非線性系統及其絕妙的數學結構(第2卷英文)/國外優秀數學著作原版系列
內容大鋼
本書包含了17篇由非線性系統不同方面的頂尖專家撰寫的特邀論文，包括常微分方程和偏微分方程、差分方程、離散或晶格方程、非交換方程和矩陣方程以及時滯方程。本書內容被分為三個主要部分：A部分為可積性，Lax對與對稱性，包含7篇論文，作者主要解決了如何檢測可積系統的基本問題，以及非線性系統對稱方法的使用問題；B部分為代數與幾何方法，包含7篇論文，作者描述了獲得非線性系統顯式解和（或）描述系統解的結構的不同方法；C部分為應用，包含了三篇論文，作者為了解決特殊的非線性問題應用了不同的方法。
作者介紹
編者:(墨)諾伯特·歐拉//(意)瑪麗亞·克拉拉·努奇|責編:張嘉芮//李蘭靜
目錄
Preface
The Authors
A1. Reciprocal transformations and their role in the integrability and classification of PDEs
1. Introduction
2. Fundamentals
3. Reciprocal transformations as a way to identify and classify PDEs
4. Reciprocal transformations to derive Lax pairs
5. A Miura-reciprocal transformation
6. Conclusions
A2. Contact Lax pairs and associated (3+l)-dimensional integrable dispersionless systems
1. Introduction
2. Isospectral versus nonisospectral Lax pairs
3. Lax representations for dispersionless systems in (I+I)D and (2+1)D
4. Lax reprcsentations for dispersionless systems in (3+l)D
5. R-matrix approach for dispersionless systems with nonisospectral
Lax representations
A3. Lax pairs for edge-constrained Boussinesq systems of partial difference equations
1. Introduction
2. Gauge equivalence of Lax pairs for PDEs and PAEs
3. Derivation of Lax pairs for Boussinesq systems
4. Gauge and gauge-like equivalences of Lax pairs
5. Application to generalized Hietarinta systems
6. Summary of results
7. Software implementation and conclusions
A4. Lie point symmetries of delay ordinary differential equations
1. Introduction
2. Illustrating example
3. Formulation of the problem for first-order DODEs
4. Construction of invariant first-order DODSs
5. First-order linearDODSs
6. Lie symmetry classification of first-order nonlinear DODSs
7. Exact solutions of the DODSs
8. Higher order DODSs
9. Traffic flow micro-model equation
10. Conclusions
A5. The symmetry approach to integrability: recent advances
1. Introduction
2. The symmetry approach to integrability
3. Integrable non-abelian equations
4. Non-evolutionary systems
A6. Evolution of the concept of h-symmetry and main applications
1. Introduction
2. Basic notions on Lie point symmetries and C∞- symmetries of ODEs
3. Analytical applications of C~-symmetries
4. Extensions and geometric interpretations of C∞-symmetries
A7. Heir-equations for partial differential equations: a 25-year review
1. Introduction
2. Constructing the heir-equations
3. Symmetry solutions of heir-equations
4. Zhdanov's conditional Lie-B~cklund symmetries and heir-equations
5. Nonclassical symmetries as special solutions of heir-equations
6. Final remarks
B1. Coupled nonlinear SchrSdinger equations: spectra and instabilities of plane waves
1. Introduction
2. Spectra
3. Dispersion relation and instability
4. Conclusions
A. Case r = 0
B. Polynomials: a tool box
B2. Rational solutions of Painlev systems
1. Introduction
2. Dressing chains and Painlevd systems
3. Hermite T-functions
4. Hermite-type rational solutions
5. Cyclic Maya diagrams
6. Examples of Hermite-type rational solutions
B3. Cluster algebras and discrete integrability
1. Introduction
2. Cluster algebras: definition and examples
3. Cluster algebras with periodicity
4. Algebraic entropy and tropical dynamics
5. Poisson and symplectic structures
6. Discrete Painlev6 equations from coefficient mutation
7. Conclusions
B4. A review of elliptic difference Painlev6 equations
1. Introduction
2. E-lattice
3. The initial-value space of the RCG equation
4. Cremona isometrics
5. Birational actions of the Cremona isometrics for the Jacobi setting
6. Special solutions of the RCG equation
A. A-lattice
B. General elliptic difference equations
B5. Linkage mechanisms governed by integrable deformations of discrete space curves
1. Introduction
2. A mathematical model of linkage
3. Hinged network and discrete space curve
4. Deformation of discrete curves
5. Extreme Kaleidocycles
B6. The Cauchy problem of the Kadomtsev=Petviashvili hierarchy and infinite-dimensional groups
1. Introduction
2. Diffeologies, Fr51icher spaces and the Ambrose-Singer theorem
3. Infinite-dimensional Lie groups and pseudodifferential operators
4. The Cauchy problem for the KP hierarchy
5. A non-formal KP hierarchy
BT. Wronskian solutions of integrable systems
1. Introduction
2. Preliminary
3. The KdV equation
4. The mKdV equation
5. The AKNS and reductions
6. Discrete case: the lpKdV equation
7. Conclusions
C1. Global gradient catastrophe in a shallow water model: evolution unfolding by stretched coordinates
1. Introduction
2. Exact solutions
3. Evolution beyond the gradient catastrophe
4. Discussion and conclusions
C2. Vibrations of an elastic bar isospectral deformations and modified Ca
[