The Lax representation for an integrable class of relativistic dynamical systems

We exhibit the Lax pair for the class of relativistic dynamical systems recently introduced by Ruijsenaars and Schneider, whose equations of motion read, whereB is an arbitrary constant and the Weierstrass elliptic function.For the 3 academic years 1983–1986     

Lax representation with spectral parameter on a torus for integrable particle systems

Complete integrability is proved for the most general class of systems of interacting particles on a straight line with the Hamiltonian including elliptic functions of coordinates, depending on seven arbitrary parameters and having the structure defined by the root systems of the classical Lie algebras. The Lax representation for them depends on the spectral parameter given on a complex torus /, where is the lattice of periods of the Jacobi functions dependent on the Hamiltonian parameters. The possibility of constructing explicit solutions to the equations of motion is discussed.     

Graded Lie algebras,representation theory,integrable mappings and systems. Non-abelian case

The exactly integrable systems connected with a semisimple series A for arbitrary grading are presented in an explicit form. Their general solutions are expressed in terms of the matrix elements of various fundamental representations of the An groups. The simplest example of such systems is the generalized Toda chain with matrices of arbitrary dimensions in each point of the lattice.     

Quantizations and integrable systems

Quantizations of parametrize realizations of universal dynamical systems whose integrability is proved and hamiltonian form established. For the case of the Moyal bracket, the analogs of the KdV equation, Boussinesq system, the first nontrivial flow in the KP hierarchy, and the KP equation, are computed.     

Contact flows and integrable systems

We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold–Liouville theorem: the system need not be integrable on the whole phase space, while the invariant hypersurface is foliated on an invariant Lagrangian tori. In the second part of the paper we consider contact systems with constraints. As an example, the Reeb flows on Brieskorn manifolds are considered.     

Dynamical simulation of integrable and nonintegrable models in the Heisenberg picture

The numerical simulation of quantum many-body dynamics is typically limited by the linear growth of entanglement with time. Recently numerical studies have shown that for 1D Bethe-integrable models the simulation of local operators in the Heisenberg picture can be efficient. Using the spin-1/2 XX chain as generic example of an integrable model that can be mapped to free fermions, we provide a simple explanation for this. We show furthermore that the same reduction of complexity applies to operators that have a high-temperature autocorrelation function which decays slower than exponential, i.e., with a power law. Thus efficient simulability may already be implied by a single conservation law as we will illustrate numerically for the spin-1 XXZ model.     

Ren-integrable and ren-symmetric integrable systems

A new type of symmetry, ren-symmetry, describing anyon physics and corresponding topological physics, is proposed. Ren-symmetry is a generalization of super-symmetry which is widely applied in super-symmetric physics such as super-symmetric quantum mechanics, super-symmetric gravity, super-symmetric string theory, super-symmetric integrable systems and so on. Supersymmetry and Grassmann numbers are, in some sense, dual conceptions, and it turns out that these conceptions coincide for the ren sit...     

Additional symmetries for integrable equations and conformal algebra representation

We present a regular procedure for constructing an infinite set of additional (spacetime variables explicitly dependent) symmetries of integrable nonlinear evolution equations (INEEs). In our method, additional symmetry equations arise together with their L-A pairs, so that they are integrable themselves. This procedure is based on a modified dressing method. For INEEs in 1+1 dimensions, some appropriate symmetry equations are shown to form the vector fields on a circle S ¹ algebra representation. In contrast to the so-called isospectral deformations, these symmetries result from conformal transformations of the associated linear problem spectrum. For INEEs in 2+1 dimensions, the commutation relations for symmetry equations are shown to coincide with operators , with integer m, p. Some additional results about Kac-Moody algebra applications are presented.     

Solitons and chaos in nearly integrable systems

We describe some properties of a two-dimensional non-linear mapping which has been derived from a differential equation model of turbulence. For certain parameter values the mapping displays chaotic behaviour. Owing to the peculiar way in which the phase space expands and contracts, parts of the attractor follow the oscillatory invariant curves associated with homoclinic points in non-integrable systems.     

Diffusion in multicomponent systems

Some consequences of the thermodynamics of irreversible processes on the diffusion coefficients are discussed. The corresponding inequalities for the diffusion coefficients are derived.Dedicated to Professor Miroslav Trlifaj on the occasion of his sixtieth birthday.     

Contrast in multicomponent systems

The forward scattering of radiation by multicomponent systems is expressed in terms of the chemical potentials and of the collision lengths associated with the components.Contrast lengths can be defined in various unequivalent ways. The formula giving the scattering cross section in terms of such contrasts is derived geometrically. This approach allows a discussion concerning the best definition of the contrast lengths.     

Deterministic diffusion in almost integrable systems

Diffusion processes are usually associated with randomness in the system. In this paper we show that deterministic diffusion processes can also occur in systems with zero entropy that mimic chaos with any precision without being mathematically chaotic. A random walk model is used to predict the behavior of the diffusion coefficient. (c) 1996 American Institute of Physics.     

Deformed Ginibre ensembles and integrable systems

We consider three Ginibre ensembles (real, complex and quaternion-real) with deformed measures and relate them to known integrable systems by presenting partition functions of these ensembles in form of fermionic expectation values. We also introduce double deformed Dyson–Wigner ensembles and compare their fermionic representations with those of Ginibre ensembles.     

Factorization in integrable systems with impurity

This article is based on recent works done in collaboration with M. Mintchev, E. Ragoucy and P. Sorba. It aims at presenting the latest developments in the subject of factorization for integrable field theories with a reflecting and transmitting impurity. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Hamiltonian group actions and integrable systems

The purpose of this paper is to show how some ideas from the theory of Hamiltonian systems with symmetry can be applied to dynamical systems like the nonperiodic Toda Lattice of Moser to yield striking information about their trajectories.Our presentation of these ideas is as self-contained as possible. We try to keep abstract machinery in its proper place, and emphasize the concrete computations which are implicit in some parts of modern symplectic geometry. Most of the results we shall derive are known in one form or another. The only novelties, perhaps, are some proofs and examples.     

Dynamical layer decoupling in a stripe-ordered high-T(c) superconductor

In the stripe-ordered state of a strongly correlated two-dimensional electronic system, under a set of special circumstances, the superconducting condensate, like the magnetic order, can occur at a nonzero wave vector corresponding to a spatial period double that of the charge order. In this case, the Josephson coupling between near neighbor planes, especially in a crystal with the special structure of La(2-x)Ba(x)CuO(4), vanishes identically. We propose that this is the underlying cause of the dynamical decoupling of the layers recently observed in transport measurements at x = 1/8.     

Classical integrable systems and gauge field theories

In this review we consider the Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimensions. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider the Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL (N, C) and explain the Symplectic Hecke correspondence for these systems. The text was submitted by the author in English.