Showing papers on "Integrable system published in 2002"

The Bethe-Ansatz for N=4 Super Yang-Mills

Abstract: We derive the one loop mixing matrix for anomalous dimensions in N=4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.

Classification of integrable equations on quad-graphs. The consistency approach

Abstract: A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three-dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three-leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three-dimensional integrable systems and to the quantum context are also discussed.

Integrable systems on quad-graphs

Abstract: Discrete (lattice) systems constitute a well-established part of the theory of integrable systems. They came up already in the early days of the theory (see, e.g. [11, 12]), and took gradually more and more important place in it (cf. a review in [18]). Nowadays many experts in the field agree that discrete integrable systems are in many respects even more fundamental than the continuous ones. They play a prominent role in various applications of integrable systems such as discrete differential geometry (see, e.g., a review in [9]). Traditionally, independent variables of discrete integrable systems are considered as belonging to a regular square lattice Z (or its multidimensional analogs Z). Only very recently, there appeared first hints on the existence of a rich and meaningful theory of integrable systems on nonsquare lattices and, more generally, on arbitrary graphs. The relevant publications are almost exhausted by [2, 3, 5, 6, 16, 20, 21, 22]. We define integrable systems on graphs as flat connections with the values in loop groups. This is very natural definition, and experts in discrete integrable systems will not only immediately accept it, but might even consider it trivial. Nevertheless, it crystallized only very recently, and seems not to appear in the literature before [3, 5, 6]. (It should be noted that a different framework for integrable systems on graphs is being developed by Novikov with collaborators [16, 20, 21].) We were led to considering such systems by our (with Hoffmann) investigations of circle patterns as objects of discrete complex analysis: in [5, 6] we demonstrated that certain classes of circle patterns with the combinatorics of regular hexagonal lattice

Stability of the Camassa-Holm solitons

Abstract: We consider the stability problem of the solitary wave solutions of a completely integrable equation that arises as a model for the unidirectional propagation of shallow water waves. We prove that the solitary waves possess the spectral properties of solitons and that their shapes are stable under small disturbances.

Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion

Abstract: We show that the methodology based on the generalized inverse scattering transform (IST) concept provides a systematic way to discover the novel exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain or absorption. The fundamental innovation of the present approach is to notice that it is possible both to allow for a variable spectral parameter with new dependent variables and to apply of the famous "moving in time focuses" concept of the self-focusing theory to the IST formalism. We show that for nonlinear optics this algorithm is a useful tool to design novel dispersion managed fiber transmission lines and soliton lasers. Fundamental soliton management regimes are predicted.

The matching conditions of controlled Lagrangians and IDA-passivity based control

Abstract: This paper discusses the matching conditions resulting from the controlled Lagrangians method and the interconnection and damping assignment passivity based control (IDA-PBC) method. Both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an Euler±Lagrange, respectively Hamiltonian, system. In the context of mechanical systems with symmetry, the original controlled Lagrangians method is reviewed, and an interpretation of the matching assumptions in terms of the matching of kinetic and potential energy is given. Secondly, both methods are applied to the general class of underactuated mechanical systems and it is shown that the controlled Lagrangians method is contained in the IDA-PBC method. The $\lambda$-method as described in recent papers for the controlled Lagrangians method, transforming the matching conditions (a set of non-linear PDEs) into a set of linear PDEs, is discussed. The method is used to transform the matching conditions obtained in the IDA-PBC method into a set of quadratic and linear PDEs. Finally, the extra freedom obtained in the IDA-PBC method (with respect to the controlled Lagrangians method) is used to discuss the integrability of the closed-loop system. Explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms.

A direct method for integrable couplings of TD hierarchy

Abstract: A direct method for establishing integrable couplings is proposed in this paper. As an example illustration, integrable couplings of TD hierarchy are obtained by constructing a suitable transformation of Lax pairs and a new Lie algebra.

