Showing papers on "Integrable system published in 2008"
TL;DR: In this paper, a two-component integrable system of coupled equations was derived in the context of shallow water theory and it was shown that while small initial data develop into global solutions, for some initial data wave breaking occurs.
441 citations
TL;DR: In this paper, the anomalous dimensions for scalar operators for a three-dimensional Chern-Simons theory were studied and the mixing matrix at two-loop order was derived for an integrable Hamiltonian of an SU(4) spin chain with sites alternating between the fundamental and the anti-fundamental representations.
Abstract: We study the anomalous dimensions for scalar operators for a three-dimensional Chern-Simons theory recently proposed in arXiv:0806.1218. We show that the mixing matrix at two-loop order is that for an integrable Hamiltonian of an SU(4) spin chain with sites alternating between the fundamental and the anti-fundamental representations. We find a set of Bethe equations from which the anomalous dimensions can be determined and give a proposal for the Bethe equations to the full superconformal group of OSp(2,2|6).
380 citations
TL;DR: In this article, a new integrable partial differential equation found by Vladimir Novikov admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic.
Abstract: We present a new integrable partial differential equation found by Vladimir Novikov. Like the Camassa-Holm and Degasperis-Procesi equations, this new equation admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic. We give a matrix Lax pair for V Novikov's equation, and show how it is related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. Infinitely many conserved quantities are found, as well as a bi-Hamiltonian structure. The latter is used to obtain the Hamiltonian form of the finite-dimensional system for the interaction of N peakons, and the two-body dynamics (N = 2) is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao.
324 citations
16 Dec 2008
TL;DR: In this article, the authors propose to use quadric nets in quadrics, special classes of discrete surfaces, and Integrable circle patterns to find solutions of selected exercises for classical differential geometry problems.
Abstract: Classical differential geometry Discretization principles. Multidimensional nets Discretization principles. Nets in quadrics Special classes of discrete surfaces Approximation Consistency as integrability Discrete complex analysis. Linear theory Discrete complex analysis. Integrable circle patterns Foundations Solutions of selected exercises Bibliography Notations Index.
300 citations
Posted Content•
TL;DR: In this paper, a new integrable partial differential equation found by Vladimir Novikov admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic.
Abstract: We present a new integrable partial differential equation found by Vladimir Novikov. Like the Camassa-Holm and Degasperis-Procesi equations, this new equation admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic. We give a matrix Lax pair for V. Novikov's equation, and show how it is related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. Infinitely many conserved quantities are found, as well as a bi-Hamiltonian structure. The latter is used to obtain the Hamiltonian form of the finite-dimensional system for the interaction of $N$ peakons, and the two-body dynamics (N=2) is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao.
294 citations
TL;DR: In this paper, the 1-soliton solution of K (m, n ) equation with the generalized evolution term in it was obtained and the solitary wave ansatz was used to obtain the exact solution.
204 citations
TL;DR: In this article, the centralised superalgebra psu(2|2) �R 3 was considered for the integrable structures of the one-dimensional Hubbard model and of the planar AdS/CFT correspondence.
Abstract: The centrally extended superalgebra psu(2|2) �R 3 was shown to play an important role for the integrable structures of the one-dimensional Hubbard model and of the planar AdS/CFT correspondence. Here we consider its quantum deformation Uq(psu(2|2) �R 3 ) and derive the fundamentalR-matrix. From the latter we deduce an integrable spin-chain Hamiltonian with three independent parameters and the corresponding Bethe equations to describe the spectrum on periodic chains. We relate our Hamiltonian to a two-parametric Hamiltonian proposed by Alcaraz and Bariev which can be considered a quantum deformation of the one-dimensional Hubbard model.
203 citations
TL;DR: In this paper, the eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the g l ( K | M ) -invariant R-matrix obey the Hirota bilinear difference equation.
177 citations
TL;DR: In this article, the authors considered form factors of integrable quantum field theories in finite volume, and extended their investigation to matrix elements with disconnected pieces, and gave a new method for generating a low temperature expansion, which they test for the one point function up to third order.
164 citations
TL;DR: In this article, two methods for approximating the stabilizing solution of the Hamilton-Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory.
Abstract: In this paper, two methods for approximating the stabilizing solution of the Hamilton-Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton-Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.
138 citations
TL;DR: In this paper, the authors apply the Sklyanin method of separation of variables to the reflection algebra underlying the open spin-1 2 XXX chain with non-diagonal boundary fields.
TL;DR: In this paper, a new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg-de Vries equation with a source.
Abstract: A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg–de Vries equation with a source. A Lax representation and an auto-Backlund transformation are found for the new equation, and its traveling wave solutions and generalized symmetries are studied.
