李-巴克兰-达布变换(英文版)(精装)

    书籍
    数学书籍
《李-巴克兰-达布变换(英文版)(精装)》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(英文版)(精装)》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(英文版)(精装)》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。

《李-巴克兰-达布变换(英文版)(精装)》
Chapter I Introduction
Chapter 2 A Brief Account on Backlund Transformations
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 Transformations 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

　　One of the mathematical miracles of the 20th century was the discovery of a group of nonlinear wave equations being integrable. These integrable systems are the in- finite dimensional counterpart of the finite dimensional integrable Hamiltonian systems of classical mechanics. Icons of integrable systems are the KdV equa- tion, sine-Gordon equation, nonlinear Schrodinger equation etc. The beauty of the integrable theory is reflected by the explicit formulas of nontrivial solutions to the integrable systems. These explicit solutions bear the iconic names of soliton, multi-soliton, breather, quasi-periodic orbit, homoelinic orbit (the focus of this book) etc. There are several ways now available for obtaining these explicit solu- tions: B/icklund transformation, Darboux transformation, and inverse scattering transform. The clear connection among these transforms is still an open question although they are certainly closely related. These transformations can be regarded as the counterpart of the canonical transformation of the finite dimensional inte- grable Hamiltonian system. B~icklund transformation originated from a quest for Lie's second type invariant transformation rather than his tangent transformation. That brings the title of this book: Lie-B~icklund-Darboux Transformations which refer to both Biieklund transformations and Darboux transformations.
　　The most famous mathematical miracle of the 20th century was probably the discovery of chaos. When the finite dimensional integrable Hamiltonian systems are under perturbations, their regular solutions can turn into chaotic solutions. For such near integrable systems, existence of chaos can sometimes be proved mathematically rigorously. Following the same spirit, one may attempt to prove the existence of chaos for near integrable nonlinear wave equations viewed as near integrable Hamiltonian partial differential equations. This has been accomplished as summarized in the book [69]. The key ingredients in this theory of chaos in partial differential equations are the explicit formulas for the homoclinie orbit and Melnikov integral. The first author's taste is to use Darboux transformation to obtain the homoclinic orbit and Melnikov integral. This will be the focus of the first part of this book.
　　The second author's taste is to use Darboux transformation in a diversity of applications especially in higher spatial dimensions. The range of applications crosses many different fields of physics. This will be the focus of the second part of this book. This book is a result of the second author's several visits at University of Missouri as a Miller scholar.
　　The first author would like to thank his wife Sherry and his son Brandon, and the second author would like to thank his wife Alla and his son Valerian, for their loving support during this work.