Showing papers on "Integrable system published in 1996"
26 May 1996
TL;DR: In this paper, the authors used algebraic Bethe Ansatz for solving integrable models and showed how it works in detail on the simplest example of spin 1/2 XXX magnetic chain.
Abstract: I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated more superficially, only the specific details are given. Several parameters, appearing in these generalizations: spin $s$, anisotropy parameter $\ga$, shift $\om$ in the alternating chain, allow to include in our treatment most known examples of soliton theory, including relativistic model of Quantum Field Theory.
814 citations
TL;DR: A simple scaling argument shows that most integrable evolutionary systems, which are known to admit a bi-Hamiltonian structure, are, in fact, governed by a compatible trio of Hamiltonian structures, and it is demonstrated how their recombination leads toIntegrable hierarchies endowed with nonlinear dispersion that supports compactons, or cusped and/or peaked solitons.
Abstract: A simple scaling argument shows that most integrable evolutionary systems, which are known to admit a bi-Hamiltonian structure, are, in fact, governed by a compatible trio of Hamiltonian structures. We demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons (solitary-wave solutions having compact support), or cusped and/or peaked solitons. A general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified Korteweg--de Vries equation, the nonlinear Schr\"odinger equation, the integrable Boussinesq system used to model the two-way propagation of shallow water waves, and the Ito system of coupled nonlinear wave equations. These hierarchies include a remarkable variety of interesting integrable nonlinear differential equations. \textcopyright{} 1996 The American Physical Society.
730 citations
TL;DR: In this article, the authors constructed the quantum versions of the monodromy matrices of KdV theory, called as T-operators, which act in highest weight Virasoro modules.
Abstract: We construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as “T-operators,” act in highest weight Virasoro modules. TheT-operators depend on the spectral parameter λ and their expansion around λ=∞ generates an infinite set of commuting Hamiltonians of the quantum KdV system. TheT-operators can be viewed as the continuous field theory versions of the commuting transfermatrices of integrable lattice theory. In particular, we show that for the values\(c = 1 - 3\frac{{3(2n + 1)^2 }}{{2n + 3}}\),n=1,2,3 .... of the Virasoro central charge the eigenvalues of theT-operators satisfy a closed system of functional equations sufficient for determining the spectrum. For the ground-state eigenvalue these functional equations are equivalent to those of the massless Thermodynamic Bethe Ansatz for the minimal conformal field theoryM2,2n+3; in general they provide a way to generalize the technique of the Thermodynamic Bethe Ansatz to the excited states. We discuss a generalization of our approach to the cases of massive field theories obtained by perturbing these Conformal Field Theories with the operator Φ1,3. The relation of theseT-operators to the boundary states is also briefly described.
598 citations
TL;DR: In this paper, an integrable theory for perturbation equations engendered from small disturbances of solutions is developed, which includes various integrability properties of the perturbations, such as hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures, and provides us with a method of generating hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones.
Abstract: An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations, such as hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures, and provides us with a method of generating hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones. The resulting perturbation equations give rise to a sort of integrable coupling of soliton equations. Two examples (MKdV hierarchy and KP equation) are carefully carried out.
342 citations
TL;DR: In this paper, an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the thermodynamic Bethe ansatz equations for its ground state, was proposed.
331 citations
256 citations
TL;DR: In this article, Hitchin's system on the cotangent bundle of the moduli space of stable bundles on a curve is described, which is integrable analytically, but not algebraically: the Liouville tori are the intermediate Jacobians of a family of Calabi-Yau manifolds.
Abstract: This is the expanded text of a series of CIME lectures. We present an algebro-geometric approach to integrable systems, starting with those which can be described in terms of spectral curves. The prototype is Hitchin's system on the cotangent bundle of the moduli space of stable bundles on a curve. A variant involving meromorphic Higgs bundles specializes to many familiar systems of mathematics and mechanics, such as the geodesic flow on an ellipsoid and the elliptic solitons. We then describe some systems in which the spectral curve is replaced by various higher dimensional analogues: a spectral cover of an arbitrary variety, a Lagrangian subvariety in an algebraically symplectic manifold, or a Calabi-Yau manifold. One peculiar feature of the CY system is that it is integrable analytically, but not algebraically: the Liouville tori (on which the system is linearized) are the intermediate Jacobians of a family of Calabi-Yau manifolds. Most of the results concerning these three types of non-curve-based systems are quite recent. Some of them, as well as the compatibility between spectral systems and the KP hierarchy, are new, while other parts of the story are scattered through several recent preprints. As best we could, we tried to maintain the survey style of this article, starting with some basic notions in the field and building gradually to the recent developments.
