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Showing papers on "Integrable system published in 2009"


Journal ArticleDOI
TL;DR: In this paper, an infinite family of exactly solvable and integrable potentials on a plane is introduced, and all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family.
Abstract: An infinite family of exactly solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvable and integrable generalization of the family is found.

200 citations


Journal ArticleDOI
TL;DR: In this paper, the integrability of nonlinear hyperbolic equations on quad-graphs is defined as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent.
Abstract: We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on ℤ2. The fields are associated with the vertices and an equation of the form Q(x 1, x 2, x 3, x 4) = 0 relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices ℤ N . We classify integrable equations with complex fields x and polynomials Q multiaffine in all variables. Our method is based on the analysis of singular solutions.

190 citations


Journal ArticleDOI
TL;DR: In this article, the explicit formulas for multipeakon solutions of Novikovs cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang.
Abstract: Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm equation which has nonlinear terms that are cubic, rather than quadratic, and which admits peaked soliton solutions (peakons). In this paper, the explicit formulas for multipeakon solutions of Novikovs cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the Degasperis-Procesi equation.

188 citations


Journal ArticleDOI
TL;DR: In this article, 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 function of the interacting one-dimensional Bose gas. At leading order, the results confirm the Luttinger liquid and conformal field theory predictions.

180 citations


Posted Content
TL;DR: In this article, the explicit formulas for multipeakon solutions of Novikov's cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang.
Abstract: Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm equation, admitting peaked soliton (peakon) solutions, which has nonlinear terms that are cubic, rather than quadratic. In this paper, the explicit formulas for multipeakon solutions of Novikov's cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the Degasperis-Procesi equation.

169 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the non-equilibrium time evolution of an integrable field theory in 1+1 dimensions after a sudden variation of a global parameter of the Hamiltonian.
Abstract: We study the non equilibrium time evolution of an integrable field theory in 1+1 dimensions after a sudden variation of a global parameter of the Hamiltonian. For a class of quenches defined in the text, we compute the long times limit of the one point function of a local operator as a series of form factors. Even if some subtleties force us to handle this result with care, there is a strong evidence that for long times the expectation value of any local operator can be described by a generalized Gibbs ensemble with a different effective temperature for each eigenmode.

154 citations


Journal ArticleDOI
TL;DR: An analytical and numerical study of pairwise peakon interactions for the CH2 system shows a different asymptotic feature, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, the MCH2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes.
Abstract: The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow a dependence on the average density as well as the pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in the velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. Numerical results for the MCH2 system are given and compared with the pure CH2 case. These numerics show that the modification in the MCH2 system to introduce the average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for the MCH2 system shows a different asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, the MCH2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes.

149 citations


Journal ArticleDOI
TL;DR: In this paper, a new chapter in the theory of wave turbulence is introduced, called Turbulence in Integrable Systems (TIS), where all systems integrable by Inverse Scattering Method (ISM) are separated in two classes: strongly and weakly integrability.
Abstract: Nonlinear wave systems integrable by Inverse Scattering Method (ISM) could demonstrate a complex behavior that demands the statistical description. The theory of this description composes a new chapter in the theory of wave turbulence—Turbulence in Integrable Systems. All systems integrable by ISM are separated in two classes: strongly and weakly integrable. Systems of both classes have infinite array of motion constants but only for strongly integrable systems this array is complete. As a result, the scattering is trivial in these systems. It means that all the collision terms in kinetic equations of arbitrary high order are identically zero. The examples of strongly integrable systems are: KdV, NLSE, and KP-2 equations. In strongly integrable systems one can choose as initial data a statistically homogenous random field with a given pair correlation function such that this function is invariant in time. The spatial spectrum of an equilibrium state can be chosen in an arbitrary way. In weakly integrable systems (KP-1, three-wave system, etc) the kinetic equations are nontrivial. They have infinite but incomplete set of motion constants. These kinetic equations have infinite amount of Rayley–Jeans-type stationary solutions, though their general stationary solutions are not explored yet.

