Showing papers on "Integrable system published in 1984"

The quantum inverse scattering method approach to correlation functions

Abstract: The inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with theR-matrix of XXX-type. These models include the one-dimensional Bose-gas and the Heisenberg XXX-model. The algebraic questions of the problem are considered.

The sine‐Gordon equations: Complete and partial integrability

Abstract: The sine–Gordon equation in one space‐one time dimension is known to possess the Painleve property and to be completely integrable. It is shown how the method of ‘‘singular manifold’’ analysis obtains the Backlund transform and the Lax pair for this equation. A connection with the sequence of higher‐order KdV equations is found. The ‘‘modified’’ sine–Gordon equations are defined in terms of the singular manifold. These equations are shown to be identically Painleve. Also, certain ‘‘rational’’ solutions are constructed iteratively. The double sine–Gordon equation is shown not to possess the Painleve property. However, if the singular manifold defines an ‘‘affine minimal surface,’’ then the equation has integrable solutions. This restriction is termed ‘‘partial integrability.’’ The sine–Gordon equation in (N+1) variables (N space, 1 time) where N is greater than one is shown not to possess the Painleve property. The condition of partial integrability requires the singular manifold to be an ‘‘Einstein space with null scalar curvature.’’ The known integrable solutions satisfy this constraint in a trivial manner. Finally, the coupled KdV, or Hirota–Satsuma, equations possess the Painleve property. The associated ‘‘modified’’ equations are derived and from these the Lax pair is found.

A proof of Kolmogorov’s theorem on invariant tori using canonical transformations defined by the Lie method

Abstract: In this paper a proof is given of Kolmogorov’s theorem on the existence of invariant tori in nearly integrable Hamiltonian systems. The scheme of proof is that of Kolmogorov, the only difference being in the way canonical transformations near the identity are defined. Precisely, use is made of the Lie method, which avoids any inversion and thus any use of the implicit-function theorem. This technical fact eliminates a spurious ingredient and simplifies the establishment of a central estimate.

On the complete and partial integrability of non-Hamiltonian systems

Abstract: The methods of singularity analysis are applied to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model. Complete integrability is defined and new completely integrable systems are discovered by means of the Painleve property. In all these cases we obtain integrals, which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation (ODE) solved by Painleve transcendents. Relaxing the Painleve property we find many partially integrable cases whose movable singularities are poles at leading order, with In( t - t 0 ) terms entering at higher orders. In an N th order, generalized Rossler model a precise relation is established between the partial fulfillment of the Painleve conditions and the existence of N - 2 integrals of the motion.

New Conserved Quantities and Test for Regular Spectra

Abstract: When Einstein-Brillouin-Keller quantization is possible, it applies to all conserved dynamical variables (not only to the Hamiltonian) and in particular to the time average of any dynamical variable. Thus, for an integrable system of $n$ degrees of freedom, the eigenvalues of $n$ independent constants of motion form a locally regular $n$-dimensional lattice, in the limit $h\ensuremath{\rightarrow}0$. Failure of doing that may be an indication of quantum chaos.

A new characterization of completely integrable systems

Abstract: A characterization of separability, projectability and integrability of dynamical systems in terms of the spectral properties of invariant mixed tensor fields with vanishing Nijenhuis tensor is given. In addition, some preliminary results on the inverse problem (from Liouville integrability to Lax representation) are illustrated.

The algebraic quantisation of the Birkhoff-Gustavson normal form

Abstract: The author develops an algebraic quantisation method for the Birkhoff-Gustavson normal form. For this purpose the Weyl quantisation rule is used. The method developed here for multidimensional systems allows one to calculate the energy levels and the transition probabilities. The author gives a brief review of the normal form, derives some of its general properties, and finds a general analytic solution for the fourth-degree normal form for Hamiltonians of two degrees of freedom. In particular, this includes the Henon-Heiles system. The author compares the results of specific examples with other works. The question of canonically invariant quantisation, the relation to the quantum mechanical perturbation theory and the question of chaotic behaviour and quantum stochasticity are discussed. The author shows that the operators corresponding to the formal integrals of the motion are also quantum mechanical integrals. If the normal form accidentally terminates, so that the classical system is integrable, then this implies quantum integrability of the normal-form Hamiltonian.

