Showing papers on "Integrable system published in 1992"
TL;DR: In this article, a Lagrangian variational formulation of twist maps is proposed to compute the flux escaping from regions bounded by partial barriers formed from minimizing orbits, which form a scaffold in the phase space and constrain the motion of remaining orbits.
Abstract: Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond to Hamiltonians for which the velocity is a monotonic function of the canonical momentum. Twist maps have a Lagrangian variational formulation. One-parameter families of twist maps typically exhibit the full range of possible dynamics-from simple or integrable motion to complex or chaotic motion. One class of orbits, the minimizing orbits, can be found throughout this transition; the properties of the minimizing orbits are discussed in detail. Among these orbits are the periodic and quasiperiodic orbits, which form a scaffold in the phase space and constrain the motion of the remaining orbits. The theory of transport deals with the motion of ensembles of trajectories. The variational principle provides an efficient technique for computing the flux escaping from regions bounded by partial barriers formed from minimizing orbits. Unsolved problems in the theory of transport include the explanation for algebraic tails in correlation functions, and its extension to maps of more than two dimensions.
627 citations
TL;DR: One-dimensional many-body integrable systems in terms of a new set of phase space variables involving exchange operators assuming a decoupled form greatly simplifies the derivation of the conserved charges and the proof of their commutativity at the quantum level.
Abstract: We formulate one-dimensional many-body integrable systems in terms of a new set of phase space variables involving exchange operators. The Hamiltonian in these variables assumes a decoupled form. This greatly simplifies the derivation of the conserved charges and the proof of their commutativity at the quantum level.
433 citations
TL;DR: In this paper, the integrability of the system of PDE for dependence on coupling parameters of the (tree-level) primary partition function in massive topological field theories, being imposed by the associativity of the perturbed primary chiral algebra, is proved.
391 citations
Book•
01 Jan 1992
TL;DR: In this article, the basic mechanisms that determine the dynamical evolution of both classical and quantum systems in sufficient generality to include nonlinear phenomena are presented, in as simple and coherent a manner as possible.
Abstract: The subjects treated here are part of an active and rapidly growing field of research that touches on the foundations of physics and chemistry. Specifically, the book presents, in as simple and coherent a manner as possible, the basic mechanisms that determine the dynamical evolution of both classical and quantum systems in sufficient generality to include nonlinear phenomena. The book begins with a discussion of Noether's Theorem, integrability, KAM theory, and a definition of chaotic behaviour; it continues with a detailed discussion of area-preserving maps, integrable quantum systems, spectral properties, path integrals, and periodically driven systems; and concludes by showing how to apply the ideas to stochastic systems.
382 citations
TL;DR: In this paper, the concept of mutually integrable dynamical variables is proposed, which leads to the quadratic Askey-Wilson algebra QAW(3), which is the dynamical symmetry algebra for all problems where the most general "classical" polynoials arise.
216 citations
TL;DR: The Serret-Frenet equations are equivalent to the Ablowitz-Kaup-Newell-Segur scattering problem at zero eigenvalue, which identifies those evolution equations, integrable or not, that can describe the motion of curves.
Abstract: Recently discovered connections between integrable evolution equations and the motion of curves are based on the following fact: The Serret-Frenet equations are equivalent to the Ablowitz-Kaup-Newell-Segur (AKNS) scattering problem at zero eigenvalue. This equivalence identifies those evolution equations, integrable or not, that can describe the motion of curves.
211 citations
TL;DR: In this article, boundary conditions compatible with integrability were obtained for two dimensional models by solving the factorizability equations for the reflection matrices $K^{\pm}(\theta)$.
Abstract: Boundary conditions compatible with integrability are obtained for two dimensional models by solving the factorizability equations for the reflection matrices $K^{\pm}(\theta)$. For the six vertex model the general solution depending on four arbitrary parameters is found. For the $A_{n-1}$ models all diagonal solutions are found. The associated integrable magnetic Hamiltonians are explicitly derived.
