Inverse problems and nonlinear evolution equations: solutions, Darboux matrices and Weyl-Titchmarsh functions
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
De Gruyter
2013
|
| Series: | De Gruyter studies in mathematics
47 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XIII, 341 S. Ill. 25 cm |
| ISBN: | 9783110258608 9783112205006 |
Staff View
MARC
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| 100 | 1 | |a Sachnovič, Aleksandr L. |e Verfasser |0 (DE-588)1041634536 |4 aut | |
| 245 | 1 | 0 | |a Inverse problems and nonlinear evolution equations |b solutions, Darboux matrices and Weyl-Titchmarsh functions |c Alexander L. Sakhnovich ; Lev. A. Sakhnovich ; Inna Ya. Roitberg |
| 264 | 1 | |a Berlin [u.a.] |b De Gruyter |c 2013 | |
| 300 | |a XIII, 341 S. |b Ill. |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a De Gruyter studies in mathematics |v 47 | |
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| 650 | 0 | 7 | |a Darboux-Transformation |0 (DE-588)4273823-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtlineare Evolutionsgleichung |0 (DE-588)4221363-0 |2 gnd |9 rswk-swf |
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Record in the Search Index
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IMAGE 1
CONTENTS
PREFACE- VII
NOTATION - IX
0 INTRODUCTION 1
1 PRELIMINARIES 13
1.1 SIMPLE TRANSFORMATIONS AND EXAMPLES 13
1.1.1 DIRAC-TYPE SYSTEMS AS A SUBCLASS OF CANONICAL SYSTEMS 13 1.1.2
SCHRODINGER SYSTEMS AS A SUBCLASS OF CANONICAL SYSTEMS 18 1.1.3 GAUGE
TRANSFORMATIONS OF THE DIRAC SYSTEMS 19 1.2 .S -NODES AND WEYL FUNCTIONS
22
1.2.1 ELEMENTARY PROPERTIES OF S -NODES 22 1.2.2 CONTINUAL FACTORIZATION
-- 24 1.2.3 CANONICAL SYSTEMS AND REPRESENTATION OF THE 5-NODES 27
1.2.4 ASYMPTOTICS OF THE WEYL FUNCTIONS, A SPECIAL CASE 30 1.2.5
FACTORIZATION OF THE OPERATORS S 36
1.2.6 WEYL FUNCTIONS OF DIRAC AND SCHRODINGER SYSTEMS 38
2 SELF-ADJOINT DIRAC SYSTEM: RECTANGULAR MATRIX POTENTIALS 44 2.1 SQUARE
MATRIX POTENTIALS: SPECTRAL AND WEYL THEORIES 45 2.1.1 SPECTRAL AND WEYL
FUNCTIONS: DIRECT PROBLEM 45
2.1.2 SPECTRAL AND WEYL FUNCTIONS: INVERSE PROBLEM 48 2.2 WEYL THEORY
FOR DIRAC SYSTEM WITH A RECTANGULAR MATRIX POTENTIAL 2.2.1 DIRECT
PROBLEM-49 2.2.2 DIRECT AND INVERSE PROBLEMS: EXPLICIT SOLUTIONS 56
2.3 RECOVERY OF THE DIRAC SYSTEM: GENERAL CASE 61 2.3.1 REPRESENTATION
OF THE FUNDAMENTAL SOLUTION 62 2.3.2 WEYL FUNCTION: HIGH ENERGY
ASYMPTOTICS 66 2.3.3 INVERSE PROBLEM AND BORG-MARCHENKO-TYPE UNIQUENESS
THEOREM 69
2.3.4 WEYL FUNCTION AND POSITIVITY OF S 73
3 SKEW-SELF-ADJOINT DIRAC SYSTEM: RECTANGULAR MATRIX POTENTIALS 79 3.1
DIRECT PROBLEM 80
3.2 THE INVERSE PROBLEM ON A FINITE INTERVAL AND SEMIAXIS 83 3.3 SYSTEM
WITH A LOCALLY BOUNDED POTENTIAL 94
4 LINEAR SYSTEM AUXILIARY TO THE NONLINEAR OPTICS EQUATION 101 4.1
DIRECT AND INVERSE PROBLEMS 102
HTTP://D-NB.INFO/1021548499
IMAGE 2
XII - CONTENTS
4.1.1 BOUNDED POTENTIALS 102
4.1.2 LOCALLY BOUNDED POTENTIALS 106
4.1.3 WEYL FUNCTIONS 115
4.1.4 SOME GENERALIZATIONS 117
4.2 CONDITIONS ON THE POTENTIAL AND ASYMPTOTICS OF GENERALIZED WEYL (GW)
FUNCTIONS- 118 4.2.1 PRELIMINARIES. BEALS-COIFMAN ASYMPTOTICS 118 4.2.2
INVERSE PROBLEM AND BORG-MARCHENKO-TYPE RESULT 120 4.3 DIRECT AND
INVERSE PROBLEMS: EXPLICIT SOLUTIONS 123
5 DISCRETE SYSTEMS - 126 5.1 DISCRETE SELF-ADJOINT DIRAC SYSTEM 126
5.1.1 DIRAC SYSTEM AND SZEGO RECURRENCE 127 5.1.2 WEYL THEORY: DIRECT
