On Bianchi permutability of Bäcklund transformations for asymmetric quad-equations

We prove the Bianchi permutability (existence of superposition principle) of Bäcklund transformations for asymmetric quad-equations. Such equations and their Bäcklund transformations form 3D consistent systems of a priori different equations. We perform this proof by using 4D consistent systems of quad-equations, the structural insights through biquadratics patterns and the consideration of super-consistent eight-tuples of quad-equations on decorated cubes.     

High-Dimensional Schwarzian Derivatives and Painlevé Integrable Models

Because all the known integrable models possess Schwarzian forms with Mobious transformation invariance,it may be one of the best ways to find new integrable models starting from some suitable Mobious transformation invariant equations. In this paper, we study the Painlevé integrability of some special (3+1)-dimensional Schwarzian models.     

Self-similar potentials and Ising models

A new link between soliton solutions of integrable nonlinear equations and one-dimensional Ising models is established. Translational invariance of the spin lattice associated with the KdV equation is related to self-similar potentials of the Schrödinger equation. This gives antiferromagnets with exponentially decaying interaction between the spins. The partition function is calculated exactly for a uniform magnetic field and two discrete values of the temperature.     

Integrable discrete autonomous quad-equations admitting,as generalized symmetries,known five-point differential-difference equations

In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations.     

Real Hamiltonian forms of Hamiltonian systems

We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a given involution. The resulting subspace is isomorphic (but not symplectomorphic) to the initial phase space. Thus to each real Hamiltonian system we are able to associate another nonequivalent (real) ones. A crucial role in this construction is played by the assumed analyticity and the invariance of the Hamiltonian under the involution. We show that if the initial system is Liouville integrable, then its complexification and its real forms will be integrable again and this provides a method of finding new integrable systems starting from known ones. We demonstrate our construction by finding real forms of dynamics for the Toda chain and a family of Calogero-Moser models. For these models we also show that the involution of the complexified phase space induces a Cartan-like involution of their Lax representations.Received: 8 October 2003, Published online: 8 June 2004PACS: 02.30.Ik Integrable systems - 45.20.Jj Lagrangian and Hamiltonian mechanics     

Differentiable probabilities: A new viewpoint on spin,gauge invariance,gauge fields,and relativistic quantum mechanics

A new approach to developing formulisms of physics based solely on laws of mathematics is presented. From simple, classical statistical definitions for the observed space-time position and proper velocity of a particle having a discrete spectrum of internal states we derive u generalized Schrödinger equation on the space-time manifold. This governs the evolution of an N component wave function with each component square integrable over this manifold and is structured like that for a charged particle in an electromagnetic field but also includes SU(N) gauge field couplings. This construction reveals a new hasis for gauge invariance and new insight into the appearance of spin and other such properties in relativistic quantum mechanics and suggests a new charged particle model.     

Test of CPT and Lorentz invariance from muonium spectroscopy

Following a suggestion from Kostelecky et al., we evaluated a test of CPT and Lorentz invariance from the microwave spectroscopy of muonium. Hamiltonian terms beyond the standard model violating CPT and Lorentz invariance would contribute frequency shifts deltanu(12) and deltanu(34) to nu(12) and nu(34), the two transitions involving muon spin flip, which were precisely measured in ground state muonium in a strong magnetic field of 1.7 T. The shifts would be indicated by anticorrelated oscillations in nu(12) and nu(34) at the Earth's sidereal frequency. No time dependence was found in nu(12) or nu(34) at the level of 20 Hz, limiting the size of some CPT and Lorentz-violating parameters at the level of 2x10(-23) GeV.     

Quantum groups and generalized statistics in integrable models

The paper deals with the integrable massive models of quantum field theory. It is shown that generalized statistics of physical particles is closely connected with the invariance under quantum groups. This invariance provides the possibility to construct quasi-local operators (parafermions) possessing generalized statistics which interpolates the physical particles. For the particular examples of SG, RSG models and scaling 3-state Potts model the parafermions are described completely (all their matrix elements in the space of states are presented).     

Euclidean invariance in statistical mechanics of classical continuous system

It is shown that equilibrium states of classical particles with short range interactions are Euclidean invariant whenever their correlation functions have a clustering which is integrable. The relation between invariance and clustering is analysed for spatial rotations and internal rotational degrees of freedom. The analysis is then extended to the case of long range interactions, including the Coulomb force and jellium systems.     

