Showing papers on "Integrable system published in 1988"

Boundary conditions for integrable quantum systems

Abstract: A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.

Action-angle maps and scattering theory for some finite-dimensional integrable systems. I. The pure soliton case

Abstract: We construct an action-angle transformation for the Calogero-Moser systems with repulsive potentials, and for relativistic generalizations thereof. This map is shown to be closely related to the wave transformations for a large classl of Hamiltonians, and is shown to have remarkable duality properties. All dynamics inl lead to the same scattering transformation, which is obtained explicitly and exhibits a soliton structure. An auxiliary result concerns the spectral asymptotics of matrices of the formM exp(tD) ast→∞. It pertains to diagonal matricesD whose diagonal elements have pairwise different real parts and to matricesM for which certain principal minors are non-zero.

Liouville model on the lattice

Abstract: Liouville equation is put on the lattice in a completely integrable way The classical version is investigated in details and a lattice deformation of the Virasoro algebra is obtained The quantum version still lacks a satisfactory definition of the Hamiltonian

Integrable and chaotic motions of four vortices II. Collision dynamics of vortex pairs

Abstract: The interaction of two vortex pairs is investigated analytically and by numerical experiments from the vantage point of dynamical-systems theory. Vortex pairs can escape to infinity, so the phase space of this system is unbounded in contrast to that of four identical vortices investigated previously. Chaotic motion is nevertheless possible both for ‘bound states’ of the system and for ‘scattering states’. For the bound states standard Poincare section techniques suffice. For scattering states chaos appears as complex structure in the numerically generated plot of scattering angle against impact parameter. Interpretations of physical space mechanisms leading to chaos are given. Analytical characterizations of the system include a formal reduction to two degrees of freedom by canonical transformations and an identification and discussion of integrable cases of which one is apparently new.

Soliton solutions in an integrable chiral model in 2+1 dimensions

Abstract: There is a modified SU(2) chiral model in 2+1 dimensions which is integrable. It admits multisoliton solutions, in which the solitons move at constant velocity, and pass through one another without scattering or changing shape.

The non-local delta problem and (2+1)-dimensional soliton equations

Abstract: A method of constructing (2+1)-dimensional non-linear integrable equations and their solutions by means of the non-local delta problem is developed. A 'basic set' of equations is obtained by using different normalisations of the non-local delta problem and the Lagrangian of the set is found. Other integrable equations, which are degenerate cases of the basic set, are also Lagrangian.