145 citations


Journal ArticleDOI
TL;DR: In this article, a generalization of integrable peakon equations with cubic nonlinearity and the Degasperis-Procesi equation with peakon solutions is proposed, which is associated with a 3 × 3 matrix spectral problem with two potentials.
Abstract: A generalization of integrable peakon equations with cubic nonlinearity and the Degasperis–Procesi equation with peakon solutions is proposed, which is associated with a 3 × 3 matrix spectral problem with two potentials. With the aid of the zero-curvature equation, we derive a hierarchy of new nonlinear evolution equations and establish their Hamiltonian structures. The generalization is exactly a negative flow in the hierarchy and admits exact solutions with N-peakons and an infinite sequence of conserved quantities. Moreover, a reduction of this hierarchy and its Hamiltonian structures are discussed.

141 citations


Journal ArticleDOI
TL;DR: In this paper, the role of several two-component integrable systems in the classical problem of shallow water waves is described, which can be related to three different integrably generalization of the Camassa-Holm equation, the Zakharov-Ito system and the Kaup-Boussinesq system.

134 citations


Journal ArticleDOI
TL;DR: In this article, an integrable generalization of the nonlinear Schrodinger (NLS) equation was proposed for nonlinear pulse propagation in monomode optical fibers.
Abstract: The nonlinear Schrodinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation, which was first derived by means of bi-Hamiltonian methods in [1]. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as a model for nonlinear pulse propagation in monomode optical fibers when certain higher-order nonlinear effects are taken into account; (b) We show that the equation is equivalent, up to a simple change of variables, to the first negative member of the integrable hierarchy associated with the derivative NLS equation; (c) We analyze traveling-wave solutions.

Journal ArticleDOI
TL;DR: In this paper, the integration of the generalized Radhakrishnan, Kundu, Lakshmanan equation to obtain the 1-soliton solution was carried out using the solitary wave ansatz.

Journal ArticleDOI
TL;DR: In this paper, an infinite family of exactly-solvable and integrable rational potentials on a plane is introduced and the underlying algebraic structure of the new potentials is revealed as well as its hidden algebra.
Abstract: An infinite family of exactly-solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvable and integrable generalization of the family is found.


Journal ArticleDOI
TL;DR: In this paper, a general method for solving integrable two-dimensional relativistic sigma models in a finite size periodic box is proposed, which is based on the integrably Hirota dynamics that follows from the Y-system.
Abstract: We propose, using the example of the O(4) sigma model, a general method for solving integrable two dimensional relativistic sigma models in a finite size periodic box. Our starting point is the so-called Y-system, which is equivalent to the thermodynamic Bethe ansatz equations of Yang and Yang. It is derived from the Zamolodchikov scattering theory in the cross channel, for virtual particles along the non-compact direction of the space-time cylinder. The method is based on the integrable Hirota dynamics that follows from the Y-system. The outcome is a nonlinear integral equation for a single complex function, valid for an arbitrary quantum state and accompanied by the finite size analogue of Bethe equations. It is close in spirit to the Destri-deVega (DdV) equation. We present the numerical data for the energy of various states as a function of the size, and derive the general Luscher-type formulas for the finite size corrections. We also re-derive by our method the DdV equation for the SU(2) chiral Gross-Neveu model.

Journal ArticleDOI
TL;DR: In this paper, the dynamics of dark optical solitons governed by the nonlinear Schrodinger's equation with power law nonlinearity with time-dependent coefficients are discussed.

Journal ArticleDOI
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.

