Author | Leznov, A. N. author |
---|---|
Title | Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems [electronic resource] / by A. N. Leznov, M. V. Saveliev |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1992 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8638-3 |
Descript | XVII, 292 p. online resource |
Background of the theory of Lie algebras and Lie groups and their representations -- ยง 1.1 Lie algebras and Lie groups -- ยง 1.2 ?-graded Lie algebras and their classification -- ยง 1.3 sl(2)-subalgebras of Lie algebras -- ยง 1.4 The structure of representations -- ยง 1.5 A parametrization of simple Lie groups -- ยง 1.6 The highest vectors of irreducible representations of semisimple Lie groups -- ยง 1.7 Superalgebras and superspaces -- Representations of complex semisimple Lie groups and their real forms -- ยง 2.1 Infinitesimal shift operators on semisimple Lie groups -- ยง 2.2 Casimir operators and the spectrum of their eigenvalues -- ยง 2.3 Representations of semisimple Lie groups -- ยง 2.4 Intertwining operators and the invariant bilinear form -- ยง 2.5 Harmonic analysis on semisimple Lie groups -- ยง 2.6 Whittaker vectors -- A general method of integrating two-dimensional nonlinear systems -- ยง 3.1 General method -- ยง 3.2 Systems generated by the local part of an arbitrary graded Lie algebra -- ยง 3.3 Generalization for systems with fermionic fields -- ยง 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 -- ยง 4.1 The generalized (finite nonperiodic) Toda lattice -- ยง 4.2 Complete integration of the two-dimensionalized system of Lotka-Volterra-type equations (difference KdV) as the Bรคcklund transformation of the Toda lattice -- ยง 4.3 String-type systems (nonabelian versions of the Toda system) -- ยง 4.4 The case of a generic Lie algebra -- ยง 4.5 Supersymmetric equations -- ยง 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 -- ยง 5.1 Lie-Bรคcklund transformations. The characteristic algebra and defining equations of exponential systems -- ยง 5.2 Systems of type (3.2.8), their characteristic algebra and local integrals -- ยง 5.3 A complete description of Lie-Bรคcklund algebras for the diagonal exponential systems of rank 2 -- ยง 5.4 The Lax-type representation of systems (3.2.8) and explicit solution of the corresponding initial value (Cauchy) problem -- ยง 5.5 The Bรคcklund transformation of the exactly integrable systems as a corollary of a contraction of the algebra of their internal symmetry -- ยง 5.6 Application of the methods of perturbation theory in the search for explicit solutions of exactly integrable systems (the canonical formalism) -- ยง 5.7 Perturbation theory in the Yang-Feldmann formalism -- ยง 5.8 Methods of perturbation theory in the one-dimensional problem -- ยง 5.9 Integration of nonlinear systems associated with infinite-dimensional Lie algebras -- Scalar Lax-pairs and soliton solutions of the generalized periodic Toda lattice -- ยง 6.1 A group-theoretical meaning of the spectral parameter and the equations for the scalar LA-pair -- ยง 6.2 Soliton solutions of the sine-Gordon equation -- ยง 6.3 Generalized Bargmann potentials -- ยง 6.4 Soliton solutions for the vector representation of Ar -- Exactly integrable quantum dynamical systems -- ยง 7.1 The Hamiltonian (canonical) formalism and the Yang-Feldmann method -- ยง 7.2 Basics from perturbation theory -- ยง 7.3 One-dimensional generalized Toda lattice with fixed end-points -- ยง 7.4 The Liouville equation -- ยง 7.5 Multicomponent 2-dimensional models. 1 -- ยง 7.6 Multicomponent 2-dimensional models. 2 -- Afterword