PROBLEMS 130
5.1.3 WEYL THEORY: INVERSE PROBLEMS 138
5.2 DISCRETE SKEW-SELF-ADJOINT DIRAC SYSTEM 142 5.3 GBDT FOR THE
DISCRETE SKEW-SELF-ADJOINT DIRAC SYSTEM 156 5.3.1 MAIN RESULTS 157
5.3.2 THE FUNDAMENTAL SOLUTION 160
5.3.3 WEYL FUNCTIONS: DIRECT AND INVERSE PROBLEMS 164 5.3.4 ISOTROPIC
HEISENBERG MAGNET 171
6 INTEGRABLE NONLINEAR EQUATIONS 177 6.1 COMPATIBILITY CONDITION AND
FACTORIZATION FORMULA 178 6.1.1 MAIN RESULTS 178
6.1.2 PROOF OF THEOREM 6.1 179
6.1.3 APPLICATION TO THE MATRIX "FOCUSING" MODIFIED KORTEWEG-DE VRIES
(MKDV) 181
6.1.4 SECOND HARMONIC GENERATION: GOURSAT PROBLEM 185 6.2 SINE-GORDON
THEORY IN A SEMISTRIP 188
6.2.1 COMPLEX SINE-GORDON EQUATION: EVOLUTION OF THE WEYL FUNCTION AND
UNIQUENESS OF THE SOLUTION 189 6.2.2 SINE-GORDON EQUATION IN A SEMISTRIP
193 6.2.3 UNBOUNDED SOLUTIONS IN THE QUARTER-PLANE 207
7 GENERAL GBDT THEOREMS AND EXPLICIT SOLUTIONS OF NONLINEAR EQUATIONS
210 7.1 EXPLICIT SOLUTIONS OF THE NONLINEAR OPTICS EQUATION 210 7.2 GBDT
FOR LINEAR SYSTEM DEPENDING RATIONALLY ON Z 212
7.3 EXPLICIT SOLUTIONS OF NONLINEAR EQUATIONS 221
IMAGE 3
CONTENTS - XIII
8 SOME FURTHER RESULTS ON INVERSE PROBLEMS AND GENERALIZED
BACKLUND-DARBOUX TRANSFORMATION (GBDT) 230 8.1 INVERSE PROBLEMS AND THE
EVOLUTION OF THE WEYL FUNCTIONS 230 8.2 GBDT FOR ONE AND SEVERAL
VARIABLES 234
9 SLIDING INVERSE PROBLEMS FOR RADIAL DIRAC AND SCHRODINGER EQUATIONS
9.1 INVERSE AND HALF-INVERSE SLIDING PROBLEMS 242
9.1.1 MAIN DEFINITIONS AND RESULTS 242
9.1.2 RADIAL SCHRODINGER EQUATION AND QUANTUM DEFECT 248 9.1.3 DIRAC
EQUATION AND QUANTUM DEFECT 252 9.1.4 PROOFS OF THEOREMS 9.10 AND 9.14
256 9.1.5 DIRAC SYSTEM ON A FINITE INTERVAL 257
9.2 SCHRODINGER AND DIRAC EQUATIONS WITH COULOMB-TYPE POTENTIALS 9.2.1
ASYMPTOTICS OF THE SOLUTIONS: SCHRODINGER EQUATION 260 9.2.2 ASYMPTOTICS
OF THE SOLUTIONS: DIRAC SYSTEM 261
APPENDICES 265
A GENERAL-TYPE CANONICAL SYSTEM: PSEUDOSPECTRAL AND WEYL FUNCTIONS 267
A.L SPECTRAL AND PSEUDOSPECTRAL FUNCTIONS 268
A.L.L BASIC NOTIONS AND RESULTS 268
A.1.2 DESCRIPTION OF THE PSEUDOSPECTRAL FUNCTIONS 272 A.1.3 POTAPOV'S
INEQUALITIES AND PSEUDOSPECTRAL FUNCTIONS 283 A.1.4 DESCRIPTION OF THE
SPECTRAL FUNCTIONS 290 A.2 SPECIAL CASES 297
A.2.1 POSITIVITY-TYPE CONDITION 297
A.2.2 CONTINUOUS ANALOGS OF ORTHOGONAL POLYNOMIALS 301
B MATHEMATICAL SYSTEM THEORY 304
C KREIN'S SYSTEM 306
D OPERATOR IDENTITIES CORRESPONDING TO INVERSE PROBLEMS 308 D.L OPERATOR
IDENTITY: THE CASE OF SELF-ADJOINT DIRAC SYSTEM 309 D.2 OPERATOR
IDENTITY FOR SKEW-SELF-ADJOINT DIRAC SYSTEM 312 D.3 FAMILIES OF POSITIVE
OPERATORS 313
D.4 SEMISEPARABLE OPERATORS S 314
D.5 OPERATORS WITH D-DIFFERENCE KERNELS 317
E SOME BASIC THEOREMS-320
BIBLIOGRAPHY 323
INDEX 339
242
259 |
| any_adam_object | 1 |
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| author_GND | (DE-588)1041634536 (DE-588)121081265 |
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| author_role | aut aut aut |
| author_sort | Sachnovič, Aleksandr L. |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV041272348 |
| illustrated | Illustrated |
| indexdate | 2024-08-03T00:55:41Z |
| institution | BVB |
| isbn | 9783110258608 9783112205006 |
| language | English |
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| physical | XIII, 341 S. Ill. 25 cm |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | De Gruyter |
| record_format | marc |