Journal ArticleDOI
TL;DR: In this article, a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions is presented, and the closed chain asymptotic Bethe equations for longrange spin chains transforming under a generic symmetry algebra are derived.
Abstract: We present a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions. Based on arbitrary shortrange (e.g. nearest neighbor) integrable spin chains, it allows us to construct an infinite set of conserved long-range charges. We explain the moduli space of deformation parameters by different classes of generating operators. The rapidity map and dressing phase in the long-range Bethe equations are a result of these deformations. The closed chain asymptotic Bethe equations for long-range spin chains transforming under a generic symmetry algebra are derived. Notably, our construction applies to generalizations of standard nearest neighbor chains such as alternating spin chains. We also discuss relevant properties for its application to planar D = 4, N = 4 and D = 3, N = 6 supersymmetric gauge theories. Finally, we present a map between long-range and inhomogeneous spin chains delivering more insight into the structures of these models, as well as their limitations at wrapping order.

Journal ArticleDOI
TL;DR: Investigation of Painlevé integrability of a generalized nonautonomous one-dimensional nonlinear Schrödinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems.
Abstract: We investigate Painleve integrability of a generalized nonautonomous one-dimensional nonlinear Schrodinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painleve analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painleve integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.

Journal ArticleDOI
TL;DR: In this paper, an exact solitary wave solution of the Korteweg-de Vries equation with power law nonlinearity with time-dependent coefficients of the nonlinear as well as the dispersion terms was obtained.
Abstract: This paper obtains an exact solitary wave solution of the Korteweg–de Vries equation with power law nonlinearity with time-dependent coefficients of the nonlinear as well as the dispersion terms. In addition, there are time-dependent damping and dispersion terms. The solitary wave ansatz is used to carry out the analysis. It is only necessary for the time-dependent coefficients to be Riemann integrable. As an example, the solution of the special case of cylindrical KdV equation falls out.

Journal ArticleDOI
TL;DR: In this article, the Lenard scheme is used to construct a hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution.
Abstract: We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms \omega_j, j in Z_+, of the variational complex \Omega. If these forms are exact, i.e. \omega_j are variational derivatives of some local functionals \int h_j, then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex \Omega is exact, provided that the algebra of functions V is "normal"; in particular, for arbitrary V, any closed form in \Omega becomes exact if we add to V a finite number of antiderivatives. We demonstrate on the examples of KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.

Journal ArticleDOI
TL;DR: The ABACUS C++ library as mentioned in this paper provides the means of calculating dynamical correlation functions of some important observables in systems such as Heisenberg spin chains and one-dimensional atomic gases.
Abstract: Recent developments in the theory of integrable models have provided the means of calculating dynamical correlation functions of some important observables in systems such as Heisenberg spin chains and one-dimensional atomic gases. This article explicitly describes how such calculations are generally implemented in the ABACUS C++ library, emphasizing the universality in treatment of different cases coming as a consequence of unifying features within the Bethe ansatz.

Journal ArticleDOI
TL;DR: In this paper, the form-factor approach was combined with the method of intertwining operator, which allows explicit summation over intermediate states and analysis of the time evolution of non-local density-density correlation functions.
Abstract: In this paper we study nonequilibrium dynamics of one dimensional Bose gas from the general perspective of dynamics of integrable systems. After outlining and critically reviewing methods based on inverse scattering transform, intertwining operators, q-deformed objects, and extended dynamical conformal symmetry, we focus on the form-factor based approach. Motivated by possible applications in nonlinear quantum optics and experiments with ultracold atoms, we concentrate on the regime of strong repulsive interactions. We consider dynamical evolution starting from two initial states: a condensate of particles in a state with zero momentum and a condensate of particles in a gaussian wavepacket in real space. Combining the form-factor approach with the method of intertwining operator we develop a numerical procedure which allows explicit summation over intermediate states and analysis of the time evolution of non-local density-density correlation functions. In both cases we observe a tendency toward formation of crystal-like correlations at intermediate time scales.

Journal ArticleDOI
TL;DR: In this paper, the eigenvectors of the modified transition matrix of the Bethe ansatz were analyzed and the results obtained by de Gier and Essler were recovered and a physical interpretation of the exceptional points were given.
Abstract: The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, which is still integrable: the spectrum of this new matrix can also be described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in previous works, the nature of the excitations and the full structure of the eigenvectors remained unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe ansatz developed for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points. The overlap of this approach with other tools such as the matrix ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points.