TL;DR: In this article, the anomalous dimensions for scalar operators for a three-dimensional Chern-Simons theory were studied and the mixing matrix at two-loop order was derived for an integrable Hamiltonian of an SU(4) spin chain with sites alternating between the fundamental and the anti-fundamental representations.
Abstract: We study the anomalous dimensions for scalar operators for a three-dimensional Chern-Simons theory recently proposed in arXiv:0806.1218. We show that the mixing matrix at two-loop order is that for an integrable Hamiltonian of an SU(4) spin chain with sites alternating between the fundamental and the anti-fundamental representations. We find a set of Bethe equations from which the anomalous dimensions can be determined and give a proposal for the Bethe equations to the full superconformal group of OSp(2,2|6).
TL;DR: In this paper, a method to derive the long-distance asymptotic behavior of correlation functions of integrable models in the framework of the algebraic Bethe ansatz is presented.
Abstract: We describe a method to derive, from first principles, the long-distance asymptotic behavior of correlation functions of integrable models in the framework of the algebraic Bethe ansatz. We apply this approach to the longitudinal spin- spin correlation function of the XXZ Heisenberg spin-1/2 chain (with magnetic field) in the disordered regime as well as to the density-density correlation func- tion of the interacting one-dimensional Bose gas. At leading order, the results confirm the Luttinger liquid and conformal field theory predictions.
TL;DR: In this article, an integrable generalization of the nonlinear Schrodinger (NLS) equation was derived by one of the authors using bi-Hamiltonian methods, in the same way that the Camassa Holm equation is related to the KdV equation.
Abstract: We consider an integrable generalization of the nonlinear Schrodinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d) Analyze solitons.
TL;DR: The use of classical and quantum third-order integrals for two-dimensional superintegrable systems with one third order and one lower order integral of motion is reviewed in this article.
Abstract: Two-dimensional superintegrable systems with one third-order and one lower order integral of motion are reviewed. The fact that Hamiltonian systems with higher order integrals of motion are not the same in classical and quantum mechanics is stressed. New results on the use of classical and quantum third-order integrals are presented in sections 5 and 6.
TL;DR: The Hirota’s bilinear method is used to obtain multiple-soliton solutions for this completely integrable equation and highlights a variety of multi-Solitary wave and multi-singular solitary solutions of the Gardner–KP equation.
TL;DR: In this article, the authors introduce new global symplectic invariants for semitoric integrable systems, which encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system.
Abstract: Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce.
TL;DR: Two new multi-symplectic formulations for the Camassa-Holm equation are presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals.
Book•
02 Jun 2008
TL;DR: Integrable Hamiltonian Hierarchies: Spectral Methods- The Lax Representation and the AKNS Approach- The Direct Scattering Problem for theZakharov-Shabat System- The Inverse Scattering problem for the Zakharov Shabat system- The Generalized Fourier Transforms- Fundamental Properties of the solvable NLEEs- Hierarchyies of Hamiltonian structures- The NLEE and the Gauge Transformations- The Classical r-Matrix Method- Integrability and Nijenhuis Tensors- Smooth Manifolds
Abstract: Integrable Hamiltonian Hierarchies: Spectral Methods- The Lax Representation and the AKNS Approach- The Direct Scattering Problem for theZakharov-Shabat System- The Inverse Scattering Problem for the Zakharov-Shabat System- The Generalized Fourier Transforms- Fundamental Properties of the solvable NLEEs- Hierarchies of Hamiltonian structures- The NLEEs and the Gauge Transformations- The Classical r-Matrix Method- Integrable Hamiltonian Hierarchies: Geometric Theory of the Recursion Operators- Smooth Manifolds- Hamiltonian Dynamics- Vector-Valued Differential Forms- Integrability and Nijenhuis Tensors- Poisson-Nijenhuis structures Related to the Generalized Zakharov-Shabat System- Linear Bundles of Lie Algebras and Compatible Poisson Structures
TL;DR: In this article, the antiferromagnetic critical point of the Potts model on the square lattice was identified as a staggered integrable six-vertex model and the integrability of this model was investigated.
TL;DR: In this article, an inverse scattering approach to defects in classical integrable field theories is presented, where the contribution of the defect to all orders is explicitely identified in terms of a defect matrix.
Abstract: We present an inverse scattering approach to defects in classical integrable field theories. Integrability is proved systematically by constructing the generating function of the infinite set of modified integrals of motion. The contribution of the defect to all orders is explicitely identified in terms of a defect matrix. The underlying geometric picture is that those defects correspond to Backlund transformations localized at a given point. A classification of defect matrices as well as the corresponding defect conditions is performed. The method is applied to a collection of well-known integrable models and previous results are recovered (and extended) directly as special cases. Finally, a brief discussion of the classical r-matrix approach in this context shows the relation to inhomogeneous lattice models and the need to resort to lattice regularizations of integrable field theories with defects.
