Showing papers on "Integrable system published in 2006"

The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics

Abstract: The asymmetric simple exclusion process (ASEP) plays the role of a paradigm in non-equilibrium statistical mechanics. We review exact results for the ASEP obtained by the Bethe ansatz and put emphasis on the algebraic properties of this model. The Bethe equations for the eigenvalues of the Markov matrix of the ASEP are derived from the algebraic Bethe ansatz. Using these equations we explain how to calculate the spectral gap of the model and how global spectral properties such as the existence of multiplets can be predicted. An extension of the Bethe ansatz leads to an analytic expression for the large deviation function of the current in the ASEP that satisfies the Gallavotti–Cohen relation. Finally, we describe some variants of the ASEP that are also solvable by the Bethe ansatz.

Quantization of models with non-compact quantum group symmetry: Modular XXZ magnet and lattice sinh-Gordon model

Abstract: We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of ) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R-matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called -operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous spacetime. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.

Spiky strings and giant magnons on S5

Abstract: Recently, classical solutions for strings moving in AdS5 x S5 have played an important role in understanding the AdS/CFT correspondence. A large set of them were shown to follow from an ansatz that reduces the solution of the string equations of motion to the study of a well-known integrable 1-d system known as the Neumann-Rosochatius (NR) system. However, other simple solutions such as spiky strings or giant magnons in S5 were not included in the NR ansatz. We show that, when considered in the conformal gauge, these solutions can be also accomodated by a version of the NR-system. This allows us to describe in detail a giant magnon solution with two additional angular momenta and show that it can be interpreted as a superposition of two magnons moving with the same speed. In addition, we consider the spin chain side and describe the corresponding state as that of two bound states in the infinite SU(3) spin chain. We construct the Bethe ansatz wave function for such bound state.

On a Camassa-Holm type equation with two dependent variables

Abstract: We consider a generalization of the Camassa–Holm (CH) equation with two dependent variables, called CH2, introduced in a paper by Liu and Zhang (Liu S-Q and Zhang Y 2005 J. Geom. Phys. 54 427–53). We briefly provide an alternative derivation of it based on the theory of Hamiltonian structures on (the dual of) a Lie algebra. The Lie algebra involved here is the same algebra as underlies the NLS hierarchy. We study the structural properties of the hierarchy defined by the CH2 equation within the bi-Hamiltonian theory of integrable PDEs, and provide its Lax representation. Then we explicitly discuss how to construct classes of solutions, both of peakon and of algebro-geometrical type. Finally we sketch the construction of a class of singular solutions, defined by setting to zero one of the two dependent variables.

A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces

Abstract: In this paper, we give a KAM theorem for a class of infinite dimensional nearly integrable Hamiltonian systems. The theorem can be applied to some Hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi–periodic solutions and its local Birkhoff normal form. The applications to the higher dimensional beam equations and the higher dimensional Schrodinger equations with nonlocal smooth nonlinearity are also given in this paper.

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour

Abstract: Hamiltonian perturbations of the simplest hyperbolic equation ut + a(u) ux = 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE.

Bethe eigenvectors of higher transfer matrices

Abstract: We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra . The models are defined on a tensor product of irreducible -modules. For each model, there exist N one-parameter families of commuting operators on , called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz, are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.

Spiky strings and giant magnons on S 5

Abstract: Recently, classical solutions for strings moving in AdS5 × S5 have played an important role in understanding the AdS/CFT correspondence. A large set of them were shown to follow from an ansatz that reduces the solution of the string equations of motion to the study of a well-known integrable 1-d system known as the Neumann-Rosochatius (NR) system. However, other simple solutions such as spiky strings or giant magnons in S5 were not included in the NR ansatz. We show that, when considered in the conformal gauge, these solutions can be also accomodated by a version of the NR-system. This allows us to describe in detail a giant magnon solution with two additional angular momenta and show that it can be interpreted as a superposition of two magnons moving with the same speed. In addition, we consider the spin chain side and describe the corresponding state as that of two bound states in the infinite SU(3) spin chain. We construct the Bethe ansatz wave function for such bound state.

Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

Abstract: The tetrahedron equation is a three-dimensional generalization of the Yang–Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this 'hidden third dimension'. In this paper, we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra , where the rank n coincides with the size of the hidden dimension. These models are related to an anisotropic deformation of the sl(n)-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact 'rank-size' duality relation for the nested Bethe Ansatz solution for these models. Note also that the above solution of the tetrahedron equation arises in the quantization of the 'resonant three-wave scattering' model, which is a well-known integrable classical system in 2 + 1 dimensions.

Exact soliton solutions and nonlinear modulation instability in spinor Bose-Einstein condensates

Abstract: We find one-, two-, and three-component solitons of the polar and ferromagnetic (FM) types in the general (non-integrable) model of a spinor (three-component) model of the Bose-Einstein condensate (BEC), based on a system of three nonlinearly coupled Gross-Pitaevskii equations. The stability of the solitons is studied by means of direct simulations, and, in a part, analytically, using linearized equations for small perturbations. Global stability of the solitons is considered by means of the energy comparison. As a result, ground-state and metastable soliton states of the FM and polar types are identified. For the special integrable version of the model, we develop the Darboux transformation (DT). As an application of the DT, analytical solutions are obtained that display full nonlinear evolution of the modulational instability (MI) of a continuous-wave (CW) state seeded by a small spatially periodic perturbation. Additionally, by dint of direct simulations, we demonstrate that solitons of both the polar and FM types, found in the integrable system, are structurally stable, i.e., they are robust under random changes of the relevant nonlinear coefficient in time.

Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.

Abstract: The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg--de Vries and nonlinear Schr\"odinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.

Yang-Baxter maps and symmetries of integrable equations on quad-graphs

Abstract: A connection between the Yang-Baxter relation for maps and the multidimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations. We use the classification results by Adler, Bobenko, and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multifield integrable equations.

Long-range gl(n) Integrable Spin Chains and Plane-Wave Matrix Theory

Abstract: Quantum spin chains arise naturally from perturbative large-N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbor type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of gl(n). We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our results to plane-wave matrix theory and show that the Hamiltonian in a closed sector is not of this form and therefore not integrable beyond the first perturbative order. This also implies that the complete model is not integrable.

Coupled KdV equations derived from two-layer fluids

Abstract: Some types of coupled Korteweg de-Vries (KdV) equations are derived from a two-layer fluid system. In the derivation procedure, an unreasonable y-average trick (usually adopted in the literature) is removed. The derived models are classified by means of the Painleve test. Three types of τ-function and multiple soliton solutions of the models are explicitly given via the exact solutions of the usual KdV equation. It is also discovered that a non-Painleve integrable coupled KdV system can have multiple soliton solutions.

Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei

Abstract: Quantum phase transitions affecting the structure of ground and excited states of integrable systems with the Mexican-hat type potential are shown to be related to a singular torus of classical orbits passing the point of unstable equilibrium. As a specific example, we consider nuclear collective vibrations described by the O(6)–U(5) transitional Hamiltonian of the interacting boson model. While all states with zero values of the O(5) invariant undergo a continuous phase transition when crossing the energy of unstable equilibrium, the other states evolve in an analytic way.

A simple and unified approach to identify integrable nonlinear oscillators and systems

Abstract: In this paper, we consider a generalized second-order nonlinear ordinary differential equation (ODE) of the form x+(k1xq+k2)x+k3x2q+1+k4xq+1+λ1x=0, where ki’s, i=1,2,3,4, λ1, and q are arbitrary parameters, which includes several physically important nonlinear oscillators such as the simple harmonic oscillator, anharmonic oscillator, force-free Helmholtz oscillator, force-free Duffing and Duffing–van der Pol oscillators, modified Emden-type equation and its hierarchy, generalized Duffing–van der Pol oscillator equation hierarchy, and so on, and investigate the integrability properties of this rather general equation. We identify several new integrable cases for arbitrary value of the exponent q,q∊R. The q=1 and q=2 cases are analyzed in detail and the results are generalized to arbitrary q. Our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature. T...

