Showing papers on "Integrable system published in 1986"

Chaos and nonlinear dynamics of single‐particle orbits in a magnetotaillike magnetic field

Abstract: The properties of charged-particle motion in Hamiltonian dynamics are studied in a magnetotaillike magnetic field configuration. It is shown by numerical integration of the equation of motion that the system is generally nonintegrable and that the particle motion can be classified into three distinct types of orbits: bounded integrable orbits, unbounded stochastic orbits, and unbounded transient orbits. It is also shown that different regions of the phase space exhibit qualitatively different responses to external influences. The concept of 'differential memory' in single-particle distributions is proposed. Physical implications for the dynamical properties of the magnetotail plasmas and the possible generation of non-Maxwellian features in the distribution functions are discussed.

Additional symmetries for integrable equations and conformal algebra representation

Abstract: We present a regular procedure for constructing an infinite set of additional (spacetime variables explicitly dependent) symmetries of integrable nonlinear evolution equations (INEEs). In our method, additional symmetry equations arise together with their L-A pairs, so that they are integrable themselves. This procedure is based on a modified ‘dressing’ method. For INEEs in 1+1 dimensions, some appropriate symmetry equations are shown to form the vector fields on a circle S1 algebra representation. In contrast to the so-called isospectral deformations, these symmetries result from conformal transformations of the associated linear problem spectrum. For INEEs in 2+1 dimensions, the commutation relations for symmetry equations are shown to coincide with operators \(\lambda ^m \partial _\lambda \), with integer m, p. Some additional results about Kac-Moody algebra applications are presented.

Exact integrability of the one-dimensional Hubbard model.

Abstract: The 1D Hubbard model is shown to be an exactly integrable system. A "covering" model of 2D statistical mechanics which I proposed recently was shown to provide a one-parameter family of transfer matrices, commuting with the Hamiltonian of the Hubbard model. I show in this work that any two transfer matrices of a family commute mutually. At the root of the commutation relation is the ubiquitous Yang-Baxter factorization condition. The form of the $R$ operator is displayed explicitly.

Integrable graded magnets

Abstract: Solutions of the graded Yang-Baxter equations are constructed which are invariant relative to the general linear and orthosymplectic supergroups. The Hamiltonians and other higher integrals (the transfer matrix) of spin systems on a finite lattice connected with the solutions found are diagonalized. A generalization of the Yang-Baxter equation to the case of the δ-commutative, G-graded Zamolodchikov algebra is presented.

Are all the equations of the Kadomtsev–Petviashvili hierarchy integrable?

Abstract: The Kadomtsev–Petviashvili (KP) hierarchy is an infinite set of nonlinear partial differential equations in which the number of independent variables increases indefinitely as one proceeds down the hierarchy. Since these equations were obtained as part of a group theoretical approach to soliton equations it would appear that the KP hierarchy provides integrable scalar equations with an arbitrary number of independent variables. It is shown, by investigating a specific equation in 3+1 dimensions, that the higher equations in the KP hierarchy are only integrable in a conditional sense. The equation under study, taken in isolation, does not pass certain well‐known and reliable integrability tests. Thus, applying Painleve analysis, we find that solutions exist, allowing movable critical points. Furthermore, solitary wave solutions are shown to exist that do not behave like solitons in multiple collisions. On the other hand, if the dependence of a solution on the first 2+1 variables is restricted by the fact that it should also satisfy the KP equation itself, then the integrability conditions in the other dimensions are satisfied. ‘‘Conditional integrability’’ thus means that linear techniques will provide only those solutions of equations in the hierarchy that simultaneously satisfy lower equations in the same hierarchy.

Stability of Motions near Resonances in Quasi-Integrable Hamiltonian Systems

Abstract: Nekhoroshev's theorem on the stability of motions in quasi-integrable Hamiltonian systems is revisited. At variance with the proofs already available in the literature, we explicitly consider the case of weakly perturbed harmonic oscillators; furthermore we prove the confinement of orbits in resonant regions, in the general case of nonisochronous systems, by using the elementary idea of energy conservation instead of more complicated mechanisms. An application of Nekhoroshev's theorem to the study of perturbed motions inside resonances is also provided.

GL/sub 3/-invariant solutions of the Yang-Baxter equation and associated quantum systems

Abstract: GL3-invariant, finite-dimensional solutions of the Yang-Baxter equations acting in the tensor product of two irreducible representations of the group GL3 are investigated. A number of relations are obtained for the transfer matrices which demonstrate the connection of representation theory and the Bethe Ansatz in GL3-invariant models. Some of the most interesting quantum and classical integrable systems connected with GL3-invariant solutions of the Yang-Baxter equation are presented.

Bäcklund transformation and the Painlevé property

Abstract: When a differential equation possesses the Painleve property it is possible (for specific equations) to define a Backlund transformation (by truncating an expansion about the ‘‘singular’’ manifold at the constant level term). From the Backlund transformation, it is then possible to derive the Lax pair, modified equations and Miura transformations associated with the ‘‘completely integrable’’ system under consideration. In this paper completely integrable systems are considered for which Backlund transformations (as defined above) may not be directly defined. These systems are of two classes. The first class consists of equations of Toda lattice type (e.g., sine–Gordon, Bullough–Dodd equations). We find that these equations can be realized as the ‘‘minus‐one’’ equation of sequences of integrable systems. Although the ‘‘Backlund transformation’’ may or may not exist for the ‘‘minus‐one’’ equation, it is shown, for specific sequences, that the Backlund transformation does exist for the ‘‘positive’’ equations of the sequence. This, in turn, allows the derivation of Lax pairs and the recursion operation for the entire sequence. The second class of equations consists of sequences of ‘‘Harry Dym’’ type. These equations have branch point singularities, and, thus, do not directly possess the Painleve property. Yet, by a process similar to the ‘‘uniformization’’ of algebraic curves, their solutions may be parametrically’’ represented by ‘‘meromorphic’’ functions. For specific systems, this is shown to provide a natural extension of the Painleve property.