Integrability and Nonintegrability in Geometry and Mechanics

Abstract: 1. Some Equations of Classical Mechanics and Their Hamiltonian Properties.- 1. Classical Equations of Motion of a Three-Dimensional Rigid Body.- 1.1. The Euler-Poisson Equations Describing the Motion of a Heavy Rigid Body around a Fixed Point.- 1.2. Integrable Euler, Lagrange, and Kovalevskaya Cases.- 1.3. General Equations of Motion of a Three-Dimensional Rigid Body.- 2. Symplectic Manifolds.- 2.1. Symplectic Structure in a Tangent Space to a Manifold.- 2.2. Symplectic Structure on a Manifold.- 2.3. Hamiltonian and Locally Hamiltonian Vector Fields and the Poisson Bracket.- 2.4. Integrals of Hamiltonian Fields.- 2.5. The Liouville Theorem.- 3. Hamiltonian Properties of the Equations of Motion of a Three-Dimensional Rigid Body.- 4. Some Information on Lie Groups and Lie Algebras Necessary for Hamiltonian Geometry.- 4.1. Adjoint and Coadjoint Representations, Semisimplicity, the System of Roots and Simple Roots, Orbits, and the Canonical Symplectic Structure.- 4.2. Model Example: SL(n, ?) and sl(n, ?).- 4.3. Real, Compact, and Normal Subalgebras.- 2. The Theory of Surgery on Completely Integrable Hamiltonian Systems of Differential Equations.- 1. Classification of Constant-Energy Surfaces of Integrable Systems. Estimation of the Amount of Stable Periodic Solutions on a Constant-Energy Surface. Obstacles in the Way of Smooth Integrability of Hamiltonian Systems.- 1.1. Formulation of the Results in Four Dimensions.- 1.2. A Short List of the Basic Data from the Classical Morse Theory.- 1.3. Topological Surgery on Liouville Tori of an Integrable Hamiltonian System upon Varying Values of a Second Integral.- 1.4. Separatrix Diagrams Cut out Nontrivial Cycles on Nonsingular Liouville Tori.- 1.5. The Topology of Hamiltonian-Level Surfaces of an Integrable System and of the Corresponding One-Dimensional Graphs.- 1.6. Proof of the Principal Classification Theorem 2.1.2.- 1.7. Proof of Claim 2.1.1.- 1.8.Proof of Theorem 2.1.1. Lower Estimates on the Number of Stable Periodic Solutions of a System.- 1.9. Proof of Corollary 2.1.5.- 1.10 Topological Obstacles for Smooth Integrability and Graphlike Manifolds. Not each Three-Dimensional Manifold Can be Realized as a Constant-Energy Manifold of an Integrable System.- 1.11. Proof of Claim 2.1.4.- 2. Multidimensional Integrable Systems. Classification of the Surgery on Liouville Tori in the Neighbourhood of Bifurcation Diagrams.- 2.1. Bifurcation Diagram of the Momentum Mapping for an Integrable System. The Surgery of General Position.- 2.2. The Classification Theorem for Liouville Torus Surgery.- 2.3. Toric Handles. A Separatrix Diagram is Always Glued to a Nonsingular Liouville Torus Tn Along a Nontrivial (n - 1)-Dimensional Cycle Tn-1.- 2.4. Any Composition of Elementary Bifurcations (of Three Types) of Liouville Tori Is Realized for a Certain Integrable System on an Appropriate Symplectic Manifold.- 2.5. Classification of Nonintegrable Critical Submanifolds of Bott Integrals.- 3. The Properties of Decomposition of Constant-Energy Surfaces of Integrable Systems into the Sum of Simplest Manifolds.- 3.1. A Fundamental Decomposition Q = mI +pII +qIII +sIV +rV and the Structure of Singular Fibres.- 3.2. Homological Properties of Constant-Energy Surfaces.- 3. Some General Principles of Integration of Hamiltonian Systems of Differential Equations.- 1. Noncommutative Integration Method.- 1.1. Maximal Linear Commutative Subalgebras in the Algebra of Functions on Symplectic Manifolds.- 1.2. A Hamiltonian System Is Integrable if Its Hamiltonian is Included in a Sufficiently Large Lie Algebra of Functions.- 1.3. Proof of the Theorem.- 2. The General Properties of Invariant Submanifolds of Hamiltonian Systems.- 2.1. Reduction of a System on One Isolated Level Surface.- 2.2. Further Generalizations of the Noncommutative Integration Method.- 3. Systems Completely Integrable in the Noncommutative Sense Are Often Completely Liouville-Integrable in the Conventional Sense.- 3.1. The Formulation of the General Equivalence Hypothesis and its Validity for Compact Manifolds.- 3.2. The Properties of Momentum Mapping of a System Integrable in the Noncommutative Sense.- 3.3. Theorem on the Existence of Maximal Linear Commutative Algebras of Functions on Orbits in Semisimple and Reductive Lie Algebras.- 3.4. Proof of the Hypothesis for the Case of Compact Manifolds.- 3.5. Momentum Mapping of Systems Integrable in the Noncommutative Sense by Means of an Excessive Set of Integrals.- 3.6. Sufficient Conditions for Compactness of the Lie Algebra of Integrals of a Hamiltonian System.- 4. Liouville Integrability on Complex Symplectic Manifolds.- 4.1. Different Notions of Complex Integrability and Their Interrelation.- 4.2. Integrability on Complex Tori.- 4.3. Integrability on K3-Type Surfaces.- 4.4. Integrability on Beauville Manifolds.- 4.5.Symplectic Structures Integrated without Degeneracies.- 4. Integration of Concrete Hamiltonian Systems in Geometry and Mechanics. Methods and Applications.- 1. Lie Algebras and Mechanics.- 1.1. Embeddings of Dynamic Systems into Lie Algebras.- 1.2. List of the Discovered Maximal Linear Commutative Algebras of Polynomials on the Orbits of Coadjoint Representations of Lie Groups.- 2. Integrable Multidimensional Analogues of Mechanical Systems Whose Quadratic Hamiltonians are Contained in the Discovered Maximal Linear Commutative Algebras of Polynomials on Orbits of Lie Algebras.- 2.1. The Description of Integrable Quadratic Hamiltonians.- 2.2. Cases of Complete Integrability of Equations of Various Motions of a Rigid Body.- 2.3. Geometric Properties of Rigid-Body Invariant Metrics on Homogeneous Spaces.- 3. Euler Equations on the Lie Algebra so(4).- 4. Duplication of Integrable Analogues of the Euler Equations by Means of Associative Algebra with Poincare Duality.- 4.1. Algorithm for Constructing Integrable Lie Algebras.- 4.2. Frobenius Algebras and Extensions of Lie Algebras.- 4.3. Maximal Linear Commutative Algebras of Functions on Contractions of Lie Algebras.- 5. The Orbit Method in Hamiltonian Mechanics and Spin Dynamics of Superfluid Helium-3.- 5. Nonintegrability of Certain Classical Hamiltonian Systems.- 1. The Proof of Nonintegrability by the Poincare Method.- 1.1. Perturbation Theory and the Study of Systems Close to Integrable.- 1.2. Nonintegrability of the Equations of Motion of a Dynamically Nonsymmetric Rigid Body with a Fixed Point.- 1.3. Separatrix Splitting.- 1.4. Nonintegrability in the General Case of the Kirchhoff Equations of Motion of a Rigid Body in an Ideal Liquid.- 2. Topological Obstacles for Complete Integrability.- 2.1. Nonintegrability of the Equations of Motion of Natural Mechanical Systems with Two Degrees of Freedom on High-Genus Surfaces.- 2.2. Nonintegrability of Geodesic Flows on High-Genus Riemann Surfaces with Convex Boundary.- 2.3. Nonintegrability of the Problem of n Gravitating Centres for n > 2.- 2.4. Nonintegrability of Several Gyroscopic Systems.- 3. Topological Obstacles for Analytic Integrability of Geodesic Flows on Non-Simply-Connected Manifolds.- 4. Integrability and Nonintegrability of Geodesic Flows on Two-Dimensional Surfaces, Spheres, and Tori.- 4.1. The Holomorphic 1-Form of the Integral of a Geodesic Flow Polynomial in Momenta and the Theorem on Nonintegrability of Geodesic Flows on Compact Surfaces of Genus g > 1 in the Class of Functions Analytic in Momenta.- 4.2. The Case of a Sphere and a Torus.- 4.3. The Properties of Integrable Geodesic Flows on the Sphere.- 6. A New Topological Invariant of Hamiltonian Systems of Liouville-Integrable Differential Equations. An Invariant Portrait of Integrable Equations and Hamiltonians.- 1. Construction of the Topological Invariant.- 2. Calculation of Topological Invariants of Certain Classical Mechanical Systems.- 3. Morse-Type Theory for Hamiltonian Systems Integrated by Means of Non-Bott Integrals.- References.