Averaging and Vibrational Control of Mechanical Systems

Abstract: This paper investigates averaging theory and oscillatory control for a large class of mechanical systems. A link between averaging and controllability theory is presented by relating the key concepts of averaged potential and symmetric product. Both analysis and synthesis results are presented within a coordinate-free framework based on the theory of affine connections. The analysis focuses on characterizing the behavior of mechanical systems forced by high amplitude high frequency inputs. The averaged system is shown to be an affine connection system subject to an appropriate forcing term. If the input codistribution is integrable, the subclass of systems with Hamiltonian equal to "kinetic plus potential energy" is closed under the operation of averaging. This result precisely characterizes when the notion of averaged potential arises and how it is related to the symmetric product of control vector fields. Finally, a notion of vibrational stabilization for mechanical systems is introduced, and sufficient conditions are provided in the form of linear matrix equality and inequality tests.

Lectures on supersymmetric Yang-Mills theory and integrable systems

Abstract: We present a series of four self-contained lectures on the following topics; (I) An introduction to 4-dimensional 1 ≤ N ≤ 4 supersymmetric Yang-Mills theory, including particle and field contents, N = 1 and N = 2 superfield methods and the construction of general invariant Lagrangians; (II) A review of holomorphicity and duality in N = 2 super-Yang-Mills, of Seiberg-Witten theory and its formulation in terms of Riemann surfaces; (III) An introduction to mechanical Hamiltonian integrable systems; such as the Toda and Calogero-Moser systems associated with general Lie algebras; a review of the recently constructed Lax pairs with spectral parameter for twisted and untwisted elliptic Calogero-Moser systems; (IV) A review of recent solutions of the Seiberg-Witten theory for general gauge algebra and adjoint hypermultiplet content in terms of the elliptic Calogero-Moser integrable systems

Compatible Poisson Brackets on Lie Algebras

Abstract: We discuss the relationship between the representation of an integrable system as an L-A-pair with a spectral parameter and the existence of two compatible Hamiltonian representations of this system. We consider examples of compatible Poisson brackets on Lie algebras, as well as the corresponding integrable Hamiltonian systems and Lax representations.

Superconducting correlations in metallic nanoparticles: Exact solution of the BCS model by the algebraic Bethe ansatz

Abstract: Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.

Algebro-geometric solution of the 2+1 dimensional Burgers equation with a discrete variable

Abstract: The quasiperiodic solution of the 2+1 dimensional Burgers equation with a discrete variable is obtained through three steps: (a) decomposition into a symplectic map plus two finite-dimensional Hamiltonian systems; (b) straightening out of both the discrete and the continuous flows in the Jacobian variety; (c) inversion into the original variables. Inner relation with the modified Kadomtsev–Petviashvili equation is presented. The explicit theta function solutions for these two 2+1 integrable models are given.

Discrete dynamical systems with W(A(1)m-1 × A(1)n-1) symmetry

Abstract: We give a birational realization of affine Weyl group of type A(1)m−1 × A(1)n−1. We apply this representation to construct some discrete integrable systems and discrete Painleve equations. Our construction has a combinatorial counterpart through the ultra-discretization procedure.

A List of 1 + 1 Dimensional Integrable Equations and Their Properties

Abstract: This paper contains a list of known integrable systems. It gives their recursion-, Hamiltonian-, symplectic- and cosymplectic operator, roots of their symmetries and their scaling symmetry.

Integrable discretizations of derivative nonlinear Schrödinger equations

Abstract: We propose integrable discretizations of derivative nonlinear Schrodinger (DNLS) equations such as the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reductions, we obtain novel integrable discretizations of the nonlinear Schrodinger (NLS), modified KdV (mKdV), mixed NLS, matrix NLS, matrix KdV, matrix mKdV, coupled NLS, coupled Hirota, coupled Sasa–Satsuma and Burgers equations. We also discuss integrable discretizations of the sine-Gordon equation, the massive Thirring model and their generalizations.

Integrable systems in three-dimensional Riemannian geometry

Abstract: In this paper we introduce a new infinite-dimensional pencil of Hamiltonian structures. These Poisson tensors appear naturally as the ones governing the evolution of the curvatures of certain flows of curves in 3-dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arclength-preserving geometric evolutions for which the variation of the curve is an invariant combination of the tangent, normal, and binormal vectors. Under very natural conditions, the evolution of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian and completely integrable.

Classical R-matrix theory of dispersionless systems: II. (2 + 1) dimension theory

Abstract: A systematic way of constructing (2 + 1)-dimensional dispersionless integrable Hamiltonian systems is presented. The method is based on the so-called central extension procedure and classical R-matrix applied to the Poisson algebras of formal Laurent series. Results are illustrated with the known and new (2 + 1)-dimensional dispersionless systems.