226 citations
TL;DR: In this paper, the scaling region of the Ising model in an external magnetic field at T ∼ T c and the scaling regions around the minimal model M 2,7 were investigated.
226 citations
Book•
29 Feb 1996
TL;DR: In this paper, the theory of Oscillations and Waves is discussed and a classification of self-Oscillatory systems with Lumped Parameters is presented. But this classification is restricted to the case of non-linear Oscillators and does not consider self-oscillatory systems.
Abstract: Preface. Introduction. Part I: Basic Notions and Definitions. 1. Dynamical Systems. Phase Space. Stochastic and chaotic Systems. The Number of Degrees of Freedom. 2. Hamiltonian Systems Close to Integrable. Appearance of Stochastic Motions in Hamiltonian Systems. 3. Attractors and Repellers. Reconstruction of Attractors from an Experimental Time Series. Quantitative Characteristics of Attractors. 4. Natural and Forced Oscillations and Waves. Self-Oscillations and Auto-Waves. Part II: Basic Dynamical Models of the Theory of Oscillations and Waves. 5. Conservative Systems. 6. Non-Conservative Hamiltonian Systems and Dissipative Systems. Part III: Natural (Free) Oscillations and Waves in Linear and Non-Linear Systems. 7. Natural Oscillations of Non-Linear Oscillators. 8. Natural Oscillations in Systems of Coupled Oscillators. 9. Natural Waves in Bounded and Unbounded Continuous Media. Solitons. Part IV: Forced Oscillations and Waves in Passive Systems. 10. Oscillations of a Non-Linear Oscillator Excited by an External Force. 11. Oscillations of Coupled Non-linear Oscillators Excited by an External Periodic Force. 12. Parametric Oscillations. 13. Waves in Semibounded Media Excited by Perturbations Applied to their Boundaries. Part V: Oscillations and Waves in Active Systems. Self-Oscillations and Auto-Waves. 14. Forced Oscillations and Waves in Active Non-Self-Oscillatory Systems. Turbulence. Burst Instability. Excitation of Waves with Negative Energy. 15. Mechanisms of Excitation and Amplitude Limitation of Self-Oscillations and Auto-Waves. Classification of Self-Oscillatory Systems. 16. Examples of Self-Oscillatory Systems with Lumped Parameters. I. 17. Examples of Self-Oscillatory Systems with Lumped Parameters. II. 18. Examples of self-oscillatory Systems with High Frequency Power Sources. 19. Examples of Self-Oscillatory Systems with Time Delay. 20. Examples of Continuous Self-Oscillatory Systems with Lumped Active Elements. 21. Examples of Self-Oscillatory Systems with Distributed Active Elements. 22. Periodic Actions on Self-Oscillatory Systems. Synchronization and Chaotization of Self-Oscillations. 23. Interaction between Self-Oscillatory Systems. 24. Examples of Auto- Waves and Dissipative Structures. 25. Convective Structures and Self-Oscillations in Fluid. The Onset of Turbulence. 26. Hydrodynamic and Acoustic Waves in Subsonic Jet and Separated Flows. Appendix A: Approximate Methods for Solving Linear Differential Equations with Slowly Varying Parameters. Appendix B: The Whitham Method and the Stability of Periodic Running Waves for the Klein--Gordon Equation. Bibliography. Index.
218 citations
TL;DR: In this article, an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the Thermodynamic Bethe Ansatz equations for its ground state, was proposed.
Abstract: We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the Thermodynamic Bethe Ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one- and two- particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal space approach, and we also treat some of the ultraviolet and infrared asymptotics analytically.
TL;DR: In this paper, the Bethe ansatz technique is applied for the calculation of the observables in the strong coupling region of the sine-Gordon model and the results are in the exact agreement with ones following from the sigma model action, which is a two-parameter U (1) ⊗ ( 1) symmetrical deformation of the O(4) non-finear sigma Model.
TL;DR: In this paper, the standard objects of quantum integrable models are identified with elements of classical nonlinear integrably difference equation. And the functional relation for commuting quantum transfer matrices of QIMs is identified with classical Hirota's bilinear difference equation, which is equivalent to the completely discretized classical 2D Toda lattice with open boundaries.
Abstract: Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota's equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's $Q$-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for $A_{k-1}$-type models appear as discrete time equations of motions for zeros of classical $\tau$-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained.
TL;DR: In this paper, a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with spatial coordinate compactified on a circle of circumference R, was developed.
