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KP or mKP: Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems
Boris A. Kupershmidt, University of Tennessee Space Institute, Tullahoma, TN
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Mathematical Surveys and Monographs
2000; 600 pp; hardcover
Volume: 78
ISBN-10: 0-8218-1400-1
ISBN-13: 978-0-8218-1400-0
List Price: US$126 Member Price: US$100.80
Order Code: SURV/78

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 analyzed extensively and motivations for the new notions are provided.

Graduate students, research mathematicians, and physicists interested in global analysis and analysis on manifolds.

Reviews

"This remarkable book ... is written so as to be completely self-contained with most arguments presented in great detail ... a remarkable account of a substantial body of work ... likely to remain a significant work of reference as the field continues to develop."

-- Mathematical Reviews

Continuous space-time
• The KP hierarchy
• The mKP hierarchy
• Between KP and mKP
• Noncommutative Lagrangian formalism
• Noncommutative Hamiltonian informalism
• mKP = m + KP
• The quasirelativistic KP hierarchy
• The second construction of integrals of the KP hierarchy
Discrete space-continuous time
• KP, then mKP
• The noncommutative differential-difference calculus
• The noncommutative Hamiltonian formalism over differential-difference
• Hamiltonian formalism for discrete integrable systems of KP and mKP types
• The Gibbons forms
• The hydrodynamical Ansatz
• Relativistic Toda lattice and related systems
Discrete space-time
• The idea of Lax representations and its discrete-time analog
• Systems of the KP type
• Systems of mKP type
• The Toda lattice, the relativistic Toda lattice, and related systems
Appendices
• Complexification of Hamiltonian systems
• Asymptotic expansions of Hamiltonian systems
• Variational calculus over noncommutative rings
• Hamiltonian correspondencies
• Covariant aspects of the Hamiltonian formalism
• Noncommutative solitons
• The noncommutative KP equation
• A list of scalar equations
• Open problems and conjectures