Liouville Integrable Hamiltonian Systems Research Articles

Periodic motions of the Kowalevski gyrostat in two constant fields

The case of the gyrostat motion found by A G Reyman and M A Semenov-Tian-Shansky is known as the Liouville integrable Hamiltonian system with three degrees of freedom without symmetry groups. We find the set of points at which the integral map has rank 1. This set consists of special periodic motions generating the singular points of bifurcation diagrams on iso-energetic surfaces. For such motions, all phase variables are expressed as algebraic functions of one auxiliary variable satisfying the differential equation integrable in elliptic functions of time. It is shown that the corresponding points in three-dimensional space of the integral constants belong to the intersection of two sheets of the discriminant surface of the Lax curve.

A hierarchy of Liouville integrable discrete Hamiltonian equations

Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.

A HIERARCHY OF LATTICE SOLITON EQUATIONS AND ITS HIGHER-ORDER SYMMETRY CONSTRAINT

Starting from a discrete matrix spectral problem, a hierarchy of lattice soliton equations is derived through a discrete zero curvature representation. The Hamiltonian structures are established for the resulting hierarchy. Then the higher-order symmetry constraint for the resulting hierarchy is studied. It is shown that under the higher-order symmetry constraint, each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional Liouville integrable Hamiltonian system.

Bifurcation of coisotropic invariant tori under locally Hamiltonian perturbations of integrable systems and nondegenerate deformation of symplectic structure

We study the bifurcation problem for a Cantor set of coisotropic invariant tori in the case where a Liouville-integrable Hamiltonian system undergoes locally Hamiltonian perturbations and, simultaneously, a deformation of the symplectic structure of the phase space. We consider a new case where the deformed symplectic structure generates a nondegenerate matrix of the Poisson brackets of action variables.

A model for separatrix splitting near multiple resonances

We propose a model for local dynamics of a perturbed convex real-analytic Liouvilleintegrable Hamiltonian system near a resonance of multiplicity 1 + m, m ≥ 0. Physically, the model represents a toroidal pendulum, coupled with a Liouville-integrable system of n nonlinear rotators via a small analytic potential. The global bifurcation problem is set-up for the n-dimensional isotropic manifold, corresponding to a specific homoclinic orbit of the toroidal pendulum. The splitting of this manifold can be described by a scalar function on an n-torus, whose kth Fourier coefficient satisfies the estimate O

Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint

Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given.

A New Integrable Hierarchy, Parametric Solutions and Traveling Wave Solutions

In this paper we give a new integrable hierarchy. In the hierarchy there are the following representatives: $$\begin{gathered} u_t = \partial _x^5 u^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} 3}} \right. \kern-\nulldelimiterspace} 3}} , \hfill \\ u_t = \partial _x^5 \frac{{\left( {u^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 3}} \right. \kern-\nulldelimiterspace} 3}} } \right)_{xx} - 2\left( {u^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 6}} \right. \kern-\nulldelimiterspace} 6}} } \right)_x^2 }}{u}, \hfill \\ u_{xxt} + 3u_{xx} u_x + u_{xxx} u_x = 0 \hfill \\ \end{gathered} $$ The first two are the positive members of the hierarchy, and the first equation was a reduction of an integrable (2+1)-dimensional system (see B. G. Konopelchenko and V. G. Dubrovsky, Phys. Lett. A 102 (1984), 15–17). The third one is the first negative member. All nonlinear equations in the hierarchy are shown to have 3×3 Lax pairs through solving a key 3×3 matrix equation, and therefore they are integrable. Under a constraint between the potential function and eigenfunctions, the 3×3 Lax pair and its adjoint representation are nonlinearized to be two Liouville-integrable Hamiltonian systems. On the basis of the integrability of 6N-dimensional systems we give the parametric solution of all positive members in the hierarchy. In particular, we obtain the parametric solution of the equation u t =∂5 x u −2/3. Finally, we present the traveling wave solutions (TWSs) of the above three representative equations. The TWSs of the first two equations have singularities, but the TWS of the 3rd one is continuous. The parametric solution of the 5th-order equation u t =∂5 x u −2/3 can not contain its singular TWS. We also analyse Gaussian initial solutions for the equations u t =∂5 x u −2/3, and u xxt +3u xx u x +u xxx u=0. Both of them are stable.

Vanishing twist near focus–focus points

We show that near a simple focus–focus value in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy–momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy–momentum map that is transversal to the line of constant energy. In contrast to this, we also show that the frequency map is non-degenerate for every point in a neighbourhood of a simple focus–focus point.

A hierarchy of discrete Hamiltonian equations and its binary nonlinearization by symmetry constraint

A discrete matrix spectral problem is introduced, and a hierarchy of nonlinear lattice equations is derived. It is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. An integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonlinearization of the Lax pair and adjoint Lax pair for the resulting hierarchy. The binary Bargmann symmetry constraint leads to Bäcklund transformation for the resulting nonlinear integrable lattice equations.