Integrable four vortex motion

Abstract: It follows from the Poisson brackets between constants of motion that the motion of four vortices of zero net vorticity is integrable if the total momentum vanishes. The phase space motion of this integrable case is analyzed. One stable and several unstable uniformly rotating configurations are identified.

Some constructions of integrable dynamical systems

Abstract: New constructions of integrable dynamical systems are found that admit representation as Lax matrix equations. A countable set of integrable systems is constructed which in the continuous limit turn into the Korteweg-de Vries equation. For an arbitrary space with finite measure and measure-preserving mapping differential equations are constructed on the space of measurable functions on . Here differentiation is with respect to time and the equations have a countable set of first integrals. Constructions are also given for first integrals of dynamical systems preserving certain differential forms, and new constructions of matrix differential equations having large families of first integrals. Bibliography: 18 titles.

Painlevé analysis of the damped, driven nonlinear Schrödinger equation

Abstract: In this paper we apply the Painleve tests to the damped, driven nonlinear Schrodinger equationwhere a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painleve tests only ifwhere α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrodinger equation, which is known to be completely integrable.

Analysis and solution of a nonlinear second‐order differential equation through rescaling and through a dynamical point of view

Abstract: The solutions of the equation y+yy+βy3=0, where β is a free parameter, are investigated. For β= (1)/(9) the equation is linearizable through an eight‐parameter symmetry group and is completely integrable. For β≠ (1)/(9) only two symmetries subsist, but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of β and according to the initial conditions.

When is a Hamiltonian system separable

Abstract: The Hamiltonian system given by H= 1/2 p2+V(q) with V∈C∞(Rn) is considered. A method for integrating such a system is that of separating the variables in the Hamilton–Jacobi equation. It is known that if such a separation is possible, then it can take place only when the equation is expressed in terms of generalized elliptic coordinates or in a degeneration of these. A criterion is proposed for deciding if separation is possible, and if it is, in which degeneration of elliptic coordinates it takes place.