| series | De Gruyter studies in mathematics |
| series2 | De Gruyter studies in mathematics |
| spelling | Sachnovič, Aleksandr L. Verfasser (DE-588)1041634536 aut Inverse problems and nonlinear evolution equations solutions, Darboux matrices and Weyl-Titchmarsh functions Alexander L. Sakhnovich ; Lev. A. Sakhnovich ; Inna Ya. Roitberg Berlin [u.a.] De Gruyter 2013 XIII, 341 S. Ill. 25 cm txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics 47 Literaturangaben Inverses Problem (DE-588)4125161-1 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Darboux-Transformation (DE-588)4273823-4 gnd rswk-swf Nichtlineare Evolutionsgleichung (DE-588)4221363-0 gnd rswk-swf Inverses Problem (DE-588)4125161-1 s Nichtlineare Evolutionsgleichung (DE-588)4221363-0 s Randwertproblem (DE-588)4048395-2 s Darboux-Transformation (DE-588)4273823-4 s DE-604 Sachnovič, Lev A. 1932- Verfasser (DE-588)121081265 aut Roitberg, Inna Ya. Verfasser aut Erscheint auch als Online-Ausgabe 978-3-11-025861-5 De Gruyter studies in mathematics 47 (DE-604)BV000005407 47 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4008441&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026245930&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sachnovič, Aleksandr L. Sachnovič, Lev A. 1932- Roitberg, Inna Ya Inverse problems and nonlinear evolution equations solutions, Darboux matrices and Weyl-Titchmarsh functions De Gruyter studies in mathematics Inverses Problem (DE-588)4125161-1 gnd Randwertproblem (DE-588)4048395-2 gnd Darboux-Transformation (DE-588)4273823-4 gnd Nichtlineare Evolutionsgleichung (DE-588)4221363-0 gnd |
| subject_GND | (DE-588)4125161-1 (DE-588)4048395-2 (DE-588)4273823-4 (DE-588)4221363-0 |
| title | Inverse problems and nonlinear evolution equations solutions, Darboux matrices and Weyl-Titchmarsh functions |
| title_auth | Inverse problems and nonlinear evolution equations solutions, Darboux matrices and Weyl-Titchmarsh functions |
| title_exact_search | Inverse problems and nonlinear evolution equations solutions, Darboux matrices and Weyl-Titchmarsh functions |
| title_full | Inverse problems and nonlinear evolution equations solutions, Darboux matrices and Weyl-Titchmarsh functions Alexander L. Sakhnovich ; Lev. A. Sakhnovich ; Inna Ya. Roitberg |
| title_fullStr | Inverse problems and nonlinear evolution equations solutions, Darboux matrices and Weyl-Titchmarsh functions Alexander L. Sakhnovich ; Lev. A. Sakhnovich ; Inna Ya. Roitberg |
| title_full_unstemmed | Inverse problems and nonlinear evolution equations solutions, Darboux matrices and Weyl-Titchmarsh functions Alexander L. Sakhnovich ; Lev. A. Sakhnovich ; Inna Ya. Roitberg |
| title_short | Inverse problems and nonlinear evolution equations |
| title_sort | inverse problems and nonlinear evolution equations solutions darboux matrices and weyl titchmarsh functions |
| title_sub | solutions, Darboux matrices and Weyl-Titchmarsh functions |
| topic | Inverses Problem (DE-588)4125161-1 gnd Randwertproblem (DE-588)4048395-2 gnd Darboux-Transformation (DE-588)4273823-4 gnd Nichtlineare Evolutionsgleichung (DE-588)4221363-0 gnd |
| topic_facet | Inverses Problem Randwertproblem Darboux-Transformation Nichtlineare Evolutionsgleichung |
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| volume_link | (DE-604)BV000005407 |
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