Posted Content•
TL;DR: In this paper, the initial-boundary value problems for an integrable generalization of the nonlinear Schrodinger equation formulated on the half-line were analyzed and the so-called linearizable boundary conditions, which in this case are of Robin type, were investigated.
Abstract: We analyze initial-boundary value problems for an integrable generalization of the nonlinear Schrodinger equation formulated on the half-line. In particular, we investigate the so-called linearizable boundary conditions, which in this case are of Robin type. Furthermore, we use a particular solution to verify explicitly all the steps needed for the solution of a well-posed problem.
TL;DR: In this article, a new technique, the extended homoclinic test technique, is proposed to seek periodic solitary wave solutions of integrable systems, and exact periodic solitary-wave solutions for classical KdV equation are obtained using this technique.
Abstract: A new technique, the extended homoclinic test technique, is proposed to seek periodic solitary wave solutions of integrable systems. Exact periodic solitary-wave solutions for classical KdV equation are obtained using this technique. This result shows that it is entirely possible for the (1+1)-dimensional integrable equation that there exists a periodic solitary-wave.
TL;DR: In this paper, the authors used the Painleve analysis preceded by appropriate transformations of nonlinear systems under investigation to discover two new cases in which the Pietrzyk-Kanattsikov-and Bandelow vector short pulse equation must be integrable due to the results of the painleve test.
Abstract: Using the Painleve analysis preceded by appropriate transformations of nonlinear systems under investigation, we discover two new cases in which the Pietrzyk–Kanattsikov–Bandelow vector short pulse equation must be integrable due to the results of the Painleve test. Those cases are technologically important because they correspond to the propagation of polarized ultra-short light pulses in usual isotropic silica optical fibers.
TL;DR: In this paper, the authors developed a hydrodynamic description of the classical Calogero-Sutherland liquid, which is based on a bidirectional analogue of the Benjamin-Ono equation.
Abstract: We develop a hydrodynamic description of the classical Calogero-Sutherland liquid: a Calogero-Sutherland model with an infinite number of particles and a non-vanishing density of particles. The hydrodynamic equations, being written for the density and velocity fields of the liquid, are shown to be a bidirectional analogue of Benjamin-Ono equation. The latter is known to describe internal waves of deep stratified fluids. We show that the bidirectional Benjamin-Ono equation appears as a real reduction of the modified KP hierarchy. We derive the Chiral Non-linear Equation which appears as a chiral reduction of the bidirectional equation. The conventional Benjamin-Ono equation is a degeneration of the Chiral Non-Linear Equation at large density. We construct multi-phase solutions of the bidirectional Benjamin-Ono equations and of the Chiral Non-Linear equations.
TL;DR: In this paper, the authors presented a maximally superintegrable Hamiltonian on an n-dimensional Riemannian space of nonconstant curvature, which can be interpreted as the intrinsic Smorodinsky-Winternitz system on such a space.
TL;DR: In this paper, the transmission factors and defect energies of two-dimensional integrable models are derived from the Bethe ansatz equations derived to describe the ground-state energy of diagonal defect systems on a cylinder.
TL;DR: In this article, an integrability-preserving recursion relation for the explicit construction of long-range spin chain Hamiltonians is presented, which is based on arbitrary nearest-neighbour integrable spin chains and sheds light on the moduli space of deformation parameters.
Abstract: We present an integrability-preserving recursion relation for the explicit construction of long-range spin chain Hamiltonians. These chains are generalizations of the Haldane–Shastry and Inozemtsev models and they play an important role in recent advances in string/gauge duality. The method is based on arbitrary nearest-neighbour integrable spin chains and it sheds light on the moduli space of deformation parameters. We also derive the closed chain asymptotic Bethe equations.
TL;DR: In this article, the authors investigated the effective behavior of a small transversal perturbation of order to a completely integrable stochastic Hamiltonian system, by which they mean a Stochastic differential equation whose diffusion vector fields are formed from a family of Hamiltonian functions Hi, i = 1,..., n.
Abstract: We investigate the effective behaviour of a small transversal perturbation of order to a completely integrable stochastic Hamiltonian system, by which we mean a stochastic differential equation whose diffusion vector fields are formed from a completely integrable family of Hamiltonian functions Hi, i = 1, ..., n. An averaging principle is shown to hold and the action component of the solution converges, as → 0, to the solution of a deterministic system of differential equations when the time is rescaled at 1/. An estimate for the rate of the convergence is given. In the case when the perturbation is a Hamiltonian vector field, the limiting deterministic system is constant in which case we show that the action component of the solution scaled at 1/2 converges to that of a limiting stochastic differentiable equation.