A hierarchy of discrete Hamiltonian equations and its binary nonlinearization by symmetry constraint

A discrete matrix spectral problem is introduced, and a hierarchy of nonlinear lattice equations is derived. It is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. An integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonlinearization of the Lax pair and adjoint Lax pair for the resulting hierarchy. The binary Bargmann symmetry constraint leads to Bäcklund transformation for the resulting nonlinear integrable lattice equations.

A Hierarchy of Lax Integrable Lattice Equations, Liouville Integrability and a New Integrable Symplectic Map**The project supported by the Scientific Research Award Foundation for Outstanding Young and Middle-Aged Scientists of Shandong Province of China

A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.

Spinning gas clouds: III. Solutions of minimal energy with precession

We consider the model of rotating and expanding gas cloud originally proposed by Ovsiannikov (1956 Dokl. Akad. Nauk SSSR 111 47) and Dyson (1968 J. Math. Mech. 18 91). Under the restricting assumptions of an adiabatic index γ = 5/3 and of vorticity-free motion, this has been shown (Gaffet 2001 J. Phys. A: Math. Gen. 34 2097) to be a Liouville integrable Hamiltonian system. In the present work, we consider the precessing solutions where the cloud does not retain a fixed rotation axis. Choosing for definiteness a particular set of constants of motion (which corresponds to a minimum of the energy), we show that a separation of variables occurs, and that the equations of motion are reducible to the form of a Riccati equation, whose integration merely involves an elliptic integral.

Bifurcations of first integrals in the Sokolov case

We study the phase topology of a new Liouville integrable Hamiltonian system with an additional quartic integral (the Sokolov case).

New Liouville integrable noncanonical Hamiltonian systems from the AKNS spectral problem

Liouville integrable noncanonical Hamiltonian systems with variable coefficient symplectic forms are generated from the AKNS 3×3 matrix Lax pairs by binary symmetry constraints of the AKNS hierarchy. These integrable systems provide new integrable factorization for every AKNS system in the hierarchy.

A Liouville integrable Hamiltonian system associated with a generalized Kaup–Newell spectral problem

Starting from a generalized Kaup–Newell spectral problem involving an arbitrary function, we derive a hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as Kaup–Newell equation, Chen–Lee–Liu equation, Gerdjikov–Ivanov equation, Burgers equation, modified Korteweg–de Vries equation and Sharma–Tasso–Olever equation. It is also shown that the hierarchy is integrable in Liouville's sense and possesses multi-Hamiltonian structure.

On the structure of Picard–Fuchs type equations for Liouville–Arnold integrable Hamiltonian system on cotangent phase spaces

There are studied in detail the structure properties of integral submanifold imbedding mapping for a class of algebraically Liouville integrable Hamiltonian systems on cotangent phase spaces and related with it so called Picard–Fuchs type equations. It is shown that these equations can be in general regularly constructed making use of a given a priori system of involutive invariants and proved that their solutions in the Hamolton–Jacobi separable variable case give rise exactly to the integral submanifold imbedding mapping being as known a main ingredient for Liouville–Arnold integrability by quadratures of the Hamiltonian system under regard.

Binary Bargmann symmetry constraints of soliton equations

Binary Bargmann symmetry constraints are applied to decompose soliton equations into finite-dimensional Liouville integrable Hamiltonian systems, generated from so-called constrained flows. The resulting constraints on the potentials of soliton equations give rise to involutive solutions to soliton equations, and thus the integrability by quadratures are shown for soliton equations by the constrained flows. The multi-wave interaction equations associated with the $3\times 3$ matrix AKNS spectral problem are chosen as an illustrative example to carry out binary Bargmann symmetry constraints. The Lax representations and the corresponding ${\bf r}$-matrix formulation are established for the constrained flows of the multi-wave interaction equations, and the integrals of motion generated from the Lax representations are utilized to show the Liouville integrability for the resulting constrained flows. Finally, involutive solutions to the multi-wave interaction equations are presented.

A class of Liouville-integrable Hamiltonian systems with two degrees of freedom

A class of two-dimensional Liouville-integrable Hamiltonian systems is studied. Separability of the corresponding Hamilton–Jacobi equation for these systems is shown to be equivalent to other fundamental properties of Hamiltonian systems, such as the existence of the Lax and bi-Hamiltonian representations of certain fixed types. Applications to physical models, including the Calogero–Moser model, an integrable case of the Hénon-Heiles potential and the nonperiodic Toda lattice are presented.

Do symmetry constraints nonlinearize spectral problems into Hamiltonian systems?

Symmetry constraints provide a way for constructing Liouville integrable finite-dimensional Hamiltonian systems. An important question is whether symmetry constraints always yield Hamiltonian systems through nonlinearization of spectral problems of soliton systems. The AKNS spectral problem is taken as an illustrative example to give a negative answer. A sort of symmetry constraints is introduced for the AKNS system of nonlinear Schrödinger equations, which generates non-Hamiltonian systems from the AKNS spectral problem.

SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS BY BINARY NONLINEARIZATION

In this paper, the translation of the Lax pairs of the Levi equations is presented. Then a symmetry constraint for the Levi equations is given by means of binary nonlinearization method. The spatial part and the temporal parts of the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrainted as finite dimensional Liouville integrable Hamiltonian systems. Finally, the involutive solutions of the Levi equations are presented.