Showing papers on "Integrable system published in 1999"

A family of complex potentials with real spectrum

Abstract: We consider a two-parameter non-Hermitian quantum mechanical Hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations. Numerical investigation shows that for some values of the potential parameters the Hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.

Spectral theory of canonical differential systems : method of operator identities

Abstract: 1 Factorization of Operator-valued Transfer Functions- 11 Realization of operator-valued functions- 12 A factorization method- 13 Factorization of rational operator-valued functions- 2 Operator Identities and S-Nodes- 21 Elementary properties of S-nodes- 22 Symmetric S-nodes- 23 Inherited properties of factors- 3 Continual Factorization- 31 The main continual factorization theorem- 32 Bounded operator-valued functions- 4 Spectral Problems on the Half-line- 41 Basic notions of spectral theory- 42 Direct and inverse spectral problems- 43 Livsic-Brodski? nodes and the spectral theory of canonical systems- 5 Spectral Problems on the Line- 51 Spectral data of a canonical system- 52 Spectral problems and S-nodes- 53 The inverse spectral problem- 6 Weyl-Titchmarsh Functions of Periodic Canonical Systems- 61 Multipliers and their behavior- 62 Weyl-Titchmarsh functions- 63 Singular points of the Weyl-Titchmarsh matrix function- 7 Division of Canonical Systems into Subclasses- 71 An effective solution of the inverse problem- 72 Two principles of dividing a class of canonical systems into subclasses- 8 Uniqueness Theorems- 81 Monodromy matrix and uniqueness theorems- 82 Spectral data and uniqueness theorems- 9 Weyl Discs and Points- 91 Basic notions- 92 Symmetric operators and deficiency indices- 93 Weyl-Titchmarsh matrix functions on the line- 94 Weyl-Titchmarsh matrix function of a system with shifted argument- 10 A Class of Canonical Systems- 101 Asymptotic formulas- 102 Spectral analysis- 103 Transformed canonical systems- 104 Dirac-type systems- 105 An inverse problem- 106 On the limit Titchmarsh-Weyl function- 11 Classical Spectral Problems- 111 Generalized string equation (direct spectral problem)- 112 Matrix Sturm-Liouville equation (direct spectral problem)- 113 Inverse spectral problem- 12 Nonlinear Integrable Equations and the Method of the Inverse Spectral Problem- 121 Evolution of the spectral data- 122 Some classical nonlinear equations- 123 On the unique solvability of the mixed problem- 124 A hierarchy of nonlinear equations and asymptotic behavior of Weyl-Titchmarsh functions- Comments- References

On construction of recursion operators from Lax representation

Abstract: In this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdV-type systems of integrable equations.

Coupled nonlinear Schrodinger equations with cubic-quintic nonlinearity: Integrability and soliton interaction in non-Kerr media

Abstract: We propose an integrable system of coupled nonlinear Schrodinger equations with cubic-quintic terms describing the effects of quintic nonlinearity on the ultrashort optical soliton pulse propagation in non-Kerr media. Lax pairs, conserved quantities and exact soliton solutions for the proposed integrable model are given. The explicit form of two solitons are used to study soliton interaction showing many intriguing features including inelastic (shape changing or intensity redistribution) scattering. Another system of coupled equations with fifth-degree nonlinearity is derived, which represents vector generalization of the known chiral-soliton bearing system.

Integrable semi-discretization of the coupled nonlinear Schrödinger equations

Abstract: A system of semi-discrete coupled nonlinear Schrodinger equations is studied. To show the complete integrability of the model with multiple components, we extend the discrete version of the inverse scattering method for the single-component discrete nonlinear Schrodinger equation proposed by Ablowitz and Ladik. By means of the extension, the initial-value problem of the model is solved. Further, the integrals of motion and the soliton solutions are constructed within the framework of the extension of the inverse scattering method.

