Integrable Models of (1+1)-Dimensional Dilaton Gravity Coupled to Scalar Matter

We describe a class of explicitly integrable models of (1+1)-dimensional dilaton gravity coupled to scalar fields in sufficient detail. The equations of motion of these models reduce to systems of Liouville equations with energy and momentum constraints. We construct the general solution of the equations and constraints in terms of chiral moduli fields explicitly and briefly discuss some extensions of the basic integrable model. These models can be related to higher-dimensional supergravity theories, but we mostly consider them independently of such interpretations. We also briefly review other integrable models of two-dimensional dilaton gravity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 115–131, January, 2006.     

Unified description of cosmological and static solutions in affine generalized theories of gravity: Vecton-scalaron duality and its applications

We briefly describe the simplest class of affine theories of gravity in multidimensional space-times with symmetric connections and their reductions to two-dimensional dilaton-vecton gravity field theories. The distinctive feature of these theories is the presence of an absolutely neutral massive (or tachyonic) vector field (vecton) with an essentially nonlinear coupling to the dilaton gravity. We emphasize that the vecton field in dilaton-vecton gravity can be consistently replaced by a new effectively massive scalar field (scalaron) with an unusual coupling to the dilaton gravity. With this vecton-scalaron duality, we can use the methods and results of the standard dilaton gravity coupled to usual scalars in more complex dilaton-scalaron gravity theories equivalent to dilaton-vecton gravity. We present the dilaton-vecton gravity models derived by reductions of multidimensional affine theories and obtain one-dimensional dynamical systems simultaneously describing cosmological and static states in any gauge. Our approach is fully applicable to studying static and cosmological solutions in multidimensional theories and also in general one-dimensional dilaton-scalaron gravity models. We focus on general and global properties of the models, seeking integrals and analyzing the structure of the solution space. In integrable cases, it can be usefully visualized by drawing a “topological portrait” resembling the phase portraits of dynamical systems and simply exposing the global properties of static and cosmological solutions, including horizons, singularities, etc. For analytic approximations, we also propose an integral equation well suited for iterations.     

Integrable boundary conditions for (2+1)-dimensional models of mathematical physics

We consider the question of integrable boundary-value problems in the examples of the two-dimensional Toda chain and Kadomtsev-Petviashvili equation. We discuss the problems that are integrable from the standpoints of two basic definitions of integrability. As a result, we propose a method for constructing a hierarchy of integrable boundary-value problems where the boundaries are cylindric surfaces in the space of three variables. We write explicit formulas describing wide classes of solutions of these boundary-value problems for the two-dimensional Toda chain and Kadomtsev-Petviashvili equation.     

Dimensional reduction of gravity and relation between static states,cosmologies, and waves

We introduce generalized dimensional reductions of an integrable (1+1)-dimensional dilaton gravity coupled to matter down to one-dimensional static states (black holes in particular), cosmological models, and waves. An unusual feature of these reductions is that the wave solutions depend on two variables: space and time. They are obtained here both by reducing the moduli space (available because of complete integrability) and by a generalized separation of variables (also applicable to nonintegrable models and to higher-dimensional theories). Among these new wavelike solutions, we find a class of solutions for which the matter fields are finite everywhere in space-time, including infinity. These considerations clearly demonstrate that a deep connection exists between static states, cosmologies, and waves. We argue that it should also exist in realistic higher-dimensional theories. Among other things, we also briefly outline the relations existing between the low-dimensional models that we discuss here and the realistic higher-dimensional ones. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 422–452, December, 2007.     

Explicit solutions of an integrable boundary value problem for the two-dimensional toda lattice

We obtain an explicit solution of the integrable boundary value problem for the two-dimensional Toda lattice using the inverse scattering method. We interpret the integrability property in terms of the corresponding linear problem, the Gel’fand-Levitan-Marchenko equation, and the dressing procedure. The simplest initial solutions of the boundary value problem become new nontrivial solutions after the dressing procedure is applied.     

Integrable discretization and numerical simulations of the generalized coupled integrable dispersionless equations

In this paper, we study the generalized coupled integrable dispersionless (GCID) equations and construct two integrable discrete analogues including a semi-discrete system and a full-discrete one. The results are based on the relations among the GCID equations, the sine-Gordon equation and the two-dimensional Toda lattice equation. We also present the N-soliton solutions to the semi-discrete and fully discrete systems in the form of Casorati determinant. In the continuous limit, we show that the fully discrete GCID equations converge to the semi-discrete GCID equations, then further to the continuous GCID equations. By using the integrable semi-discrete system, we design two numerical schemes to the GCID equations and carry out several numerical experiments with solitons and breather solutions.     

System of equations for stimulated combination scattering and the related double periodic <Emphasis Type="Italic">A</Emphasis><Stack><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><Superscript>(1)</Superscript></Stack> Toda chains

We consider a system of equations describing stimulated combination scattering of light. We show that solutions of this system are expressed in terms of two solutions of the sine-Gordon equation that are related to each other by a Bäcklund transformation. We also show that this system is integrable and admits a Zakharov-Shabat pair. In the general case, the system of equations for the Bäcklund transformation of periodic A _n ⁽¹⁾ Toda chains is also shown to be integrable and to have a Zakharov-Shabat pair.     

