figure Eight Structures

5.8 A Melnikov Function for Landau-Lifshitz-Gilbert Equation

Chapter 6 Vector Nonlinear Schrodinger Equations

6.1 Physical Background

6.2 Lax Pair

6.3 Linearized Equations

6.4 Homoclinic Orbits and Figure Eight Structures

6.5 A Melnikov Vector

Chapter 7 Derivative Nonlinear Schrodinger Equations

7.1 Physical Background

7.2 Lax Pair

7.3 Darboux Transformations

7.4 Floquet Theory

7.5 Strange Tori

7.6 Whisker of the Strange T2

7.7 Whisker of the Circle

7.8 Diffusion

7.9 Diffusion Along the Strange T2

7.10 Diffusion Along the Whisker of the Circle

Chapter 8 Discrete Nonlinear Schrodinger Equation

8.1 Background

8.2 Hamiltonian Structure

8.3 Lax Pair and Floquet Theory

8.4 Examples of Floquet Spectra

8.5 Melnikov Vectors

8.6 Darboux Transformations

8.7 Homoclinic Orbits and Melnikov Vectors

Chapter 9 Davey-Stewartson II Equation

9.1 Background

9.2 Linear Stability

9.3 Lax Pair and Darboux Transformations

9.4 Homoclinic Orbits

9.5 Melnikov Vectors

9.5.1 Melnikov Integrals

9.5.2 An Example

9.6 Extra Comments

Chapter 10 Acoustic Spectral Problem

10.1 Physical Background

10.2 Connection with Linear Schrodinger Operator

10.3 Discrete Symmetries of the Acoustic Problem

……

Chapter 11 SUSY and Spectrum Reconstructions

Chapter 12 Darboux Transformations for Dirac Equation

Chapter 13 Moutard Transformations for the 2D and 3D

Chapter 14 BLP Equation

Chapter 15 Goursat Equation

Chapter 16 Links Among Integrable Systems

Bibliography

Index

《李-巴克兰-达布变换》提出了无限维动力系统、偏微分方程、数学物理交叉学科尖端领域的处理某些议题的新方法。书中的第一部分着重介绍了第一作者在达布变换和同宿轨道以及建立可积偏微分方程梅尔尼科夫积分方面取得的成果。第二部分则专注第二作者将达布变换应用于物理领域的工作。
　　《李-巴克兰-达布变换》的特点在于第一作者及合作者发展的用达布变换建立可积系统中同宿轨道、梅尔尼科夫积分及梅尔尼科夫向量的崭新方法。可积系统（也叫孤立子方程）是有限维可积哈密顿系统在无限维的对应物，而上述所说的崭新方法所展示的是无限维相空间结构。
　　《李-巴克兰-达布变换》可供数学、物理及其他相关学科领域的高年级本科生、研究生及该领域的专家参考。