On the thermodynamic Bethe ansatz equation in the sinh-Gordon model

Abstract: Two implicit periodic structures in the solution of the sinh-Gordon thermodynamic Bethe ansatz (TBA) equation are considered The analytic structure of the solution as a function of complex θ is studied to some extent both analytically and numerically The results hint at how the CFT integrable structures can be relevant in the sinh-Gordon and staircase models More motivations are figured out for subsequent studies of the massless sinh-Gordon (ie Liouville) TBA equation

Construction of mappings for Hamiltonian systems and their applications

Abstract: Basics of Hamiltonian Mechanics- Perturbation Theory for Nearly Integrable Systems- Mappings for Perturbed Systems- Method of Canonical Transformation for Constructing Mappings- Mappings Near Separatrix Theory- Mappings Near Separatrix Examples- The KAM Theory Chaos Nontwist and Nonsmooth Maps- Rescaling Invariance of Hamiltonian Systems Near Saddle Points- Chaotic Transport in Stochastic Layers- Magnetic Field Lines in Fusion Plasmas- Mapping of Field Lines in Ergodic Divertor Tokamaks- Mappings of Magnetic Field Lines in Poloidal Divertor Tokamaks- Miscellaneous

New integrable system of 2dim fermions from strings on AdS(5) x S**5

Abstract: We consider classical superstrings propagating on AdS5 × S5 space-time. We consistently truncate the superstring equations of motion to the so-called (1|1) sector. By fixing the uniform gauge we show that physical excitations in this sector are described by two complex fermionic degrees of freedom and we obtain the corresponding lagrangian. Remarkably, this lagrangian can be cast in a two-dimensional Lorentz-invariant form. The kinetic part of the lagrangian induces a non-trivial Poisson structure while the hamiltonian is just the one of the massive Dirac fermion. We find a change of variables which brings the Poisson structure to the canonical form but makes the hamiltonian nontrivial. The hamiltonian is derived as an exact function of two parameters: the total S5 angular momentum J and string tension λ; it is a polynomial in 1/J and in (λ')1/2 where λ' = λ/J2 is the effective BMN coupling. We identify the string states dual to the gauge theory operators from the closed (1|1) sector of = 4 SYM and show that the corresponding near-plane wave energy shift computed from our hamiltonian perfectly agrees with that recently found in the literature. Finally we show that the hamiltonian is integrable by explicitly constructing the corresponding Lax representation.

Birkhoff Normal Form and Hamiltonian PDEs

Abstract: These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some perturbations technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centered on the Birkhoff normal form theorem that we first proved in finite dimension. Then, after giving some exemples of Hamiltonian PDEs, we present an abstract Birkhoff normal form theorem in infinite dimension and discuss the dynamical consequences for Hamiltonian PDEs.

Exact solution of the open XXZ chain with general integrable boundary terms at roots of unity

Abstract: We propose a Bethe-ansatz-type solution of the open spin-1/2 integrable XXZ quantum spin chain with general integrable boundary terms and bulk anisotropy values iπ/(p+1), where p is a positive integer. All six boundary parameters are arbitrary, and need not satisfy any constraint. The solution is in terms of generalized T–Q equations, having more than one Q function. We find numerical evidence that this solution gives the complete set of 2N transfer matrix eigenvalues, where N is the number of spins.

Algebraic Representation of Correlation Functions in Integrable Spin Chains

Abstract: Taking the XXZ chain as the main example, we give a review of an algebraic representation of correlation functions in integrable spin chains obtained recently. We rewrite the previous formulas in a form which works equally well for the physically interesting homogeneous chains. We discuss also the case of quantum group invariant operators and generalization to the XYZ chain.

Partial preservation of frequencies in KAM theory

Abstract: We consider perturbations of moderately degenerate integrable or partially integrable Hamiltonian systems, so that unperturbed invariant n-tori with prescribed frequencies or frequency ratios do not persist, but there is preservation of, say, the first d < n frequencies or their ratios. Lagrangian and lower dimensional tori are treated in a unified way. The proofs are very simple and follow Herman's idea of 1990: we introduce external parameters to remove degeneracies and then eliminate these parameters making use of a suitable number-theoretical lemma concerning Diophantine approximations of dependent quantities. Parallel results for reversible, volume preserving and dissipative systems are also presented.