Algebraic and spectral methods for nonlinear wave equations:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Harlow, Essex
Longman
1990
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Pitman monographs and surveys in pure and applied mathematics
49 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 417 S. graph. Darst. |
| ISBN: | 0582037204 |
Internformat
MARC
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| 100 | 1 | |a Asano, Naruyoshi |d 1940- |e Verfasser |0 (DE-588)1145717993 |4 aut | |
| 245 | 1 | 0 | |a Algebraic and spectral methods for nonlinear wave equations |c N. Asano and Y. Kato |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Harlow, Essex |b Longman |c 1990 | |
| 300 | |a XVII, 417 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Pitman monographs and surveys in pure and applied mathematics |v 49 | |
| 650 | 7 | |a Equation d'onde - Solutions numériques |2 ram | |
| 650 | 7 | |a Equations aux dérivées partielles - Solutions numériques |2 ram | |
| 650 | 7 | |a Ondes nonlinéaires |2 ram | |
| 650 | 4 | |a Differential equations, Partial |x Numerical solutions | |
| 650 | 4 | |a Inverse scattering transform | |
| 650 | 4 | |a Nonlinear waves | |
| 650 | 4 | |a Wave equation |x Numerical solutions | |
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Datensatz im Suchindex
| _version_ | 1804118540919242752 |
|---|---|
| adam_text | Contents
Preface x
Introduction xi
1 Point Transformation 1
1.1 Local coordinate transformation 1
One parameter transformation group 1
Multi parameter group 3
Invariant and partially invariant manifolds 7
1.2 Prolongation of the point transformation 12
Point transformation in jet space 12
Differential invariants 19
1.3 Analysis of PDEs and their solutions 22
Invariant groups for PDEs 23
Transformations for the evolution equations 25
Point transformation for equivalent PDEs 32
Invariant and partially invariant solutions 33
Further reading 40
2 Higher Order Transformation 41
2.1 Point transformation on Jk(M,N) 41
Tangent transformation on Jk(M,N) 41
Contact transformation 45
2.2 Lie Backlund transformation 47
Transformation on f (M,N) 48
Formal groups 52
Invariant PDEs 54
Isomorphism for equivalent systems 56
Examples of LB groups 59
2.3 Backlund transformation I 62
A formulation of the BT 62
Evolution equations of second order 68
2.4 Examples and applications of BT 73
Examples of BT 73
Applications of BT 76
Further reading 80
v
CONTENTS
3 Recursion Operator and Lax Pair 82
3.1 Recursion operator and Lax representation 82
Recursion operator for symmetry 82
Examples of recursion operators 84
Symmetry algebra for the isentropic flow equation 86
3.2 Construction of recursion operator 91
Algebras 91
Conditions for the existence 97
Construction of the operators 99
3.3 Recursion operator for matrix evolution equations 107
Evolution equations 107
Hereditary property of L 110
3.4 The Wahlquist Estabrook method 115
Differential forms 115
WE prolongation method 119
3.5 Backlund transformation II 125
BT via the recursion operator 125
BT via the gauge transformation 130
Further reading 135
4 Variational and Hamiltonian Formlaisms 137
4.1 Variational calculus 137
Euler Lagrange equation 137
Image of Eu ler operator 140
4.2 Conservation laws and symmetries 144
Conservation law 144
Conservation laws and symmetries 146
4.3 Hamiltonian Formalism 150
Poisson bracket 150
Determination of H 153
Symmetry and conservation law 158
Further reading 162
5 Inverse Scattering Method 163
5.1 Lax theory and the Zakharov Shabat Ablowitz Kaup Newell Segur
system 163
Solitons and the Lax theory 163
Zakharov Shabat Ablowitz Kaup Newell Segur system 165
5.2 Jost solutions and the scattering matrix 168
Zakharov Shabat equation 169
vi
CONTENTS
Integral equations for the Zakharov Shabat equation 170
Classes of matrix functions 174
Jost solutions 175
Scattering matrix of the Jost solutions 179
5.3 Inverse scattering problem 183
Marchenko equation 183
Inverse scattering tran sf orm 188
Scattering data 189
Kernel of the Marchenko equation 193
Solution of the Marchenko equation 196
5.4 Inverse scattering method 200
