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Bibliographic Details
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2009
Series:	Lecture Notes in Physics 767
Subjects:	Integrables System Integrierbarkeit
Online Access:	Inhaltsverzeichnis
Physical Description:	XIII, 339 S.
ISBN:	9783540881100 9783540881117

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MARC


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adam_text	CONTENTS INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A.V. MIKHAILOV REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1 SYMMETRIES OF DIFFERENTIAL EQUATIONS AND THE PROBLEM OF INTEGRABILITY 19 A.V. MIKHAILOV AND V.V. SOKOLOV 1.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2 SYMMETRIES AND FIRST INTEGRALS OF FINITE-DIMENSIONAL DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 BASIC CONCEPTS OF THE SYMMETRY APPROACH . . . . . . . . . . . . . . . . . . . 32 1.4 MODIFICATIONS AND GENERALIZATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.5 SHORT DESCRIPTION OF SOLVED CLASSIFICATION PROBLEMS AND REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2 NUMBER THEORY AND THE SYMMETRY CLASSIFICATION OF INTEGRABLE SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 J.A. SANDERS AND J.P. WANG 2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.2 THE SYMBOLIC METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.3 AN IMPLICIT FUNCTION THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.4 SYMMETRY-INTEGRABLE EVOLUTION EQUATIONS . . . . . . . . . . . . . . . . . . . . 98 2.5 EVOLUTION SYSTEMS WITH K COMPONENTS . . . . . . . . . . . . . . . . . . . . . . . 105 2.6 ONE SYMMETRY DOES NOT IMPLY INTEGRABILITY . . . . . . . . . . . . . . . . . . . 108 2.7 CONCLUDING REMARKS, OPEN PROBLEMS AND FURTHER DEVELOPMENT . . 113 2.8 SOME IRREDUCIBILITY RESULTS BY F. BEUKERS . . . . . . . . . . . . . . . . . . . . . 114 2.9 THE FILTERED LIE ALGEBRA VERSION OF THE IMPLICIT FUNCTION THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3 FOUR LECTURES: DISCRETIZATION AND INTEGRABILITY. DISCRETE SPECTRAL SYMMETRIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 S.P. NOVIKOV 3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 XI XII CONTENTS 3.2 CONTINUOUS AND DISCRETE SPECTRAL SYMMETRIES OF 1D SYSTEMS AND SPECTRAL THEORY OF OPERATORS. 1D CONTINUOUS SCHR¨ ODINGER OPERATOR AND ITS DISCRETE ANALOGUE . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.3 2D SCHR¨ ODINGER OPERATOR. DISCRETE SPECTRAL SYMMETRIES, SPECTRAL THEORY OF THE SELECTED ENERGY LEVEL AND SPACE/LATTICE DISCRETIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.4 DISCRETIZATION OF THE 2D SCHR¨ ODINGER OPERATORS AND LAPLACE TRANSFORMATIONS ON THE SQUARE AND EQUILATERAL LATTICES . . . . . . . . . 128 3.5 2D MANIFOLDS WITH THE COLORED BLACKWHITE TRIANGULATION. INTEGRABLE SYSTEMS ON A TRIVALENT TREE. . . . . . . . . . . . . . . . . . . . . . . . 133 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4 SYMMETRIES OF SPECTRAL PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A. SHABAT 4.1 LIE-TYPE SYMMETRIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.2 DISCRETE SYMMETRIES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5 NORMAL FORM AND SOLITONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Y. HIRAOKA AND Y. KODAMA 5.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.2 PERTURBED KDV EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.3 CONSERVED QUANTITIES AND N -SOLITON SOLUTIONS . . . . . . . . . . . . . . . . . 180 5.4 SYMMETRY AND THE PERTURBED EQUATION . . . . . . . . . . . . . . . . . . . . . . . . 184 5.5 NORMAL FORM THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.6 INTERACTIONS OF SOLITARY WAVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.7 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6 MULTISCALE EXPANSION AND INTEGRABILITY OF DISPERSIVE WAVE EQUATIONS . 215 A. DEGASPERIS 6.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.2 NONLINEAR SCHR¨ ODINGER-TYPE MODEL EQUATIONS AND INTEGRABILITY . . 225 6.3 HIGHER ORDER TERMS AND INTEGRABILITY . . . . . . . . . . . . . . . . . . . . . . . . . 234 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 7 PAINLEV´ E TESTS, SINGULARITY STRUCTURE AND INTEGRABILITY . . . . . . . . . . . . . . 245 A.N.W. HONE 7.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.2 PAINLEV´ E ANALYSIS FOR ODES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.3 THE ABLOWITZRAMANISEGUR CONJECTURE . . . . . . . . . . . . . . . . . . . . . 254 7.4 THE WEISSTABORCARNEVALE PAINLEV´ E TEST . . . . . . . . . . . . . . . . . . . . 257 7.5 TRUNCATION TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.6 WEAK PAINLEV´ E TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 7.7 OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 CONTENTS XIII 8 HIROTAS BILINEAR METHOD AND ITS CONNECTION WITH INTEGRABILITY . . . . . . . 279 J. HIETARINTA 8.1 WHY THE BILINEAR FORM? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8.2 FROM NONLINEAR TO BILINEAR (KDV) . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 8.3 MULTISOLITON SOLUTIONS FOR THE KDV CLASS . . . . . . . . . . . . . . . . . . . . . 283 8.4 SOLITON SOLUTION FOR THE MKDV AND SG CLASS . . . . . . . . . . . . . . . . . . 290 8.5 THE NONLINEAR SCHR¨ ODINGER EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . 294 8.6 HIERARCHIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 8.7 BILINEAR B¨ ACKLUND TRANSFORMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 8.8 THE THREE-SOLITON CONDITION AS AN INTEGRABILITY TEST . . . . . . . . . . . . 306 8.9 FROM BILINEAR TO NONLINEAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 8.10 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 9 INTEGRABILITY OF THE QUANTUM XXZ HAMILTONIAN . . . . . . . . . . . . . . . . . . . . 315 T. MIWA 9.1 INTEGRABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 9.2 SYMMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
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spelling	Integrability A. V. Mikhailov (ed.) Berlin [u.a.] Springer 2009 XIII, 339 S. txt rdacontent n rdamedia nc rdacarrier Lecture Notes in Physics 767 Integrables System (DE-588)4114032-1 gnd rswk-swf Integrierbarkeit (DE-588)4474751-2 gnd rswk-swf Integrables System (DE-588)4114032-1 s DE-604 Integrierbarkeit (DE-588)4474751-2 s Mikhailov, Alexander 1953- Sonstige (DE-588)136866492 oth Lecture Notes in Physics 767 (DE-604)BV000003166 767 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017024684&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Integrability Lecture Notes in Physics Integrables System (DE-588)4114032-1 gnd Integrierbarkeit (DE-588)4474751-2 gnd
subject_GND	(DE-588)4114032-1 (DE-588)4474751-2
title	Integrability
title_auth	Integrability
title_exact_search	Integrability
title_full	Integrability A. V. Mikhailov (ed.)
title_fullStr	Integrability A. V. Mikhailov (ed.)
title_full_unstemmed	Integrability A. V. Mikhailov (ed.)
title_short	Integrability
title_sort	integrability
topic	Integrables System (DE-588)4114032-1 gnd Integrierbarkeit (DE-588)4474751-2 gnd
topic_facet	Integrables System Integrierbarkeit
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017024684&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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