Solitary Waves for N Coupled Nonlinear Schrodinger Equations

Abstract: A hierarchy of exact analytic solitary-wave solutions for $N$ coupled nonlinear Schr\"odinger equations for which the nonlinear coupling parameters can change continuously and cover many regions is presented. Besides their potentially many practical applications to optical communication and multispecies Bose-Einstein condensates for couplings outside the special integrable cases, these analytically solvable cases for special initial conditions supplement and provide important links to and among the integrable cases.

Discrete integrable geometry and physics

Abstract: I: GEOMETRY 1. A. Bobenko & U. Pinkall Discretization of Surfaces and Integrable Systems 2. U. Hertrich-Jeromin, T. Hoffmann & U. Pinkall A Discrete Version of the Darboux Transform for Isothermic Surfaces Darboux Transform for Isothermic Surfaces 3. T. Hoffmann Discrete Amsler Surfaces and a Discrete Painleve III Equation 4. T. Hoffmann Discrete cmc Surfaces and Discrete Holomorphic Maps 5. A. Bobenko & W. Schief Discrete Indefinite Affine Spheres 6. A. Doliwa & P. M. Santini Geometry of Discrete Curves and Lattices and Integrable Difference Equations II: CLASSICAL SYSTEMS 7. Y. Suris R-matrices and Integrable Discretizations 8. F. Nijhoff Discrete Painleve Equations and Symmetry Reduction on the Lattice 9. N. Kutz Lagrangian Description of Doubly Discrete Sine-Gordon Type Models III: QUANTUM SYSTEMS 10. J. Kellendonk, N. Kutz & R. Seiler Spectra of Quantum Integrals 11. L. Faddeev & A. Volkov Algebraic Quantization of Integrable Models in Discrete Space-Time 12. R. Kashaev & N. Reshetikhin Affine Toda Field Theory as a Three-Dimensional Integrable System 13. R. Kashaev Quantum Hyperbolic Invariants of Knots 14. T. Richter & R. Seiler Charge Transport in the Discretized Landau Model

From the special 2 + 1 Toda lattice to the Kadomtsev-Petviashvili equation

Abstract: The nonlinearization of the eigenvalue problems associated with the Toda hierarchy and the coupled Korteweg-de Vries (KdV) hierarchy leads to an integrable symplectic map S and an integrable Hamiltonian system (H0), respectively. It is proved that S and (H0) have the same integrals {Hk}. The quasi-periodic solution of the (2 + 1)-dimensional Kadomtsev-Petviashvili equation is split into three Hamiltonian systems (H0),(H1),(H2), while that of the special (2 + 1)-dimensional Toda equation is separated into (H0),(H1) plus the discrete flow generated by the symplectic map S. A clear evolution picture of various flows is given through the `window' of Abel-Jacobi coordinates. The explicit theta-function solutions are obtained by resorting to this separation technique.

Quantum monodromy in integrable systems

Abstract: Let P1(h),...,Pn(h) be a set of commuting self-adjoint h-pseudo-differential operators on an n-dimensional manifold. If the joint principal symbol p is proper, it is known from the work of Colin de Verdiere [6] and Charbonnel [3] that in a neighbourhood of any regular value of p, the joint spectrum locally has the structure of an affine integral lattice. This leads to the construction of a natural invariant of the spectrum, called the quantum monodromy. We present this construction here, and show that this invariant is given by the classical monodromy of the underlying Liouville integrable system, as introduced by Duistermaat [9]. The most striking application of this result is that all two degree of freedom quantum integrable systems with a focus-focus singularity have the same non-trivial quantum monodromy. For instance, this proves a conjecture of Cushman and Duistermaat [7] concerning the quantum spherical pendulum.