On the weak-noise limit of Fokker-Planck models

Abstract: The weak-noise limit of Fokker-Planck models leads to a set of nonlinear Hamiltonian canonical equations. We show that the existence of a nonequilibrium potential in the weak-noise limit requires the existence of whiskered tori in the Hamiltonian system and, therefore, the complete integrability of the latter. A specific model is considered, where the Hamiltonian system in the weak-noise limit is not integrable. Two different perturbative solutions are constructed: the first solution describes analytically the breakdown of the whiskered tori due to the appearance of wild separatrices; the second solution allows the analytic construction of an approximate nonequilibrium potential and an asymptotic expression for the probability density in the steady state.

Hamilton-Jacobi theory for constrained systems

Abstract: We extend the Hamilton–Jacobi formulation to constrained dynamical systems. The discussion covers both the case of first‐class constraints alone and that of first‐ and second‐class constraints combined. The Hamilton–Dirac equations are recovered as characteristic of the system of partial differential equations satisfied by the Hamilton–Jacobi function.

The soliton correlation matrix and the reduction problem for integrable systems

Abstract: Integrable 1+1 dimensional systems associated to linear first-order matrix equations meromorphic in a complex parameter, as formulated by Zakharov, Mikhailov, and Shabat [1−3] (ZMS) are analyzed by a new method based upon the “soliton correlation matrix” (M-matrix). The multi-Backlund transformation, which is equivalent to the introduction of an arbitrary number of poles in the ZMS dressing matrix, is expressed by a pair of matrix Riccati equations for theM-matrix. Through a geometrical interpretation based upon group actions on Grassman manifolds, the solution of this system is explicitly determined in terms of the solutions to the ZMS linear system. Reductions of the system corresponding to invariance under finite groups of automorphisms are also solved by reducing theM-matrix suitably so as to preserve the class of invariant solutions.

A note on integration of the Emden-Fowler equation

Abstract: Using a simple change of variables, the Emden-Fowler equation, ( x v + α y ′)′ + ax v y n = 0 is shown to be integrable provided that either of the constraints ( v + α − 1) n = 3 − α + v or ( v + α − 1) n = 3 − 2 α − v is satisfied. Every integrable case generates a one parameter family of integrable Emden-Fowler equations.

Explicit integrability for Hamiltonian systems and the Painlevé conjecture

Abstract: We analyze a class of Hamiltonian systems in two dimensions for which we proved that a second constant of motion exists. It is shown that, using the two first integrals, the equations of motion can be written in a form which allows their integration by quadratures. An analysis of the equations of motion in this reduced form establishes the behavior of the solutions in the complex‐time plane. It is shown explicitly that the systems belonging to this class possess the ‘‘weak Painleve’’ property, i.e., their solutions in complex time can present singularities of a specific algebraic type.

The forced Toda lattice: An example of an almost integrable system

Abstract: A method for solving forced integrable systems is presented. The method requires the knowledge of at least one piece of information about the solution. Once this is known, one may then construct the remainder of the solution. In this sense these systems are ‘‘almost integrable.’’ The forced semi‐infinite Toda lattice is used as an example and to illustrate the method.

Integrable systems as nonlinear realizations of infinite-dimensional symmetries: The Liouville equation example

Abstract: The Liouville equation is shown to have a natural interpretation in terms of the nonlinear realization of an infinite parameter conformal group in 1+1-dimensions. The relevant zero-curvature representation and Backlund transformations get a simple treatment in this approach. The proposed construction can hopefully be generalized to other integrable systems.

New Completely Integrable Multiparticle Dynamical Systems

Abstract: Complete integrability is established for some N-body dynamical problems on the motion of particle systems with a binary interaction in an external field.

Integrable graded manifolds and nonlinear equations

Abstract: A method is proposed for the classification of integrable embeddings of (2+2)-dimensional supermanifoldsV2|2 into an enveloping superspace supplied with the structure of a Lie superalgebra. The approach is first applied to the “even part” of the scheme, i.e. for the embeddings of 2-dimensional manifoldsV2 into Riemannian or non-Riemannian enveloping space. The general consideration is also illustrated by the example of superspaces supplied with the structure of the series sl(n, n+1), whose integrable supermanifolds are described by supersymmetrical 2-dimensional Toda lattice type equations. In particular, forn=1 they are described by the supersymmetrical Liouville and Sine-Gordon equations.

Homogeneous spaces with integrable g-invariant hamiltonian flows

Abstract: Examples are constructed of homogeneous spaces M with semisimple groups of motions G for which all G-invariant Hamiltonian systems on T*M are integrable. Particular examples of such include affine symmetric spaces. Bibliography: 11 titles.