Inverse Problems and Nonlinear Evolution Equations: Solutions, Darboux Matrices and Weyl–Titchmarsh Functions

✍ Scribed by Alexander L. Sakhnovich; Lev A. Sakhnovich; Inna Ya. Roitberg

• Publisher: De Gruyter
• Year: 2013
• Tongue: English
• Leaves: 356
• Series: De Gruyter Studies in Mathematics; 47
• Category: Library

✦ Synopsis

This book is based on the method of operator identities and related theory of S-nodes, both developed by Lev Sakhnovich. The notion of the transfer matrix function generated by the S-node plays an essential role.

The authors present fundamental solutions of various important systems of differential equations using the transfer matrix function, that is, either directly in the form of the transfer matrix function or via the representation in this form of the corresponding Darboux matrix, when Bäcklund–Darboux transformations and explicit solutions are considered. The transfer matrix function representation of the fundamental solution yields solution of an inverse problem, namely, the problem to recover system from its Weyl function. Weyl theories of selfadjoint and skew-selfadjoint Dirac systems, related canonical systems, discrete Dirac systems, system auxiliary to the N-wave equation and a system rationally depending on the spectral parameter are obtained in this way.

The results on direct and inverse problems are applied in turn to the study of the initial-boundary value problems for integrable (nonlinear) wave equations via inverse spectral transformation method. Evolution of the Weyl function and solution of the initial-boundary value problem in a semi-strip are derived for many important nonlinear equations. Some uniqueness and global existence results are also proved in detail using evolution formulas.

The reading of the book requires only some basic knowledge of linear algebra, calculus and operator theory from the standard university courses.

✦ Table of 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
2 Self-adjoint Dirac system: rectangular matrix potentials
2.1 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 S
3 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 result
4.3 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 formula
6.1.1 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
7.2 GBDT for linear system depending rationally on z
7.3 Explicit solutions of nonlinear equations
8 Some further results on inverse problems and generalized Bäcklund-Darboux transformation (GBDT)
8.1 Inverse problems and the evolution of the Weyl functions
8.2 GBDT for one and several variables
9 Sliding inverse problems for radial Dirac and Schrödinger equations
9.1 Inverse and half-inverse sliding problems
9.1.1 Main definitions and results
9.1.2 Radial Schrödinger equation and quantum defect
9.1.3 Dirac equation and quantum defect
9.1.4 Proofs of Theorems 9.10 and 9.14
9.1.5 Dirac system on a finite interval
9.2 Schrödinger and Dirac equations with Coulomb-type potentials
9.2.1 Asymptotics of the solutions: Schrödinger equation
9.2.2 Asymptotics of the solutions: Dirac system
Appendices
A General-type canonical system: pseudospectral and Weyl functions
A.1 Spectral and pseudospectral functions
A.1.1 Basic notions and results
A.1.2 Description of the pseudospectral functions
A.1.3 Potapov’s inequalities and pseudospectral functions
A.1.4 Description of the spectral functions
A.2 Special cases
A.2.1 Positivity-type condition
A.2.2 Continuous analogs of orthogonal polynomials
B Mathematical system theory
C Krein’s system
D Operator identities corresponding to inverse problems
D.1 Operator identity: the case of self-adjoint Dirac system
D.2 Operator identity for skew-self-adjoint Dirac system
D.3 Families of positive operators
D.4 Semiseparable operators S
D.5 Operators with D-difference kernels
E Somebasictheorems
Bibliography
Index