Abstract: We develop a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with spatial coordinate compactified on a circle of circumference R. The IQFT ``commuting transfer-matrices'' introduced by us (BLZ) for Conformal Field Theories (CFT) are generalized to non-conformal IQFT obtained by perturbing CFT with the operator $\Phi_{1,3}$. We study the models in which the fusion relations for these ``transfer-matrices'' truncate and provide closed integral equations which generalize the equations of Thermodynamic Bethe Ansatz to excited states. The explicit calculations are done for the first excited state in the ``Scaling Lee-Yang Model''.
Book•
01 Jan 1996
TL;DR: The Mumford system as discussed by the authors is a complete integrable Hamiltonian system on affine Poisson varietie, and the Mumford systems can be seen as a generalization of Hamiltonian systems on level manifolds.
Abstract: Introduction.- Integrable Hamiltonian systems on affine Poisson varietie: Affine Poisson varieties and their morphisms Integrable Hamiltonian systems and their morphisms Integrable Hamiltonian systems on other spaces.- Integrable Hamiltonian systems and symmetric products of curves: The systems and their integrability The geometry of the level manifolds .- Interludium: the geometry of Abelian varieties: Divisors and line bundles Abelian varieties Jacobi varieties Abelian surfaces of type (1,4).- Algebraic completely integrable Hamiltonian systems: A.c.i. systems Painlev analysis for a.c.i. systems The linearization of two-dimensional a.c.i. systems Lax equations.- The Mumford systems: Genesis Multi-Hamiltonian structure and symmetries The odd and the even Mumford systems The general case .- Two-dimensional a.c.i. systems and applications: The genus two Mumford systems Application: generalized Kummersurfaces The Garnier potential An integrable geodesic flow on SO(4) ...
Journal Article•
TL;DR: The classical Arnold-Liouville theorem describes the geometry of integrable Hamiltonian systems near a regular level set of the moment map as mentioned in this paper, and it can be decomposed topologically, together with the associated singular Lagrangian foliation, to a direct product of simplest (codimension 1 and codimension 2) singularities.
Abstract: The classical Arnold-Liouville theorem describes the geometry of an integrable Hamiltonian system near a regular level set of the moment map. Our results describe it near a nondegenerate singular level set: a tubular neighborhood of a connected singular nondegenerate level set, after a normal finite covering, admits a non-complete system of action-angle functions (the number of action functions is equal to the rank of the moment map), and it can be decomposed topologically, together with the associated singular Lagrangian foliation, to a direct product of simplest (codimension 1 and codimension 2) singularities. These results are essential for the global topological study of integrable Hamiltonian systems.
TL;DR: In this paper, an integral representation for multi-point Local Height Probabilities for the Andrews-Baxter-Forrester model in regime III was derived by using the bosonization technique.
TL;DR: In this article, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to 1 + 2 dimensions.
TL;DR: In this paper, a new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev-Petviashvili equation.
Abstract: A new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev–Petviashvili equation. The integrability property is explicitly demonstrated, by exhibiting the corresponding Lax pair, that is obtained by applying the reduction technique to the Lax pair of the Kadomtsev–Petviashvili equation. This model equation is likely to be of applicative relevance, because it may be considered a consistent approximation of a large class of nonlinear evolution PDEs.
TL;DR: In this article, it was shown that in the small amplitude, long wave limit there exists an explicit transformation which maps these equations to a system of two integrable equations, and that the concepts of master symmetries and of bi-Hamiltonian structures can be used to obtain similar results for other physical systems.
Abstract: The asymptotic integrability of the idealized water waves is formally established. Namely, it is shown that in the small amplitude, long wave limit there exists an explicit transformation which maps these equations to a system of two integrable equations. It is also shown that the concepts of master symmetries and of bi-Hamiltonian structures can be used to obtain similar results for other physical systems.
TL;DR: In this paper, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to $1+2$ dimensions.
Abstract: The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to $1+2$ dimensions.
TL;DR: In this paper, it was shown that the equation which describes constant mean curvature surfaces via the generalized Weierstrass - Enneper induction has Hamiltonian form, and the interpretation of well known Delaunay and do Carmo - Dajczer surfaces via an integrable finite-dimensional Hamiltonian system was established.
Abstract: It is shown that the equation which describes constant mean curvature surfaces via the generalized Weierstrass - Enneper induction has Hamiltonian form. Its simplest finite-dimensional reduction is the integrable Hamiltonian system with two degrees of freedom. This finite-dimensional system admits -action and classes of -equivalence of its trajectories are in one-to-one correspondence with different helicoidal constant mean curvature surfaces. Thus the interpretation of well known Delaunay and do Carmo - Dajczer surfaces via an integrable finite-dimensional Hamiltonian system is established.
