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Inverse Problems and Nonlinear Evolution Equations : Solutions, Darboux Matrices and Weyl-Titchmarsh Functions.
This monograph fits theclearlyneed for books with a rigorous treatment of theinverse problems for non-classical systems and that of initial-boundary-value problems for integrable nonlinear equations. The authorsdevelop a unified treatment of explicit and global solutions via the transfer matrix func...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Berlin :
De Gruyter,
2013.
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| Colección: | De Gruyter studies in mathematics.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; Notation; 0 Introduction; 1 Preliminaries; 1.1 Simple transformations and examples; 1.1.1 Dirac-type systems as a subclass of canonical systems; 1.1.2 Schrödinger systems as a subclass of canonical systems; 1.1.3 Gauge transformations of the Dirac systems; 1.2 S-nodes and Weyl functions; 1.2.1 Elementary properties of S-nodes; 1.2.2 Continual factorization; 1.2.3 Canonical systems and representation of the S-nodes; 1.2.4 Asymptotics of the Weyl functions, a special case; 1.2.5 Factorization of the operators S; 1.2.6 Weyl functions of Dirac and Schrödinger systems.
- 2 Self-adjoint Dirac system: rectangular matrix potentials2.1 Square matrix potentials: spectral and Weyl theories; 2.1.1 Spectral and Weyl functions: direct problem; 2.1.2 Spectral and Weyl functions: inverse problem; 2.2 Weyl theory for Dirac system with a rectangularmatrix potential; 2.2.1 Direct problem; 2.2.2 Direct and inverse problems: explicit solutions; 2.3 Recovery of the Dirac system: general case; 2.3.1 Representation of the fundamental solution; 2.3.2 Weyl function: high energy asymptotics; 2.3.3 Inverse problem and Borg-Marchenko-type uniqueness theorem.
- 2.3.4 Weyl function and positivity of S3 Skew-self-adjoint Dirac system: rectangular matrix potentials; 3.1 Direct problem; 3.2 The inverse problem on a finite interval and semiaxis; 3.3 System with a locally bounded potential; 4 Linear system auxiliary to the nonlinear optics equation; 4.1 Direct and inverse problems; 4.1.1 Bounded potentials; 4.1.2 Locally bounded potentials; 4.1.3 Weyl functions; 4.1.4 Some generalizations; 4.2 Conditions on the potential and asymptotics of generalized Weyl (GW) functions; 4.2.1 Preliminaries. Beals-Coifman asymptotics.
- 4.2.2 Inverse problem and Borg-Marchenko-type result4.3 Direct and inverse problems: explicit solutions; 5 Discretesystems; 5.1 Discrete self-adjoint Dirac system; 5.1.1 Dirac system and Szegö recurrence; 5.1.2 Weyl theory: direct problems; 5.1.3 Weyl theory: inverse problems; 5.2 Discrete skew-self-adjoint Dirac system; 5.3 GBDT for the discrete skew-self-adjoint Dirac system; 5.3.1 Main results; 5.3.2 The fundamental solution; 5.3.3 Weyl functions: direct and inverse problems; 5.3.4 Isotropic Heisenberg magnet; 6 Integrable nonlinear equations.
- 6.1 Compatibility condition and factorization formula6.1.1 Main results; 6.1.2 Proof of Theorem 6.1; 6.1.3 Application to the matrix "focusing" modified Korteweg-de Vries (mKdV); 6.1.4 Second harmonic generation: Goursat problem; 6.2 Sine-Gordon theory in a semistrip; 6.2.1 Complex sine-Gordon equation: evolution of the Weyl function and uniqueness of the solution; 6.2.2 Sine-Gordon equation in a semistrip; 6.2.3 Unbounded solutions in the quarter-plane; 7 General GBDT theorems and explicit solutions of nonlinear equations; 7.1 Explicit solutions of the nonlinear optics equation.