Journal ArticleDOI
TL;DR: An alternative Lagrangian definition of an integrable defect is provided and analysed in this paper, which is sufficiently broad to allow a description of defects within the Tzitzeica model, and may be generalizable.
Abstract: An alternative Lagrangian definition of an integrable defect is provided and analysed. The new approach is sufficiently broad to allow a description of defects within the Tzitzeica model, which was not possible in previous approaches, and may be generalizable. New, two-parameter, sine-Gordon defects are also described, which have characteristics resembling a pair of 'fused' defects of a previously considered type. The relationship between these defects and Backlund transformations is described and a Hamiltonian description of integrable defects is proposed.

Journal ArticleDOI
TL;DR: In this article, the 1-soliton solution of the B(m,n) equation with generalized evolution term was obtained by using the solitary wave ansatz, and the four exhaustive cases, depending on the parameters, were considered.

Journal ArticleDOI
TL;DR: In this article, the eigenvectors of the modified transition matrix are physically relevant, and an explicit expression for the EIGs is given for the eIGs and applied to the study of atypical currents.
Abstract: The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, that is still integrable: the spectrum of this new matrix can be also described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in the previous works, the nature of the excitations and the full structure of the eigenvectors were still unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe Ansatz developped for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this Ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points The overlap of this approach with other tools such as the matrix Ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points.

Journal ArticleDOI
TL;DR: In this paper, the authors discuss variational identities associated with continuous and discrete spectral problems and their applications to Hamiltonian structures of soliton equations, including the AKNS hierarchy and the Volterra lattice hierarchy associated with semisimple Lie algebras.
Abstract: This is an introductory report concerning our recent research on Hamiltonian structures. We will discuss variational identities associated with continuous and discrete spectral problems, and their applications to Hamiltonian structures of soliton equations. Our illustrative examples are the AKNS hierarchy and the Volterra lattice hierarchy associated with semisimple Lie algebras, and two hierarchies of their integrable couplings associated with non-semisimple Lie algebras. The resulting Hamiltonian structures generate infinitely many commuting symmetries and conservation laws for the four soliton hierarchies. The presented variational identities can be applied to Hamiltonian structures of other soliton hierarchies.

Journal ArticleDOI
TL;DR: In this paper, the authors introduce new global symplec- tic 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, ω) be a symplectic 4-manifold. A semitoric integrable system on (M, ω) is a pair of smooth functions J,H ∈ C ∞ (M, R) for which J generates a Hamiltonian S 1 -action and the Poisson brackets {J,H } vanish. We shall introduce new global symplec- tic 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.

Journal ArticleDOI
TL;DR: In this paper, the authors show how integrability (via the algebraic Bethe ansatz) gives one numerical access, in a nearly exact manner, to the dynamics resulting from a global interaction quench of an ensemble of fermions with pairing interactions.
Abstract: By instantaneously changing a global parameter in an extended quantum system, an initially equilibrated state will afterwards undergo a complex nonequilibrium unitary evolution whose description is extremely challenging. A nonperturbative method giving a controlled error in the long time limit remained highly desirable to understand general features of the quench induced quantum dynamics. In this paper we show how integrability (via the algebraic Bethe ansatz) gives one numerical access, in a nearly exact manner, to the dynamics resulting from a global interaction quench of an ensemble of fermions with pairing interactions (Richardson’s model). This possibility is deeply linked to the specific structure of this particular integrable model which gives simple expressions for the scalar product of eigenstates of two different Hamiltonians. We show how, despite the fact that a sudden quench can create excitations at any frequency, a drastic truncation of the Hilbert space can be carried out therefore allowing access to large systems. The small truncation error which results does not change with time and consequently the method grants access to a controlled description of the long time behavior which is a hard to reach limit with other numerical approaches.