ISBN-10:
3764326158
ISBN-13:
9783764326159
Pub. Date:
04/22/1992
Publisher:
Birkh�user Basel
Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems / Edition 1

Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems / Edition 1

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Product Details

ISBN-13: 9783764326159
Publisher: Birkh�user Basel
Publication date: 04/22/1992
Series: Progress in Mathematical Physics , #15
Edition description: 1992
Pages: 292
Product dimensions: 6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

Background of the theory of Lie algebras and Lie groups and their representations.- § 1.1 Lie algebras and Lie groups.- 1.1.1 Basic definitions.- 1.1.2 Contractions and deformations.- 1.1.3 Functional algebras.- § 1.2 ?-graded Lie algebras and their classification.- 1.2.1 Definitions.- 1.2.2 Semisimple, nilpotent and solvable Lie algebras. The Levi-Malcev theorem.- 1.2.3 Simple Lie algebras of finite growth: Classification and Dynkin-Coxeter diagrams.- 1.2.4 Root systems and the Weylgroup.- 1.2.5 A parametrization and ordering of roots of simple finite-dimensional Lie algebras.- 1.2.6 The real forms of complex simple Lie algebras.- § 1.3 sl(2)-subalgebras of Lie algebras.- 1.3.1 Embeddings of sl(2) into Lie algebras.- 1.3.2 Infinite-dimensional graded Lie algebras corresponding to embeddings of sl(2) into simple finite-dimensional Lie algebras.- 1.3.3 Explicit realization of simple finite-dimensional Lie algebras for the principal embedding of sl(2).- § 1.4 The structure of representations.- 1.4.1 Terminology.- 1.4.2 The adjoint representation.- 1.4.3 The regular representation and Casimir operators.- 1.4.4 Bases in the space of representation.- 1.4.5 Fundamental representations.- § 1.5 A parametrization of simple Lie groups.- § 1.6 The highest vectors of irreducible representations of semisimple Lie groups.- 1.6.1 Generalities.- 1.6.2 Expression for the highest matrix elements in terms of the adjoint representation.- 1.6.3 A formal expression for the highest matrix elements of the fundamental representations.- 1.6.4 Recurrence relations for the highest matrix elements of the fundamental representations.- 1.6.5 The highest matrix elements of irreducible representations expressed via generalized Euler angles.- § 1.7 Superalgebras and superspaces.- 1.7.1 Superspaces.- 1.7.2 Classical Lie superalgebras.- Representations of complex semisimple Lie groups and their real forms.- § 2.1 Infinitesimal shift operators on semisimple Lie groups.- 2.1.1 General expression of infinitesimal operators.- 2.1.2 The asymptotic domain.- § 2.2 Casimir operators and the spectrum of their eigenvalues.- 2.2.1 General formulation of the problem.- 2.2.2 Quadratic Casimir operators.- 2.2.3 Construction of Casimir operators for semisimple Lie groups.- § 2.3 Representations of semisimple Lie groups.- 2.3.1 Integral form of realization of operator-irreducible representations.- 2.3.2 The matrix elements of finite transformations.- § 2.4 Intertwining operators and the invariant bilinear form.- 2.4.1 Intertwining operators and problems of reducibility, equivalence and unitarity of representations.- 2.4.2 Construction of intertwining operators.- 2.4.3 The invariant Hermitian form.- § 2.5 Harmonic analysis on semisimple Lie groups.- 2.5.1 General method.- 2.5.2 Characters of operator-irreducible representations.- 2.5.3 Plancherel measure of the principal continuous series of unitary representations.- § 2.6 Whittaker vectors.- A general method of integrating two-dimensional nonlinear systems.- § 3.1 General method.- 3.1.1 Lax-type representation.- 3.1.2 Examples.- 3.1.3 Construction of solutions.- § 3.2 Systems generated by the local part of an arbitrary graded Lie algebra.- 3.2.1 Exactly integrable systems.- 3.2.2 Systems associated with infinite-dimensional Lie algebras.- 3.2.3 Hamiltonian formalism.- 3.2.4 Solutions of exactly integrable systems (Goursát problem).- § 3.3 Generalization for systems with fermionic fields.- § 3.4 Lax-type representation as a realization of self-duality of cylindrically-symmetric gauge fields.- Integration of nonlinear dynamical systems associated with finite-dimensional Lie algebras.- § 4.1 The generalized (finite nonperiodic) Toda lattice.- 4.1.1 Preliminaries.- 4.1.2 Construction of exact solutions on the base of the general scheme of Chapter 3.- 4.1.3 Examples.- 4.1.4 Construction of solutions without appealing to the Lax-type representation.- 4.1.4.1 Symmetry properties of the Toda lattice for the series A, B, C and the reduction procedure.- 4.1.4.2 Direct solution of the system (3.1.10) for the series A.- 4.1.4.3 Invariant generalization of the reduction scheme for arbitrary simple Lie algebras.- 4.1.5 The one-dimensional generalized Toda lattice.- 4.1.6 Boundary value problem (instantons and monopoles).- § 4.2 Complete integration of the two-dimensionalized system of Lotka-Volterra-type equations (difference KdV) as the Bäcklund transformation of the Toda lattice.- § 4.3 String-type systems (nonabelian versions of the Toda system).- § 4.4 The case of a generic Lie algebra.- § 4.5 Supersymmetric equations.- § 4.6 The formulation of the one-dimensional system (3.2.13) based on the notion of functional algebra.- Internal symmetries of integrable dynamical systems.- § 5.1 Lie-Bäcklund transformations. The characteristic algebra and defining equations of exponential systems.- § 5.2 Systems of type (3.2.8), their characteristic algebra and local integrals.- § 5.3 A complete description of Lie-Bäcklund algebras for the diagonal exponential systems of rank 2.- § 5.4 The Lax-type representation of systems (3.2.8) and explicit solution of the corresponding initial value (Cauchy) problem.- § 5.5 The Bäcklund transformation of the exactly integrable systems as a corollary of a contraction of the algebra of their internal symmetry.- § 5.6 Application of the methods of perturbation theory in the search for explicit solutions of exactly integrable systems (the canonical formalism).- § 5.7 Perturbation theory in the Yang-Feldmann formalism.- § 5.8 Methods of perturbation theory in the one-dimensional problem.- § 5.9 Integration of nonlinear systems associated with infinite-dimensional Lie algebras.- Scalar Lax-pairs and soliton solutions of the generalized periodic Toda lattice.- § 6.1 A group-theoretical meaning of the spectral parameter and the equations for the scalar LA-pair.- § 6.2 Soliton solutions of the sine-Gordon equation.- § 6.3 Generalized Bargmann potentials.- § 6.4 Soliton solutions for the vector representation of Ar.- Exactly integrable quantum dynamical systems.- § 7.1 The Hamiltonian (canonical) formalism and the Yang-Feldmann method.- § 7.2 Basics from perturbation theory.- § 7.3 One-dimensional generalized Toda lattice with fixed end-points.- 7.3.1 Schrödinger’s picture.- 7.3.2 Heisenberg’s picture (the canonical formalism).- 7.3.3 Heisenberg’s picture (Yang-Feldmann’s formalism).- § 7.4 The Liouville equation.- § 7.5 Multicomponent 2-dimensional models. 1.- § 7.6 Multicomponent 2-dimensional models. 2.- Afterword.

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