Nonlinear systems of partial differential equations in applied mathematics

Abstract: These two volumes of 47 papers focus on the increased interplay of theoretical advances in nonlinear hyperbolic systems, completely integrable systems, and evolutionary systems of nonlinear partial differential equations. The papers both survey recent results and indicate future research trends in these vital and rapidly developing branches of PDEs. The editor has grouped the papers loosely into the following five sections: integrable systems, hyperbolic systems, variational problems, evolutionary systems, and dispersive systems. However, the variety of the subjects discussed as well as their many interwoven trends demonstrate that it is through interactive advances that such rapid progress has occurred.These papers require a good background in partial differential equations. Many of the contributors are mathematical physicists, and the papers are addressed to mathematical physicists (particularly in perturbed integrable systems), as well as to PDE specialists and applied mathematicians in general.

On the integrability of systems of nonlinear ordinary differential equations with superposition principles

Abstract: A new class of ‘‘solvable’’ nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations (ODE’s) describing each member of this class possess nonlinear superposition principles. These systems of ODE’s are generally not derived from a Hamiltonian and are classified by associated pairs of Lie algebras of vector fields. In this paper, all such systems of n≤3 ODE’s are integrated in a unified way by finding explicit integrals for them and relating them all to a ‘‘pivotal’’ member of their class: the projective Riccati equations. Moreover, by perturbing two parametrically driven projective Riccati equations (thus making them nonsolvable in the above sense) evidence is discovered of chaotic behavior on the Poincare surface of section—in the form of sensitive dependence on initial conditions—near a boundary separating bounded from unbounded motion.

Complete integrability in a quantum description of chaotic systems.

Abstract: For quantum bound systems whose classical versions are nonintegrable, coupled dynamical equations for both energy levels and eigenfunctions with a nonintegrability parameter $t$ taken as "time" are shown to be a completely integrable Calogero-Moser system in 1 + 1 dimensions with internal complex vector space. Lax forms and their complete algebraic solutions are given which, in place of statistical mechanics procedures, determine possible energy spectrum and wave-function patterns at an arbitrary value of $t$.

Periodic fixed points of Bäcklund transformations and the Korteweg–de Vries equation

Abstract: A new method for studying integrable systems based on the ‘‘periodic fixed points’’ of Backlund transformations (BT’s) is presented. Normally the BT maps an ‘‘old’’ solution into a ‘‘new’’ solution and requires a known ‘‘seed’’ solution to get started. Besides this limitation, it can also be difficult to qualitatively classify the result of applying the BT several times to a known solution. By studying the periodic fixed points of the BT (regarded as a nonlinear map in a function space), integrable systems of equations of finite degree (equal to the order of the fixed point) and a method for the systematic classification of the solutions of the original system are obtained.

Sta¨kel equivalent integrable Hamiltonian systems

Abstract: The Stackel transform is a mapping of the commuting constants of the motion (corresponding to a separable coordinate system) for one completely integrable classical or quantum Hamiltonian system to the constants of the motion for another such system. Here the transform is defined and given an intrinsic characterization, and a large family of nontrivial examples is worked out of systems which are “Stackel equivalent”. Among the simplest examples are geodesic flow on an n-dimensional ellipsoid with distinct axes, which is equivalent to the motion of a mass point on the unit sphere in $R^{n + 1} $ under the influence of a quadratic potential with distinct eigenvalues, and the Kepler (Coulomb) problem in three dimensions which is equivalent to the pseudo-Coulomb problem.

Statistical mechanics of the sine-Gordon equation

Abstract: We give two fundamental methods for evaluation of classical free energies of all the integrable models admitting soliton solutions; the sine-Gordon equation is one example. Periodic boundary conditions impose integral equations for allowed phonon and soliton momenta. From these, generalized Bethe-Ansatz and functional-integration methods using action-angle variables follow. Results for free energies coincide, and coincide with those that we find by transfer-integral methods. Extension to the quantum case, and quantum Bethe Ansatz, on the lines to be reported elsewhere for the sinh-Gordon equation, is indicated.

Proof of the invariance of the Bethe-ansatz solutions under complex conjugation

Abstract: It is shown that the solutions to the systems of algebraic equations obtained when the Bethe-Ansatz is applied to the integrable XXX and XXZ magnets of arbitrary spin are invariant under complex conjugation.

A variable coefficient Korteweg–de Vries equation: Similarity analysis and exact solution. II

Abstract: A Korteweg–de Vries (KdV) equation with time‐dependent coefficients is studied in this paper. The similarity transformation for this system is investigated and an exact solution in a particular case is obtained. The Ablowitz–Ramani–Segur (ARS) conjecture is used to identify the integrability of the system. It is found that in some special cases the system may be integrable.

Super AKNS Scheme and Its Infinite Conserved Currents

Abstract: The 3×3 eigenvalue problem, in which some of the variables are the usual even quantities and others are odd, is considered. Using the methods of one of the authors Li (1982), we deduce a class of completely integrable equations, which can be proved to be pure differential equations. One of its reduced forms is given, which includes the super KdV equation as the special case and their infinite conserved currents and discussed.