Evolution of the scattering data 200
Algorithm of the inverse scattering method 206
Appearance of singularity in the solution 208
Solitons 209
Further reading 210
6 Inverse Scattering and Related Methods 211
6.1 Solutions of the Korteweg de Vries equation 211
Jost solutions 211
Wronskians of the Jost solutions 213
Zeros of the Wronskian 214
Marchenko equation 217
Solution of the Marchenko equation 220
Evolution of the scattering data 226
Solutions of the KdV equation 228
6.2 Bilinear form method 232
Bilinear equation 232
Soliton solution 233
General solution 238
6.3 Method of the Riemann Hilbert problem 241
Formulation of the Riemann Hilbert problem 242
Reduction of the Riemann Hilbert problem 244
Riemann Hilbert problem and the Marchenko equation 248
Right and left inverse scattering problems 249
6.4 Potentials and the scattering data 252
Completion in the inverse scattering transform 252
Mapping in spectral problems 259
6.5Spectralanalysis and the Marchenko equation 261
Resolvent and the spectrum 261
vii
CONTENTS
Expansion theorem 264
Spectral density and the scattering data 267
Expansion theorem and the Marchenko equation 269
6.6 Dressing method and d problem 272
Dressing method 273
Kadomtsev Petviashvili equation 277
Nonlocal Riemann Hilbert problem and 3 problem 280
Further reading 285
7 Waves with Finite or Periodic Boundary Value 288
7.1 Spectral problem of the Zakharov Shabat operator with finite tail
potential 288
Riemann surface 289
Jost solution 294
Wronskians of the Jost solutions 297
Resolvent 298
Expansion theorem 301
Parseval equation 306
Step potential 307
7.2 Inverse scattering method for finite asymptotic boundary values 310
Simliarity transformations 310
Marchenko equation 314
Solution of the Marchenko equation 321
Theorems of the inverse scattering problem 323
Mapping between potentials and scattering data 328
Inverse scattering method 330
7.3 Periodic boundary value problem of the Korteweg de Vries equation 330
Spectrum of the Schrodinger equation 331
Finite band potentials 336
Inverse spectral problem 338
Evolution of finite band potentials 342
The Jacobi inversion problem and the Riemann 0 function 344
Further reading 350
8 The Fredholm Theory and the N x N Zakharov Shabat operator 352
8.1 Fredholm integral equations 352
Integral operators 352
Convergences and regularity 354
Solution of the integral equation 357
Trace formulae 360
viii
CONTENTS
8.2 Scattering problem of the N x N Zakharov Shabat equation 363
Jost solutions and the Fredholm theory 365
Modified Jost solution 369
Fredholm determinants 373
S matrix andT matrix 375
8.3 Spectral problem of the N x N Zakharov Shabat equation 378
Asymptotic forms of the Fredholm determinants 379
Eigenfunctions of the transposed operator 382
Point spectrum and eigenfunctions 384
Resolvent 386
Expansion theorem 388
Transformation of the matrix x and the point spectrum 390
Inverse scattering problem and the inverse scattering method 390
Further reading 391
References 393
Index 411
ix
|
| any_adam_object | 1 |
| author | Asano, Naruyoshi 1940- Kato, Yusuke 1929- |
| author_GND | (DE-588)1145717993 (DE-588)141422203 |
| author_facet | Asano, Naruyoshi 1940- Kato, Yusuke 1929- |
| author_role | aut aut |
| author_sort | Asano, Naruyoshi 1940- |
| author_variant | n a na y k yk |
| building | Verbundindex |
| bvnumber | BV004360163 |
| callnumber-first | Q - Science |
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| callnumber-sort | QA 3927 |
| callnumber-subject | QA - Mathematics |
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| classification_tum | MAT 225f PHY 013f MAT 354f |
| ctrlnum | (OCoLC)20294485 (DE-599)BVBBV004360163 |
| dewey-full | 530.1/24 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.1/24 |
| dewey-search | 530.1/24 |
| dewey-sort | 3530.1 224 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV004360163 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:11:56Z |
| institution | BVB |
| isbn | 0582037204 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002707279 |
| oclc_num | 20294485 |
