Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems

The book reviews a large number of 1- and 2-dimensional equations that describe nonlinear phenomena in various areas of modern theoretical and mathematical physics. It is meant, above all, for physicists who specialize in the field theory and physics of elementary particles and plasma, for mathe­ maticians dealing with nonlinear differential equations, differential geometry, and algebra, and the theory of Lie algebras and groups and their representa­ tions, and for students and post-graduates in these fields. We hope that the book will be useful also for experts in hydrodynamics, solid-state physics, nonlinear optics electrophysics, biophysics and physics of the Earth. The first two chapters of the book present some results from the repre­ sentation theory of Lie groups and Lie algebras and their counterpart on supermanifolds in a form convenient in what follows. They are addressed to those who are interested in integrable systems but have a scanty vocabulary in the language of representation theory. The experts may refer to the first two chapters only occasionally. As we wanted to give the reader an opportunity not only to come to grips with the problem on the ideological level but also to integrate her or his own concrete nonlinear equations without reference to the literature, we had to expose in a self-contained way the appropriate parts of the representation theory from a particular point of view.

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.