KP or mKP : noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems / Boris A. Kupershmidt.

Format:

Book

Author/Creator:

Kupershmidt, Boris A., 1946- author.

Series:

Mathematical surveys and monographs ; no. 78.

Mathematical surveys and monographs, 0076-5376 ; volume 78

Language:

English

Subjects (All):

Hamiltonian systems.

Noncommutative differential geometry.

Physical Description:

1 online resource (623 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2000]

Language Note:

English

Summary:

This book develops a theory that can be viewed as a noncommutative counterpart of the following topics: dynamical systems in general and integrable systems in particular; Hamiltonian formalism; variational calculus, both in continuous space and discrete. The text is self-contained and includes a large number of exercises. Many different specific models are analysed extensively and motivations for the new notions are provided.

Contents:

""Contents""; ""Preface""; ""Acknowledgments""; ""Part A. Continuous Space Time""; ""Chapter 1. The KP Hierarchy""; ""Â1.1. The Basic Equations And Their First Properties""; ""Â1.2. Hamiltonian Formalism For The KP Hierarchy""; ""Â1.3. Quaternionic KP Hierarchy""; ""Â1.4. The KP Hierarchy With Values In Finite-Dimensional Associative Algebras""; ""Â1.5. Dressing Motions""; ""Chapter 2. The MKP Hierarchy""; ""Â2.1. Construction Of The Basic Equations And The Commutativity Of The Flows In The MKP Hierarchy""; ""Â2.2. The Hamiltonian Formalism For The MKP Hierarchy""

""Â2.3. The MKP Hierarchy With Values In Finite-Dimensional Associative Algebras""""Â2.4. The Equations Of Dispersive Water Waves""; ""Â2.5. The Burgers Hierarchy""; ""Â2.6. The Korteweg-De Vries Hierarchy""; ""Â2.7. The MKP Hierarchy Dressed Up""; ""Chapter 3. Between KP And MKP""; ""Â3.1. The Miura Map In The Language Of Lax Representations""; ""Â3.2. The Miura Map In The Language Of Wilson Equations""; ""Â3.3. The Miura Map From MKP To KP Is Hamiltonian""; ""Â3.4. From DWW To KP""; ""Â3.5. From Nonlinear Schrodinger To KP""; ""Â3.6. From Derivative NLS To MKP""

""Â3.7. Between DNLS And NLS""""Â3.8. Water Form Of Nonlinear Schrodinger""; ""Â3.9. The Real Miura Map Between The KdV And MkdV Hierarchies""; ""Â3.10. KP Factorized, Or MKP[sup(II)]""; ""Â3.11. P. S.: The Fully Nonabelian Miura Map Between The KPAnd Potential MKP Hierarchies Revisited And Found Perfectly Hamiltonian""; ""Chapter 4. Noncommutative Lagrangian Formalism""; ""Â4.1. Motivation Extracted From A Lobotomy Of The KdV Equation""; ""Â4.2. Variational Derivatives And Related Notions""; ""Â4.3. Transformation Formula For The Variational Derivatives""

""Â4.4. The Variational Complex""""Â4.5. The Residue Formula""; ""Â4.6. The Legendre Transform""; ""Â4.7. Localizations""; ""Chapter 5. Noncommutative Hamiltonian Formalism""; ""Â5.1. The Main Result Of The Hamiltonian Formalism""; ""Â5.2. Hamiltonian Maps""; ""Â5.3. Linear And Affine Hamiltonian Operators, Lie Algebras And Two-Cocycles""; ""Â5.4. The Local-Global Principle""; ""Â5.5. Hamiltonian Analog Of A Homomorphism Of Lie Algebras""; ""Chapter 6. MKP = M+ KP""; ""Â6.1. KP, MKP, KdV, And Other Equations as Noncommutative Hamiltonian Systems""

""Â6.2. Inverting The Noninvertible Miura Map Between The MKP And KP Hierarchies""""Â6.3. M2KP""; ""Â6.4. Clebsch Representations""; ""Â6.5. The Kontsevich-Type Formula""; ""Â6.6. The Third Hamiltonian Structure Of The MKP Hierarchy""; ""Chapter 7. The Quasirelativistic KP Hierarchy""; ""Â7.1. Construction Of Basic Equations And The Commutativity Of The Flows""; ""Â7.2. Hamiltonian Formalism For The Quasirelativistic Flows""; ""Â7.3. Quasirelativistic NLS Hierarchy""; ""Chapter 8. The Second Construction Of Integrals Of The KP Hierarchy""; ""Â8.1. The Wilson Formulae""

""Â8.2. The Cherednik-Flaschka Formulae""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 591-595) and index.

Description based on print version record.

ISBN:

1-4704-1305-1