Generalised Calogero-Moser models and universal Lax pair operators

Abstract: Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H3, H4, and the dihedral group I2(m), besides the well-known ones basedon crystallographic root systems, namely those associatedwith Lie algebras. Universal Lax pair operators for all of the generalisedCalogero-Moser mod els andfor any choices of the potentials are constructedas linear combinations of the reflection operators. The consistency conditions are reduced to functional equations for the coefficient functions of the reflection operators in the Lax pair. There are only four types of such functional equations corresponding to the two-dimensional sub-root systems, A2, B2, G2, and I2(m). The root type andthe minimal type Lax pairs, d erivedin our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators, a powerful tool for solving quantum Calogero-Moser models. Generalised Calogero-Moser models are integrable many-particle dynamical systems based on finite reflection groups. Finite reflection groups include the dihedral groups I2(m) and H3 and H4 together with the Weyl groups of the root systems associated with Lie algebras, called crystallographic root systems. Integrability of classical Calogero-Moser models based on the crystallographic root systems 1), 2) is shown in terms of Lax pairs. The root and the minimal type Lax pairs derived in our previous papers 3) provide a universal framework for these Calogero-Moser models, including those based on exceptional root systems and the twisted models. On the other hand, a theory of classical integrability for the models based on non-crystallographic root systems has been virtually non-existent. This is in sharp contrast with the quantum counterpart. Dunkl operators, which are useful for solving quantum Calogero-Moser models, were first explicitly constructed for the models based on the dihedral groups. 4)

The Whitham equations for optical communications: mathematical theory of NRZ

Abstract: We present a model of optical communication system for high-bit-rate data transmission in the nonreturn-to-zero (NRZ) format over transoceanic distance. The system operates in a small group velocity dispersion regime, and the model equation is given by the well-known Whitham equations describing the slow modulation of multiphase wavetrains of the (defocusing) nonlinear Schrodinger (NLS) equation. The model equation is of hyperbolic type, and NRZ pulse with certain initial phase modulation develops a shock. We then show how one can obtain a global solution by choosing an appropriate Riemann surface on which the Whitham equation is defined. We also discuss the effect of third order dispersion by using an integrable hierarchy of the NLS equation, and we give a condition to avoid a shock formation.

Dynamics of large-amplitude solitons

Abstract: The interaction and generation of solitons in nonlinear integrable systems which allow the existence of a soliton of limiting amplitude are considered. The integrable system considered is the Gardner equation, which includes the Korteweg-de Vries equation (for quadratic nonlinearity) and the modified Korteweg-de Vries equation (for cubic nonlinearity) as special cases. A two-soliton solution of the Gardner equation is derived, and a criterion, which distinguishes between different scenarios for the interaction of two solitons, is determined. The evolution of an initial pulsed disturbance is considered. It is shown, in particular, that solitons of opposite polarity appear during such evolution on the crest of a limiting soliton.

Evolution equations for quark-gluon distributions in multi-color QCD and open spin chains

Abstract: We study the scale dependence of the twist-3 quark-gluon parton distributions using the observation that in the multi-color limit the corresponding QCD evolution equations possess an additional integral of motion and turn out to be effectively equivalent to the Schrodinger equation for integrable open Heisenberg spin chain model. We identify the integral of motion of the spin chain as a new quantum number that separates different components of the twist-3 parton distributions. Each component evolves independently and its scale dependence is governed by anomalous dimension given by the energy of the spin magnet. To find the spectrum of the QCD induced open Heisenberg spin magnet we develop the Bethe Ansatz technique based on the Baxter equation. The solutions to the Baxter equation are constructed using different asymptotic methods and their properties are studied in detail. We demonstrate that the obtained solutions provide a good qualitative description of the spectrum of the anomalous dimensions and reveal a number of interesting properties. We show that the few lowest anomalous dimensions are separated from the rest of the spectrum by a finite mass gap and estimate its value.