182 citations
TL;DR: In this article, an approach for calculating the free energy as well as the correlation lengths of integrable quantum chains at arbitrary finite temperatures is presented, which is applied to critical Hamiltonians related to restricted solid-on-olid models comprising the hierarchy by Andrews, Baxter and Forrester, and generalizations hereof by the fusion procedure.
Abstract: An approach is presented for calculating the free energy as well as the correlation lengths of integrable quantum chains at arbitrary finite temperatures. The method is applied to critical Hamiltonians related to restricted solid-on-olid models comprising the hierarchy by Andrews, Baxter and Forrester, and generalizations hereof by the fusion procedure. The derived non-linear integral equations can be studied analytically in the low-temperature and high-temperature limits. The central charges and all primary conformal weights are obtained for the generalized minimal unitary series of conformal field theory and the Z N parafermion theories. Thus an extension of the thermodynamic Bethe Ansatz is realized which recently has been speculated on in the literature
167 citations
CERN1
TL;DR: In this paper, the authors generalized Sklyanin's approach of constructing open integrable quantum spin chains to the case of PT-invariant R matrices and formulated a fusion procedure for such chains.
Abstract: The authors have generalized Sklyanin's approach of constructing open integrable quantum spin chains to the case of PT-invariant R matrices. They formulate a fusion procedure for such chains. In particular, they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated. Applications of these results include constructing open integrable higher-spin chains, as well as obtaining functional equations for transfer-matrix eigenvalues, which may be solved by an analytical Bethe ansatz.
161 citations
TL;DR: In this article, a metric-independent σ-models are constructed, which generalise the membrane idea to situations where the target space has fewer dimensions than the base manifold, and invariance of solutions of field equations under arbitrary functional redefinitions of the field quantities is established.
160 citations
TL;DR: In this paper, the separation of variables is constructed for the SL(3) magnetic chain, an example of an integrable model associated to a non-hyperelliptic algebraic curve.
Abstract: There are two fundamental problems studied by the theory of hamiltonian integrable systems: integration of equations of motion, and construction of action-angle variables. A third problem, however, should be added to the list: separation of variables. Though much simpler thant the two others, it has important relations to quantum integrability. Separation of variables is constructed for theSL(3) magnetic chain—an example of an integrable model associated to a nonhyperelliptic algebraic curve.
TL;DR: In this paper, it was shown that the quantum-algebra-invariant open spin chains associated with affine Lie algebras are integrable, and that such chains do not require crossing symmetry.
Abstract: We show that the quantum-algebra-invariant open spin chains associated with the affine Lie algebras $A^{(1)}_n$ for $n>1$ are integrable. The argument, which applies to a large class of other quantum-algebra-invariant chains, does not require that the corresponding $R$ matrix have crossing symmetry.
TL;DR: In this article, the authors show that the linear systems associated with some integrable hierarchies of the soliton equations in 2+1 dimensions can be constrained to integrably hierarchical hierarchies in 1+ 1 dimensions such that submanifolds solutions of the given systems in 2 + 1 can be obtained by solving the resulting integrability systems in 1 + 1 dimensions.
Abstract: The authors show that the linear systems associated with some integrable hierarchies of the soliton equations in 2+1 dimensions can be constrained to integrable hierarchies in 1+1 dimensions such that submanifolds solutions of the given systems in 2+1 can be obtained by solving the resulting integrable systems in 1+1 dimensions. The constraints of the KP hierarchy to the AKNS and Burgers hierarchies respectively are shown in detail and the results of these for the modified KP and 2+1 dimensional analogue of the Caudrey-Dodd-Gibbon-Kotera-Sawata equations to several integrable systems in 1+1 are given.
TL;DR: In this paper, a general procedure to generate integrable Hamiltonians combining to any kind of spins distributed arbitrarily on the line is given, and a concrete application, anisotropic chains formed by spin 1/2 and spin 1 operators at alternating sites are presented and solved exactly by Bethe ansatz (BA).