Quantum versus classical integrability in Calogero-Moser systems

Abstract: Calogero–Moser systems are classical and quantum integrable multiparticle dynamics defined for any root system Δ. The quantum Calogero systems having 1/q2 potential and a confining q2 potential and the Sutherland systems with 1/sin2q potentials have 'integer' energy spectra characterized by the root system Δ. Various quantities of the corresponding classical systems, e.g. minimum energy, frequencies of small oscillations, the eigenvalues of the classical Lax pair matrices etc, at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems. To our surprise, most of these classical data are also 'integers', or they appear to be 'quantized'. To be more precise, these quantities are polynomials of the coupling constant(s) with integer coefficients. The close relationship between quantum and classical integrability in Calogero–Moser systems deserves fuller analytical treatment, which would lead to better understanding of these systems and of integrable systems in general.

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Abstract: We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of $\beta=2$ symmetry class.

Edge bifurcations for near integrable systems via evans function techniques

Abstract: When studying the linearstability of waves for near integrable systems, a fundamental problem is the location of the point spectrum of the linearized operator. Internal modes may be created upon the perturbation, i.e., eigenvalues may bifurcate out of the continuous spectrum, even if the corresponding eigenfunction is not initially localized. This phenomenon is also known as an edge bifurcation. It has recently been shown that the Evans function is a powerful tool when one wishes to detect an edge bifurcation and track the resulting eigenvalues. It has been an open question as to the role played by the solutions to the Lax pair, associated with the integrable problem, in the construction of the Evans function and the detection of edge bifurcations. Using the Zakharov--Shabat eigenvalue problem and the massive Thirring model as illustrations, we show the connection between the inverse scattering formalism and the linear stability analysis of waves. In particular, we show a direct connection between the sca...

Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems

Abstract: An n degree-of-freedom Hamiltonian system with r (1 independent first integrals which are in involution is called partially integrable Hamiltonian system and a partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi partially integrable Hamiltonian system. In the present paper, the averaged Ito and Fokker–Planck–Kolmogorov (FPK) equations for quasi partially integrable Hamiltonian systems in both cases of non-resonance and resonance are derived. It is shown that the number of averaged Ito equations and the dimension of the averaged FPK equation of a quasi partially integrable Hamiltonian system is equal to the number of independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. The technique to obtain the exact stationary solution of the averaged FPK equation is presented. The largest Lyapunov exponent of the averaged system is formulated, based on which the stochastic stability and bifurcation of original quasi partially integrable Hamiltonian systems can be determined. Examples are given to illustrate the applications of the proposed stochastic averaging method for quasi partially integrable Hamiltonian systems in response prediction and stability decision and the results are verified by using digital simulation.

Generalized r-matrix structure and algebro-geometric solution for integrable systems

Abstract: The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a generalized Lax matrix instead of usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Poisson bracket. It can be clearly seen that various nonlinear constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV, c-Toda, r-Toda, c-Levi, etc, are derived from the reduction of this structure. All these nonlinear systems have {\it r}-matrices, and are completely integrable in Liouville's sense. Furthermore, our generalized structure is developed to become an approach to obtain the algebro-geometric solutions of integrable NLEEs. Finally, the two typical examples are considered to illustrate this approach: the infinite or periodic Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary.

Perturbation theory for infinite-dimensional integrable systems on the line. A case study

Abstract: C o n t e n t s

Integrable Noncommutative Equations on Quad-graphs. The Consistency Approach

Abstract: We extend integrable systems on quad-graphs, such as the Hirota equation and the cross-ratio equation, to the noncommutative context, when the fields take values in an arbitrary associative algebra. We demonstrate that the three-dimensional consistency property remains valid in this case. We derive the noncommutative zero curvature representations for these systems, based on the latter property. Quantum systems with their quantum zero curvature representations are particular cases of the general noncommutative ones.

A Variable Separation Approach to Solve the Integrable and Nonintegrable Models: Coherent Structures of the (2+1)-Dimensional KdV Equation*

Abstract: We study the localized coherent structures of a generally nonintegrable (2+1)-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermore, in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.