Exact solution of a spin-ladder model

Abstract: An integrable spin-ladder model with nearest-neighbor exchanges and biquadratic interactions is proposed. With the Bethe ansatz solutions of the model Hamiltonian, it is found that there are three possible phases in the ground state, i.e., a rung-dimerized phase with a spin gap, and two massless phases. The possible fixed points of the system and the quantum critical behavior at the critical point ${J=J}_{+}^{c}$ are discussed.

Integrable discretizations for lattice system: local equations of motion and their hamiltonian properties

Abstract: We develop the approach to the problem of integrable discretization based on the notion of r-matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying continuous time systems. A common feature of the discretizations obtained in this approach is non-locality. We demonstrate how to overcome this drawback. Namely, we introduce the notion of localizing changes of variables and construct such changes of variables for a large number of examples, including the Toda and the relativistic Toda lattices, the Volterra and the relativistic Volterra lattices, the second flows of the Toda and of the Volterra hierarchies, the modified Volterra lattice, the Belov–Chaltikian lattice, the Bogoyavlensky lattices, the Bruschi–Ragnisco lattice. We also introduce a novel class of constrained lattice KP systems, discretize all of them, and find the corresponding localizing change of variables. Pulling back the differential equations of motion under the localizing changes of variables, we find also (sometimes novel) integrable one-parameter deformations of integrable lattice systems. Poisson properties of the localizing changes of variables are also studied: they produce interesting one-parameter deformations of the known Poisson algebras.

Expectation values of local fields for a two-parameter family of integrable models and related perturbed conformal field theories

Abstract: We calculate the vacuum expectation values of local fields for the two-parameter family of integrable field theories introduced and studied by Fateev. Using this result we propose an explicit expression for the vacuum expectation values of local operators in parafermionic sine-Gordon models and in integrable perturbed SU(2) coset conformal field theories.

Quasi-Periodic and Periodic Solutions for Systems of Coupled Nonlinear SCHR\"Odinger Equations

Abstract: We consider travelling periodic and quasiperiodic wave solutions of a set of coupled nonlinear Schr\"odimger equations. In fibre optics these equations can be used to model single mode fibers with strong birefringence and two-mode optical fibres. Recently these equations appear as modes, which describe pulse-pulse interaction in wavelength-division-multiplexed channels of optical fiber transmission systems. Two phase quasi-periodic solutions for integrable Manakov system are given in tems of two-dimensional Kleinian functions. The reduction of quasi-periodic solutions to elliptic functions is dicussed. New solutions in terms of generalized Hermite polynomilas, which are associated with two-gap Treibich-Verdier potentials are found.

Integrable geodesic flow with positive topological entropy

Abstract: An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by $C^\infty$ functions and has positive topological entropy is constructed.

KAM theorem of symplectic algorithms for Hamiltonian systems

Abstract: In this paper we prove that an analog of the celebrated KAM theorem holds for symplectic algorithms, which Channel and Scovel (1990), Feng Kang (1991) and Sanz-Serna and Calvo (1994) suggested a few years ago. The main results consist of the existence of invariant tori, with a smooth foliation structure, of a symplectic numerical algorithm when it applies to a generic integrable Hamiltonian system if the system is analytic and the time-step size of the algorithm is s ufficiently small. This existence result also implies that the algorithm, when it is applied to a generic integrable system, possesses n independent smooth invariant functions which are in involution and well-defined on the set filled by the invariant tori in the sense of Whitney. The invariant tori are just the level sets of these functions. Some quantitative results about the numerical invariant tori of the algorithm approximating the exact ones of the system are also given.

The Equations A T X \pm X T A=B

Abstract: We give necessary and sufficient conditions for the matrix equations ATX\pm XTA=B to admit solutions and give their general solution. Because A may be singular, the solution involves the generalized inverse of A. The (-) equation underlies the modern R-matrix approach to completely integrable Hamiltonian systems. This paper provides a new and algorithmic approach to constructing such R-matrices.

Seiberg-Witten Curves and Integrable Systems

Abstract: This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of Seiberg-Witten curves in terms of corresponding integrable models.