Diagonalization of the system of static Lamé equations of isotropic linear elasticity

We find a simplest representation for the general solution to the system of the static Lamé equations of isotropic linear elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lamé equations. The system of Lamé equations diagonalizes; i.e., it is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and some formula for producing new solutions which makes it possible, given a solution, to find new solutions to the static Lamé equations by derivations. In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov-Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of the two-dimensional elliptic systems.     

Properties of the Fast Diffusion Equation and Its Multidimensional Exact Solutions

We prove invariance of the fast diffusion equation in the two-dimensional coordinate space and give its reduction to a one-dimensional analog in the space variable. Using these results, we construct new exact multidimensional solutions which depend on arbitrary harmonic functions. As a consequence, we obtain new exact solutions to the well-known Liouville equation, the stationary analog of the fast diffusion equation with a linear source. We consider some generalizations to the case of systems of quasilinear parabolic equations.     

New integrable systems as a limit of the elliptic SL(N, ?) top

We consider the scaling limit of an elliptic top. This limit is a combination of a scaling of the elliptic top variables, an infinite shift of the spectral parameter, and the trigonometric limit. We give general necessary constraints on the scaling of the variables and examples of such a degeneracy. A certain subclass of limit systems is integrable in the Liouville sense, which can also be shown directly.     

Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis

We study the integrable structure recently revealed in some classical problems in the theory of functions in one complex variable. Given a simply connected domain bounded by a simple analytic curve in the complex plane, we consider the conformal mapping problem, the Dirichlet boundary problem, and the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional in the family gives a formal solution of the above problems. These functions satisfy an infinite set of dispersionless Hirota equations and are therefore tau-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda chain. In addition to our previous studies, we show that within a more general definition of the moments, this connection pertains not to a particular solution of the Hirota equations but to the hierarchy itself.     

A family of integrable differential-difference equations and its Bargmann symmetry constraint

A family of integrable differential-difference equations is constructed through discrete zero curvature equation. The Hamiltonian structures of the resulting differential-difference equations are established by the discrete trace identity. The Bargmann symmetry constraint of the resulting family is presented. Under this symmetry constraint, every differential-difference equation in the resulting family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.     

ℤ-graded loop Lie algebras, loop groups, and Toda equations

We consider Toda equations associated with twisted loop groups. Such equations are specified by ℤ-gradings of the corresponding twisted loop Lie algebras. We discuss the classification of Toda equations related to twisted loop Lie algebras with integrable ℤ-gradings. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 451–476, March, 2008.     

SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS BY BINARY NONLINEARIZATION

In this paper, the translation of the Lax pairs of the Levi equations is presented. Then a symmetry constraint for the Levi equations is given by means of binary nonlinearization method. The spatial part and the temporal parts of the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrainted as finite dimensional Liouville integrable Hamiltonian systems. Finally, the involutive solutions of the Levi equations are presented.     

A NEW INTEGRABLE SYMPLECTIC MAP ASSOCIATED WITH A DISCRETE MATRIX SPECTRAL PROBLEM

A hierarchy of lattice soliton equations is derived from a discrete matrix spectral problem. It is shown that the resulting lattice soliton equations are all discrete Liouville integrable systems. A new integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonli-nearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resulting lattice soliton equations.     

Quasi-Frobenius Algebras and Their Integrable N-Parameter Deformations Generated by Compatible N×N Metrics of Constant Riemannian Curvature

We prove that the equations describing compatible N×N metrics of constant Riemannian curvature define a special class of integrable N-parameter deformations of quasi-Frobenius (in general, noncommutative) algebras. We discuss connections with open–closed two-dimensional topological field theories, associativity equations, and Frobenius and quasi-Frobenius manifolds. We conjecture that open–closed two-dimensional topological field theories correspond to a special class of integrable deformations of associative quasi-Frobenius algebras.     

Symmetries of Integrable Difference Equations on the Quad-Graph

This paper describes symmetries of all integrable difference equations that belong to the famous Adler–Bobenko–Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations.     

Matrix models, complex geometry, and integrable systems: I

We consider the simplest gauge theories given by one-and two-matrix integrals and concentrate on their stringy and geometric properties. We recall the general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of semiclassical integrable systems solved by constructing tau functions or prepotentials. We discuss the complex curves and tau functions of one-and two-matrix models in detail. [This article was written at the request of the Editorial Board. It is based on several lectures presented at schools of mathematical physics and talks at the conference “Complex Geometry and String Theory” and the Polivanov memorial seminar.] __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 163–228, May, 2006.     

Construction of a new class of quantum integrable inhomogeneous models

Integrable inhomogenous or impurity models are usually constructed by either shifting the spectral parameter in the Lax operator or using another representation of the spin algebra. We propose a more involved general method for such construction in which the Lax operator contains generators of a novel quadratic algebra, a generalization of the known quantum algebra. In forming the monodromy matrix, we can replace any number of the local Lax operators with different realizations of the underlying algebra, which can result in spin chains with nonspin impurities causing changed coupling across the impurity sites, as well as with impurities in the form of bosonic operators. Following the same idea, we can also generate integrable inhomogeneous versions of the generalized lattice sine-Gordon model, nonlinear Schrödinger equation, Liouville model, relativistic and nonrelativistic Toda chains, etc.