Glimpses of soliton theory: the algebra and geometry of nonlinear PDEs
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2010]
|
| Series: | Student mathematical library
volume 54 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvi, 304 Seiten Illustrationen, Diagramme |
| ISBN: | 9780821852453 |
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MARC
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| 245 | 1 | 0 | |a Glimpses of soliton theory |b the algebra and geometry of nonlinear PDEs |c Alex Kasman |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2010] | |
| 264 | 4 | |c © 2010 | |
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Record in the Search Index
| _version_ | 1804143816175779841 |
|---|---|
| adam_text | Titel: Glimpses of soliton theory
Autor: Kasman, Alex
Jahr: 2010
Contents
Preface ix
Chapter 1. Differential Equations 1
§1.1. Classification of Differential Equations 4
§1.2. Can we write solutions explicitly? 5
§1.3. Differential Equations as Models of Reality
and Unreality 7
§1.4. Named Equations 8
§1.5. When are two equations equivalent? 9
§1.6. Evolution in Time 12
Problems 18
Suggested Reading 22
Chapter 2. Developing PDE Intuition 23
§2.1. The Structure of Linear Equations 23
§2.2. Examples of Linear Equations 30
§2.3. Examples of Nonlinear Equations 35
Problems 41
Suggested Reading 43
Chapter 3. The Story of Solitons 45
§3.1. The Observation 45
§3.2. Terminology and Backyard Study 46
§3.3. A Less-than-enthusiastic Response 47
§3.4. The Great Eastern 49
§3.5. The KdV Equation 49
§3.6. Early 20th Century 52
VI
§3.7. Numerical Discovery of Solitons 53
§3.8. Hints of Nonlinearity 57
§3.9. Explicit Formulas for n-soliton Solutions 59
§3.10. Soliton Theory and Applications 60
§3.11. Epilogue
62
Problems 63
Suggested Reading
65
Chapter 4. Elliptic Curves and KdV Traveling Waves 67
§4.1. Algebraic Geometry 67
§4.2. Elliptic Curves and Weierstrass p-ftmctions 68
§4.3. Traveling Wave Solutions to the KdV Equation 84
Problems 91
Suggested Reading 93
Chapter 5. KdV n-Solitons 95
§5.1. Pure n-soliton Solutions 95
§5.2. A Useful Trick: The r-function 96
§5.3. Some Experiments 99
§5.4. Understanding the 2-soliton Solution 1°3
§5.5. General Remarks and Conclusions 109
Problems 109
Suggested Reading Hl
Chapter 6. Multiplying and Factoring Differential Operators 113
§6.1. Differential Algebra H3
§6.2. Factoring Differential Operators l21
§6.3. Almost Division l24
§6.4. Application to Solving Differential Equations ^
§6.5. Producing an ODO with a Specified Kernel ^
Problems 13
Suggested Reading 13*
Chapter 7. Eigerifunctions and Jsospectraiity 13*
§7.1. Isospectral Matrices 133
§7.2. Eigenfunctions and Differential Operators 138
§7.3. Dressing for Differential Operators 140
Problems 145
Suggested Reading 147
Chapter 8. Lax Form for KdV and Other Soliton Equations 149
§8.1. KdV in Lax Form 150
§8.2. Finding Other Soliton Equations 154
§8.3. Lax Equations Involving Matrices 159
§8.4. Connection to Algebraic Geometry 164
Problems 165
Suggested Reading 171
Chapter 9. The KP Equation and Bilinear KP Equation 173
§9.1. The KP Equation 173
§9.2. The Bilinear KP Equation 181
Problems 193
Suggested Reading 195
Chapter 10. The Grassmann Cone T2,a and the Bilinear KP 197
Equation
§10.1. Wedge Products 197
§10.2. Decomposability and the Plücker Relation 200
§10.3. The Grassmann Cone 172,4 as a Geometrie Object 203
§10.4. Büinear KP as a Plücker Relation 204
§10.5. Geometrie Objects in the Solution Spaces of
Nonlinear PDEs 209
Problems 215
Suggested Reading 217
Chapter 11. Pseudo-Differential Operators and the KP 219
Hierarchy
§11.1. Motivation 219
§11.2. The Algebra of Pseudo-Differential Operators 220
Vlll________________________________________________________________
§11.3. *DOs are Not Really Operators 224
§11.4. Application to Soliton Theory 225
Problems 232
Suggested Reading 234
Chapter 12. The Grassmann Cone Tk n and the Bilinear KP 235
Hierarchy
§12.1. Higher Order Wedge Products 235
§12.2. The Bilinear KP Hierarchy 240
Problems 246
Suggested Reading 248
Chapter 13. Concluding Remarks 251
§13.1. Soliton Solutions and their Applications 251
§13.2. Algebro-Geometric Structure of Soliton Equations 252
Appendix A. Mathematica Guide 257
§A.l. Basic Input 257
§A.2. Some Notation 259
§A.3. Graphics 263
§A.4. Matrices and Vectors 265
§A.5. Trouble Shooting: Common Problems and Errors 267
Appendix B. Complex Numbers 269
§B.l. Algebra with Complex Numbers 269
§B.2. Geometry with Complex Numbers 270
§B.3. Functions and Complex Numbers 272
Problems 274
Appendix C. Ideas for Independent Projects 275
References 289
Glossary of Symbols 297
Index 301
|
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV037215696 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:53:35Z |
| institution | BVB |
| isbn | 9780821852453 |
| language | English |
| lccn | 2010024820 |
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| publisher | American Mathematical Society |
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| spelling | Kasman, Alex 1967- Verfasser (DE-588)143264443 aut Glimpses of soliton theory the algebra and geometry of nonlinear PDEs Alex Kasman Providence, Rhode Island American Mathematical Society [2010] © 2010 xvi, 304 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Student mathematical library volume 54 Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Korteweg-de-Vries-Gleichung (DE-588)4287203-0 gnd rswk-swf Korteweg-de-Vries-Gleichung (DE-588)4287203-0 s Algebraische Geometrie (DE-588)4001161-6 s Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1637-9 Student mathematical library volume 54 (DE-604)BV013184751 54 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021129676&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kasman, Alex 1967- Glimpses of soliton theory the algebra and geometry of nonlinear PDEs Student mathematical library Partielle Differentialgleichung (DE-588)4044779-0 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Korteweg-de-Vries-Gleichung (DE-588)4287203-0 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4001161-6 (DE-588)4287203-0 |
| title | Glimpses of soliton theory the algebra and geometry of nonlinear PDEs |
| title_auth | Glimpses of soliton theory the algebra and geometry of nonlinear PDEs |
| title_exact_search | Glimpses of soliton theory the algebra and geometry of nonlinear PDEs |
| title_full | Glimpses of soliton theory the algebra and geometry of nonlinear PDEs Alex Kasman |
| title_fullStr | Glimpses of soliton theory the algebra and geometry of nonlinear PDEs Alex Kasman |
| title_full_unstemmed | Glimpses of soliton theory the algebra and geometry of nonlinear PDEs Alex Kasman |
| title_short | Glimpses of soliton theory |
| title_sort | glimpses of soliton theory the algebra and geometry of nonlinear pdes |
| title_sub | the algebra and geometry of nonlinear PDEs |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Korteweg-de-Vries-Gleichung (DE-588)4287203-0 gnd |
| topic_facet | Partielle Differentialgleichung Algebraische Geometrie Korteweg-de-Vries-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021129676&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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