Abstract: A general procedure to generate new integrable Hamiltonians combining to any kind of spins distributed arbitrarily on the line is given. As a concrete application, anisotropic chains formed by spin-1/2 and spin-1 operators at alternating sites are presented and solved exactly by the Bethe ansatz (BA). The authors compute the ground-state and excitation energies and momentum. The higher-order BA equations are derived. Depending on the choice, these new Hamiltonians exhibit, or not, conformal invariance in their low energy spectrum.
TL;DR: In this paper, the Volterra model with nonlocal Poisson structure is shown to be equivalent to the special limit of the lattice sinh-Gordon model, and the corresponding quantum integrable model is constructed together with the quantum lattice deformations of the Miura transformation and the Virasoro algebra.
TL;DR: In this paper, the authors consider volume-preserving diffeomorphisms on I × Tn, where I ∈ ℝ is a closed interval and Tn is an n-dimensional torus.
Abstract: In this paper we consider certain volume-preserving diffeomorphisms on I × Tn, where I ∈ ℝ is a closed interval and Tn is an n-dimensional torus. We show that under certain non-degeneracy conditions, all of the maps sufficiently close to the integrable maps preserve a large set of n-dimensional invariant tori.
TL;DR: In this article, a finite Euler hierarchy of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for classical topological field theories is constructed.
Abstract: Finite Euler hierarchies of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for classical topological field theories are constructed. The analysis uses two main ingredients. On the one hand, there exists a generic finite Euler hierarchy for one field leading to a universal equation which generalizes the Plebanski equation of self‐dual four‐dimensional gravity. On the other hand, specific maps are introduced between field theories which provide a ‘‘triality’’ between certain classes of arbitrary field theories, classical topological field theories and generalized string and membrane theories. The universal equations, which derive from an infinity of inequivalent Lagrangians, are generalizations of certain reductions of the Plebanski and KdV equations, and could possibly define new integrable systems, thus in particular integrable membrane theories. Some classes of solutions are constructed in the general case. The general solution to some ...
TL;DR: In this article, the symmetry group of the KdV equation is shown to be at most four dimensional, and this occurs if and only if the equation is equivalent, under local point transformations, to the kdV with f = g = 1.
Abstract: The Lie point symmetries of the equation ut+f(x, t)uux+g(x,t)uxxx=0 are studied. The symmetry group is shown to be, at most, four dimensional, and this occurs if and only if the equation is equivalent, under local point transformations, to the KdV equation with f=g=1. For nine different classes of functions f and g, the symmetry group turns out to be three dimensional. Two‐dimensional and one‐dimensional symmetry groups occur for 11 and 15 classes of equations, respectively.
TL;DR: In this paper, a unified dressing method for 2+1 solvable in terms of a nonlocal Riemann-Hilbert problem is proposed, which yields a Lax pair suitable for obtaining solutions that are perturbations of an arbitrary exact solution of the given equation.
Abstract: We consider equations in 2+1 solvable in terms of a nonlocal Riemann-Hilbert problem and show that for such an equation there exists a unified dressing method which yields: (i) a Lax pair suitable for obtaining solutions that are perturbations of an arbitrary exact solution of the given equation; (ii) certain integrable generalizations of the given equation. Using this generalized dressing method large classes of solutions of these equations, including dromions and line dromions, can be obtained. The method is illustrated by using theN-wave interactions, the Davey-Stewartson I, and the Kadomtsev-Petviashvili I equations. We also show that a careful application of the usual dressing method yields a certain generalization of theN-wave interactions.
TL;DR: In this article, a hamiltonian amplitude equation, iψ x +ψ tt +2σ|ψ| 2 ψ-∈ψ xt =0 σ=± 1, ∈<<1, is introduced.
Abstract: A new hamiltonian amplitude equation, iψ x +ψ tt +2σ|ψ| 2 ψ-∈ψ xt =0 σ=±1, ∈<<1, is introduced. The equation governs certain instabilities of modulated wave-trains, and the addition of the term -∈ψ xt overcomes the ill-posedness of the unstable nonlinear Schrodinger equation. This new equation is apparently not integrable, but it is a Hamiltonian analogue of the Kuramoto-Sivashinsky equation, which arises in dissipative systems
TL;DR: In this article, Chakravarty et al. show how classical modular forms and functions appear as tau-functions for a certain integrable reduction of the self-dual Yang-Mills equations obtained by S. Chakrabarty, M. Ablowitz, and P. Clarkson.