Lattice equations, hierarchies and Hamiltonian structures

Abstract: By now many integrable systems with soliton solutions are known in two dimensions, for fields u(n1,n2) defined at the sites (nln2) of a two-dimensional (2D) lattice, for time-dependent fields u(n,t) defined at the sites n of a one-dimensional chain, and for fields u(x,t) depending on two continuous variables x and t. The relation between an integrable partial differential equation and its integrable discrete versions can be treated in the framework of the direct linearization method [1, 2] which is based on a linear integral equation with arbitrary measure and contour [3, 4]. In the treatment use is made of Backlund transformations (BT’s) which are generated by scalar multiplications of the free-wave function and/or measure in the integral equation [5]. The integrable lattice equations are obtained in the form of Bianchi identities expressing the commutativity of BT’s. The integrable equations with one or more continuous variables and their direct linearizations are obtained applying suitable continuum limits to the lattice equation and the integral equation at the same time.

The S=1 quantum spin chain with pure biquadratic exchange

Abstract: States of the pure biquadratic quantum spin chain with up to four deviations are calculated exactly for arbitrary chain length. The four-deviation states have a Bethe ansatz form and consist of two interacting two-strings. They have the same form as two-deviation states of an integrable S=1/2 XXZ model. It is found that states with more than four deviations also map onto corresponding S=1/2 states. The ground states of the two systems are not the same for finite N, but appear to become the same in the limit N to infinity .

Miura transformation on a lattice

Abstract: A proposal is made for extending the Miura transformation to a lattice, and this makes it possible to construct a completely integrable discrete variant of Liouville's equation which takes into account singular solutions

From Solitons to Knots and Links

Abstract: Development in the theory of solvable (integrable) models is reviewed. It covers from basic knowledge on completely integrable systems to recent work by the authors. First, soliton theory is briefly summarized. Through the inverse scattering method and its quantum extension, a central concept, commuting transfer matrices, and a key relation, the Yang-Baxter relation, are introduced. Second, it is shown that there exists at least ∞×∞ number of solvable models in two-dimensional statistical mechanics. Third, quantum spin chains corresponding to solvable statistical mechanical models are discussed. In particular, finite temperature extension of Baxter's formula is given. Fourth, a new approach is presented to the classification problem of knots and links. It is shown that link polynomial, topological invariant for knots and links, can be associated with any solvable model in statistical mechanics. In the presentation, universality of the soliton picture in field theory, spin systems and statistical mechanics is observed

Bi‐Hamiltonian formulation of the Kadomtsev–Petviashvili and Benjamin–Ono equations

Abstract: It was shown recently that the Kadomtsev–Petviashvili (KP) equation (an integrable equation in 2+1, i.e., in two‐spatial and one‐temporal dimensions) admits a bi‐Hamiltonian formulation. This was achieved by considering KP as a reduction of a (3+1)‐dimensional system (in the variables x,y1, y2,t). It is shown here, using the KP as a concrete example, that equations in 2+1 possess two bi‐Hamiltonian formulations and two recursion operators. Both Hamiltonian operators associated with the x direction are local; in contrast only one of the Hamiltonian operators associated with the y direction is local. Furthermore, using the Benjamin–Ono equation as a concrete example, it is shown that intergrodifferential equations in 1+1 admit an algebraic formulation analogous to that of equations in 2+1.

The two-dimensional second-order differential spectral problem: compatibility conditions, general BTs and integrable equations

Abstract: Nonlinear evolution systems in two spatial dimensions integrable by the spectral problem ( delta x2- sigma 2 delta y2+ phi 1 delta x+ phi 2 delta y+U(x, y)) psi =0 are considered. It is shown that such systems possess the matrix commutativity representation (T1M,T2M)=0 which is equivalent to the usual 'L-A-B triad' representation of the compatibility condition. General Backlund transformations (BTS) and the general form of integrable equations are found by the recursion operator method.

Nijenhuis $G$-manifolds and Lenard bicomplexes: a new approach to KP systems

Abstract: We suggest a method to extend the theory of recursion operators to integrable Hamiltonian systems in two-space dimensions, like KP systems. The approach aims to stress the conceptual unity of the theories in one and two space dimensions. A sound explanation of the appearance of bilocal operators is also given.

Multidimensional inverse scattering and nonlinear equations

Abstract: Is it possible to duplicate in higher dimensions the great success of the one-dimensional inverse scattering transform (IST) in the exact solution of integrable nonlinear equations? The question immediately splits into two subquestions: i) is it possible to develop a genuinely multidimensional inverse scattering transform? and ii) are there corresponding nonlinear equations solvable by IST? This lecture presented recent joint work with Mark Ablowitz addressing mostly the a%alytical problem i). We have introduced a general method for solving a wide class of multidimensional inverse scattering problems and treating the overdeterminacy issues involved. In [i], [2] we have applied this method to the Classical inverse scattering problems for: a) the time-dependent Schr~dinger operator in ~ × ~n