Gauge invariance and formal integrability of the Yang-Mills-Higgs equations

In the framework of the formal theory of overdetermined systems of partial differential equations, it is shown that the Yang-Mills-Higgs equations are an involutive, and hence formally integrable, system. To this end a key role is played by the gauge invariance of the theory and the resulting differential identities involving the field equations themselves. By applying a theorem of Malgrange, an existence theorem for the solutions of the Yang-Mills-Higgs field equations in the analytic context is thus obtained. The approach is within differential geometry.     

Exact solution of new integrable nineteen-vertex models and quantum spin-1 chains

New exactly solvable nineteen vertex models and related quantum spin-1 chains are solved. Partition functions, excitation energies, correlation lengths, and critical exponents are calculated. It is argued that one of the non-critical Hamiltonians is a realization of an integrable Haldane system. The finite-size spectra of the critical Hamiltonians deviate in their structure from standard predictions by conformal invariance.Work performed within the research program of the Sonderforschungsbereich 341, Köln-Aachen-Jülich     

Möbius invariant integrable lattice equations associated with KP and 2DTL hierarchies

The integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. They generate the corresponding continuous hierarchy of singular manifold equations, its Bäcklund transformations and different forms of superposition principles; their distinctive feature is invariance under the action of Möbius transformation. Geometric interpretation of these discrete equations is given.     

Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons

In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified Camassa-Holm equation pointing out that in the peakon sector (a family of non-smooth solitons) only one of them, namely the one obtained from the distributional compatibility condition, supports the time invariance of the Sobolev H¹ norm.     

A model of modulated diffusion. II. Numerical results on statistical properties

We investigate numerically the statistical properties of a model of modulated diffusion for which we have already computed analytically the diffusion coefficientD. Our model is constructed by adding a deterministic or random noise to the frequency of an integrable isochronous system. We consider in particular the central limit theorem and the invariance principle and we show that they follow wheneverD is positive and for any magnitude of the noise; we also investigate the asymptotic distribution in a case whenD=0.     

Supersymmetry and Integrability in Planar Mechanical Systems

We present an N=2-supersymmetric mechanical system whose bosonic sector, with two degrees of freedom, exhibits the most general possible supersymmetric fourth order potential, including the interesting case of SU(2) Yang–Mills theory. The Painlevé test is adopted to discuss integrability and we focus on the rôle of supersymmetry and parity invariance in two space dimensions for the attainment of integrable or non-integrable models, with some remarks on the chaotic behavior. Our result shows that, for the model studied here, the relationships among the parameters, as imposed by supersymmetry, restrict the parameter space in such a way that the reduction on its non-integrable sector is much more severe than on its integrable sector (especially on the non-separable subset of the latter), thus suggesting that supersymmetry may favor (mainly non-separable) integrability.     

New Integrable Models from the Gauge/YBE Correspondence

We introduce a class of new integrable lattice models labeled by a pair of positive integers N and r. The integrable model is obtained from the Gauge/YBE correspondence, which states the equivalence of the 4d $\mathcal {N} =1$ $S^{1}\times S^{3}/ \mathbb {Z} _{r}$ index of a large class of SU(N) quiver gauge theories with the partition function of 2d classical integrable spin models. The integrability of the model (star-star relation) is equivalent with the invariance of the index under the Seiberg duality. Our solution to the Yang-Baxter equation is one of the most general known in the literature, and reproduces a number of known integrable models. Our analysis identifies the Yang-Baxter equation with a particular duality (called the Yang-Baxter duality) between two 4d $\mathcal {N} =1$ supersymmetric quiver gauge theories. This suggests that the integrability goes beyond 4d lens indices and can be extended to the full physical equivalence among the IR fixed points.     

Bargmann Symmetry Constraint for a Family of Liouville Integrable Differential-Difference Equations

A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.     

On the Structure of the B¨acklund Transformations for the Relativistic Lattices

Abstract

The B¨acklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to distinguish effectively the integrable cases. Iterations of the B¨acklund transformations can be described in the terms of nonrelativistic lattices of the Toda type. Several multifield generalizations are presented.     

A Hierarchy of Integrable Lattice Soliton Equations and New Integrable Symplectic Map

Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Bäcklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.     

The time dependent solution of the masterequation for a laser system

Calculations of the cross sections for collision-induced transitions between Zeeman sublevels of excited alcali atoms are proposed, which are independent from dynamical models. 1. Supposing that the spin of the valence electron does not flip, the transition matrix, describing collision-induced deorientation of the orbital angular momentum, is analysed considering invariance principles only. Consequently the cross sections for depolarizing collisions are given by two dynamical parameters. 2. The full transition matrix, describing deorientation of the orbital angular momentum as well as of the spin of the valence electron, is analysed in the same way. In this general case the depolarizing cross sections are given by six dynamical parameters.