Abstract: We show how classical modular forms and functions appear as tau-functions for a certain integrable reduction of the self-dual Yang-Mills equations obtained by S. Chakravarty, M. Ablowitz, and P. Clarkson [6]. We discuss possible consequences of this novel phenomenon in integrable systems which indicate deep connections between integrable equations, group representations, modular forms, and moduli spaces.
TL;DR: In this article, a method of constructing finite-dimensional integrable systems starting from a bi-Hamiltonian hierarchy of soliton equations is introduced, where the existence of two Hamiltonian structures of the hierarchy leads to a bi−Hamiltonian formulation of the resulting finite−dimensional systems.
Abstract: A systematic method of constructing finite‐dimensional integrable systems starting from a bi‐Hamiltonian hierarchy of soliton equations is introduced. The existence of two Hamiltonian structures of the hierarchy leads to a bi‐Hamiltonian formulation of the resulting finite‐dimensional systems. The case of coupled KdV hierarchies is studied in detail. A surprising connection with separable Jacobi potentials is uncovered and described.
TL;DR: In this article, the concepts of tau -algebras and master algaes were introduced to describe time-polynomial-dependent symmetries of nonlinear integrable equations.
Abstract: For a general spectral operator, the author establishes types of algebraic structures of the spaces of the corresponding isospectral Lax operators, which essentially form the theoretical basis of the Lax operator method. Furthermore the author introduces the concepts of tau -algebras and master algebras to describe time-polynomial-dependent symmetries of nonlinear integrable equations. Finally the author applies the theory of Lax operators to the KP hierarchy of integrable equations as an illustrative example, and thus obtain the master symmetry algebra of the KP hierarchy.
TL;DR: In this paper, a method of constructing semiclassical asymptotics with complex phases is presented for multidimensional spectral problems (scalar, vector, and with operator-valued symbol) corresponding to both classically integrable and classically nonintegrable Hamiltonian systems.
Abstract: A method of constructing semiclassical asymptotics with complex phases is presented for multidimensional spectral problems (scalar, vector, and with operator-valued symbol) corresponding to both classically integrable and classically nonintegrable Hamiltonian systems. In the first case, the systems admit families of invariant Lagrangian tori (of complete dimension equal to the dimensionn of the configuration space) whose quantization in accordance with the Bohr—Sommerfeld rule with allowance for the Maslov index gives the semiclassical series in the region of large quantum numbers. In the nonintegrable case, families of Lagrangian tori with complete dimension do not exist. However, in the region of regular (nonchaotic) motion, such systems do have invariant Lagrangian tori of dimensionk
TL;DR: In this paper, a generalization of the Adler-Gel-fand-Dikii scheme is used to generate bi-Hamiltonian structures in two spatial dimensions, which can be used for the Kadomtsev-Petviashvili equation as well as other integrable equations in 2+1.
Abstract: A generalization of the Adler–Gel’fand–Dikii scheme is used to generate bi‐Hamiltonian structures in two spatial dimensions. In order to implement this scheme, a Hamiltonian theory is built over a noncommutative ring, namely the ring of formal pseudodifferential operators. Bi‐Hamiltonian structures generated in this way can be used for the Kadomtsev–Petviashvili equation as well as other integrable equations in 2+1.
Book•
01 Jan 1992
TL;DR: In this article, a geometrical meaning is given to the quantum double and other properties of quantum groups, and a multiplicative presentation of the Yangian double is analyzed in the context of 2D quantum field theory.
Abstract: Various aspects of recent works on affine quantum group symmetry of integrable 2d quantum field theory are reviewed and further clarified. A geometrical meaning is given to the quantum double, and other properties of quantum groups. Multiplicative presentations of the Yangian double are analyzed.