Posted Content•
TL;DR: In this article, the authors used algebraic Bethe Ansatz for solving integrable models and showed how it works in detail on the simplest example of spin 1/2 XXX magnetic chain.
Abstract: I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated more superficially, only the specific details are given. Several parameters, appearing in these generalizations: spin $s$, anisotropy parameter $\ga$, shift $\om$ in the alternating chain, allow to include in our treatment most known examples of soliton theory, including relativistic model of Quantum Field Theory.
Posted Content•
TL;DR: In this paper, a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric Yang-Mills theories and soliton equations is provided. But their phase spaces are Jacobian-type bundles over the leaves of a foliation in a universal configuration space.
Abstract: We provide a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric Yang-Mills theories and soliton equations. Their phase spaces are Jacobian-type bundles over the leaves of a foliation in a universal configuration space. On one hand, imbedded into finite-gap solutions of soliton equations, these symplectic forms assume explicit expressions in terms of the auxiliary Lax pair, expressions which generalize the well-known Gardner-Faddeev-Zakharov bracket for KdV to a vast class of 2D integrable models; on the other hand, they determine completely the effective Lagrangian and BPS spectrum when the leaves are identified with the moduli space of vacua of an N=2 supersymmetric gauge theory. For SU($N_c$) with $N_f\leq N_c+1$ flavors, the spectral curves we obtain this way agree with the ones derived by Hanany and Oz and others from physical considerations.
TL;DR: The symmetric space sine-Gordon models as mentioned in this paper arise by conformal reduction of ordinary 2-dim σ-models, and they are integrable exhibiting a black-hole type metric in target space.
TL;DR: In this paper, the authors derived the corresponding discrete equation, which is symmetric with respect to all permutations of the three coordinates, in the continuous limit, and associated with the BKP hierarchy.
Abstract: The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to three dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable three-dimensional lattice models, and has related them to solutions to the local YBE. The simplest Korepanov model is related to the star-triangle relation in the Ising model. In this Letter the corresponding discrete equation is derived. In the continuous limit it leads to a differential three-dimensional equation, which is symmetric with respect to all permutations of the three coordinates. A similar analysis of the star-triangle transformation in electric networks leads to the discrete bilinear equation of Miwa, associated with the BKP hierarchy.
TL;DR: In this article, an integral representation for multi-point Local Hight Probabilities for the Andrews-Baxter-Forrester model in the regime III was derived by using the bosonization technique.
Abstract: By using the bosonization technique, we derive an integral representation for multi-point Local Hight Probabilities for the Andrews-Baxter-Forrester model in the regime III. We argue that the dynamical symmetry of the model is provided by the deformed Virasoro algebra.
TL;DR: In this paper, an exactly integrable symplectic correspondence is derived which in a continuum limit leads to equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider.
Abstract: An exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider. For the discrete-time model the equations of motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2 XYZ Heisenberg magnet. We present a Lax pair, the sympletic structure and prove the involutivity of the invariants. Exact solutions are investigated in the rational and hyperbolic (trigonometric) limits of the system that is given in terms of elliptic functions. These solutions are connected with discrete soliton equations. The results obtained allow us to consider the Bethe Ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding Bethe wavefunction.
TL;DR: In this paper, the authors adapted Hitchin's integrable systems to the case of a punctured curve, and constructed a dynamical $r$-matrix for Hitchin systems for a puncture elliptic curve.
Abstract: We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of $\CC P^{1}$ and $SL_{n}$-bundles, they are equivalent to systems studied by Garnier. The corresponding quantum systems were identified by B. Feigin, E. Frenkel and N. Reshetikhin with Gaudin systems. We give a formula for the higher Gaudin operators, using results of R. Goodman and N. Wallach on the center of the enveloping algebras of affine algebras at the critical level. Finally we construct a dynamical $r$-matrix for Hitchin systems for a punctured elliptic curve, and $GL_{n}$-bundles, and (for $n=2$) the corresponding quantum system.
TL;DR: In this paper, the authors obtained determinant representation for the correlation function of the quantum nonlinear Schrodinger equation out of a free fermionic point and derived completely integrable equation and asymptotic for the quantum correlation function.
Abstract: The foundation for the theory of correlation functions of exactly solvable models is determinant representation. Determinant representation permit to describe correlation functions by classical completely integrable differential equations [Barough, McCoy, Wu]. In this paper we show that determinant represents works not only for free fermionic models. We obtained determinant representation for the correlation function $ $ of the quantum nonlinear Schr\"odinger equation, out of free fermionic point. In the forthcoming publications we shall derive completely integrable equation and asymptotic for the quantum correlation function of this model of interacting fermions.