| Description |
1 online resource (356 pages). |
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text file |
| Series |
De Gruyter Studies in Mathematics
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De Gruyter studies in mathematics.
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| Contents |
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. |
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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. |
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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. |
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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. |
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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. |
| Note |
7.2 GBDT for linear system depending rationally on z. |
| Summary |
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 function in a form due to Lev A. Sakhnovich. The book primarily addresses specialists in the field. However, it is self-contained andstarts with preliminaries and examples, and hencealso serves as an introduction for advanced graduate students in the field. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Inverse problems (Differential equations)
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Inverse problems (Differential equations) |
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Evolution equations, Nonlinear.
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Evolution equations, Nonlinear. |
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Darboux transformations.
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Darboux transformations. |
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Boundary value problems.
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Boundary value problems. |
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Matrices.
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Matrices. |
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Functions.
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Functions. |
| Genre/Form |
Electronic books.
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Electronic books.
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| Added Author |
Sakhnovich, Alexander L.
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Roitberg, Inna Ya.
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| Other Form: |
Print version: Sakhnovich, Lev A. Inverse Problems and Nonlinear Evolution Equations : Solutions, Darboux Matrices and Weyl-Titchmarsh Functions. Berlin : De Gruyter, ©2013 9783110258608 |
| ISBN |
9783110258615 (electronic book) |
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3110258617 (electronic book) |
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