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| owner | DE-12 DE-91 DE-BY-TUM DE-384 DE-739 DE-634 DE-83 DE-11 DE-188 |
| owner_facet | DE-12 DE-91 DE-BY-TUM DE-384 DE-739 DE-634 DE-83 DE-11 DE-188 |
| physical | XVII, 417 S. graph. Darst. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Longman |
| record_format | marc |
| series | Pitman monographs and surveys in pure and applied mathematics |
| series2 | Pitman monographs and surveys in pure and applied mathematics |
| spelling | Asano, Naruyoshi 1940- Verfasser (DE-588)1145717993 aut Algebraic and spectral methods for nonlinear wave equations N. Asano and Y. Kato 1. publ. Harlow, Essex Longman 1990 XVII, 417 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pitman monographs and surveys in pure and applied mathematics 49 Equation d'onde - Solutions numériques ram Equations aux dérivées partielles - Solutions numériques ram Ondes nonlinéaires ram Differential equations, Partial Numerical solutions Inverse scattering transform Nonlinear waves Wave equation Numerical solutions Inverses Streuproblem (DE-588)4027547-4 gnd rswk-swf Nichtlineare Wellengleichung (DE-588)4042104-1 gnd rswk-swf Kontinuierliche Gruppe (DE-588)4165160-1 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Nichtlineare Wellengleichung (DE-588)4042104-1 s Kontinuierliche Gruppe (DE-588)4165160-1 s DE-604 Inverses Streuproblem (DE-588)4027547-4 s Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s Kato, Yusuke 1929- Verfasser (DE-588)141422203 aut Pitman monographs and surveys in pure and applied mathematics 49 (DE-604)BV000022446 49 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002707279&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Asano, Naruyoshi 1940- Kato, Yusuke 1929- Algebraic and spectral methods for nonlinear wave equations Pitman monographs and surveys in pure and applied mathematics Equation d'onde - Solutions numériques ram Equations aux dérivées partielles - Solutions numériques ram Ondes nonlinéaires ram Differential equations, Partial Numerical solutions Inverse scattering transform Nonlinear waves Wave equation Numerical solutions Inverses Streuproblem (DE-588)4027547-4 gnd Nichtlineare Wellengleichung (DE-588)4042104-1 gnd Kontinuierliche Gruppe (DE-588)4165160-1 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4027547-4 (DE-588)4042104-1 (DE-588)4165160-1 (DE-588)4128900-6 |
| title | Algebraic and spectral methods for nonlinear wave equations |
| title_auth | Algebraic and spectral methods for nonlinear wave equations |
| title_exact_search | Algebraic and spectral methods for nonlinear wave equations |
| title_full | Algebraic and spectral methods for nonlinear wave equations N. Asano and Y. Kato |
| title_fullStr | Algebraic and spectral methods for nonlinear wave equations N. Asano and Y. Kato |
| title_full_unstemmed | Algebraic and spectral methods for nonlinear wave equations N. Asano and Y. Kato |
| title_short | Algebraic and spectral methods for nonlinear wave equations |
| title_sort | algebraic and spectral methods for nonlinear wave equations |
| topic | Equation d'onde - Solutions numériques ram Equations aux dérivées partielles - Solutions numériques ram Ondes nonlinéaires ram Differential equations, Partial Numerical solutions Inverse scattering transform Nonlinear waves Wave equation Numerical solutions Inverses Streuproblem (DE-588)4027547-4 gnd Nichtlineare Wellengleichung (DE-588)4042104-1 gnd Kontinuierliche Gruppe (DE-588)4165160-1 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Equation d'onde - Solutions numériques Equations aux dérivées partielles - Solutions numériques Ondes nonlinéaires Differential equations, Partial Numerical solutions Inverse scattering transform Nonlinear waves Wave equation Numerical solutions Inverses Streuproblem Nichtlineare Wellengleichung Kontinuierliche Gruppe Nichtlineare partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002707279&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000022446 |
| work_keys_str_mv | AT asanonaruyoshi algebraicandspectralmethodsfornonlinearwaveequations AT katoyusuke algebraicandspectralmethodsfornonlinearwaveequations |