Liouville Integrable Hamiltonian Systems Research Articles

Geometric Scattering Monodromy

In this paper we give geometric conditions so that the integral mapping of a Liouville integrable Hamiltonian system with a focus-focus equilibrium point has scattering monodromy. Using a complex version of the Morse lemma, we show that scattering monodromy is the same as the scattering monodromy of the standard focus-focus system.

Stäckel transform of Lax equations

AbstractWe construct a map between Lax equations for pairs of Liouville integrable Hamiltonian systems related by a multiparameter Stäckel transform. Using this map, we construct Lax representation for a wide class of separable systems by applying the multiparameter Stäckel transform to Lax equations of suitably chosen systems from a seed class. For a given separable system, we obtain in this way a set of nonequivalent Lax equations parameterized by an arbitrary function of the spectral parameter, as it is in the case of a related seed system.

Bifurcations of Liouville tori in a system of two vortices of positive intensity in a Bose-Einstein condensate

In this paper we consider a completely Liouville integrable Hamiltonian system with two degrees of freedom, which describes the dynamics of two vortex filaments in a Bose-Einstein condensate enclosed in a harmonic trap. For vortex pairs of positive intensity detected bifurcation of three Liouville tori into one. Such bifurcation was found in the integrable case of Goryachev-Chaplygin-Sretensky in the dynamics of a rigid body. For the integrable perturbation of the physical parameter of the intensity ratio, identified bifurcation proved to be unstable, which led to bifurcations of the type of two tori into one and vice versa.

Bifurcations of Liouville Tori in a System of Two Vortices of Positive Intensity in a Bose–Einstein Condensate

In this paper we consider a completely Liouville integrable Hamiltonian system with two degrees of freedom, which describes the dynamics of two vortex filaments in a Bose-Einstein condensate enclosed in a harmonic trap. For vortex pairs of positive intensity detected bifurcation of three Liouville tori into one. Such bifurcation was found in the integrable case of Goryachev-Chaplygin-Sretensky in the dynamics of a rigid body. For the integrable perturbation of the physical parameter of the intensity ratio, identified bifurcation proved to be unstable, which led to bifurcations of the type of two tori into one and vice versa.

DECOMPOSING A NEW NONLINEAR DIFFERENTIAL-DIFFERENCE SYSTEM UNDER A BARGMANN IMPLICIT SYMMETRY CONSTRAINT

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.

Explicit determination of certain periodic motions of a generalized two-field gyrostat

The case of motion of a generalized two-field gyrostat found by V. V. Sokolov and A.V. Tsiganov is known as a Liouville integrable Hamiltonian system with three degrees of freedom. For this system, we find some special periodic motions at which the momentum mapping has rank 1. For such motions, all phase variables can be expressed in terms of algebraic functions of a single auxiliary variable and a set of constants. This auxiliary variable satisfies a differential equation which can be integrated in elliptic functions of time. As an application, the explicit formulas of characteristic exponents for determining the Williamson type of the special periodic motions are obtained.

Finite-dimensional Liouville integrable Hamiltonian systems generated from Lax pairs of a bi-Hamiltonian soliton hierarchy by symmetry constraints

We construct finite-dimensional Hamiltonian systems by means of symmetry constraints from the Lax pairs and adjoint Lax pairs of a bi-Hamiltonian hierarchy of soliton equations associated with the 3-dimensional special linear Lie algebra, and discuss the Liouville integrability of these systems based on the existence of sufficiently many integrals of motion.

The Geometry of the Semiclassical Wave Front Set for Schrödinger Eigenfunctions on the Torus

This paper deals with the phase space analysis for a family of Schrodinger eigenfunctions ψℏ on the flat torus 𝕋n = (ℝ/2πℤ)n by the semiclassical Wave Front Set. We study those ψℏ such that WFℏ(ψℏ) is contained in the graph of the gradient of some viscosity solutions of the Hamilton-Jacobi equation. It turns out that the semiclassical Wave Front Set of such Schrodinger eigenfunctions is stable under viscous perturbations of Mean Field Game kind. These results provide a further viewpoint, and in a wider setting, of the link between the smooth invariant tori of Liouville integrable Hamiltonian systems and the semiclassical localization of Schrodinger eigenfunctions on the torus.

Two integrable systems with integrals of motion of degree four

We discuss the possibility of using second-order Killing tensors to construct Liouville-integrable Hamiltonian systems that are not Nijenhuis integrable. As an example, we consider two Killing tensors with a nonzero Haantjes torsion that satisfy weaker geometric conditions and also three-dimensional systems corresponding to them that are integrable in Euclidean space and have two quadratic integrals of motion and one fourth-order integral in momenta.

On a Poisson–Lie deformation of the BCn Sutherland system

A deformation of the classical trigonometric BCn Sutherland system is derived via Hamiltonian reduction of the Heisenberg double of SU(2n). We apply a natural Poisson–Lie analogue of the Kazhdan–Kostant–Sternberg type reduction of the free particle on SU(2n) that leads to the BCn Sutherland system. We prove that this yields a Liouville integrable Hamiltonian system and construct a globally valid model of the smooth reduced phase space wherein the commuting flows are complete. We point out that the reduced system, which contains 3 independent coupling constants besides the deformation parameter, can be recovered (at least on a dense submanifold) as a singular limit of the standard 5-coupling deformation due to van Diejen. Our findings complement and further develop those obtained recently by Marshall on the hyperbolic case by reduction of the Heisenberg double of SU(n,n).

A simple proof of the geometric fractional monodromy theorem

A simple proof of the “geometric fractional monodromy theorem” (Broer-Efstathiou-Lukina 2010) is presented. The fractional monodromy of a Liouville integrable Hamiltonian system over a loop γ ⊂ ℝ2 is a generalization of the classic monodromy to the case when the Liouville foliation has singularities over γ. The “geometric fractional monodromy theorem” finds, up to an integral parameter, the fractional monodromy of systems similar to the 1: (−2) resonance system. A handy equivalent definition of fractional monodromy is presented in terms of homology groups for our proof.

New Lax pairs of the Toda lattice and the nonlinearization under a higher-order Bargmann constraint

By starting from a new discrete spectral problem, the Toda lattice is derived through the discrete zero curvature equation. Applying the discrete variational identity to the spectral problem will also reach to the bi-Hamiltonian structure of the Toda lattice. Under a higher-order Bargmann symmetry constraint, the new Lax pairs and the adjoint Lax pairs are nonlinearized into integrable symplectic maps and finite-dimensional Liouville integrable Hamiltonian systems. Finally, a Bäcklund transformation of the Toda lattice is obtained.

Higher-Dimensional Lie Algebra and New Integrable Coupling of Discrete KdV Equation

Based on semi-direct sums of Lie suhalgehra , a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. A hierarchy of integrable coupling KdV equation with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, the Hamiltonian forms is deduced for lattice equation in the resulting hierarchy by means of the discrete variational identity — a generalized trace identity. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian systems.

The integrable coupling system of a 3 × 3 discrete matrix spectral problem

Integrable coupling with six potentials is first proposed by coupling a given 3 × 3 discrete matrix spectral problem. It is shown that coupled system of integrable equations can possess zero curvature representations and recursion operators, which yield infinitely many commuting symmetries. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebras, the Hamiltonian form is deduced for the lattice equations in the resulting hierarchy. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian system.

Integrability of Hamiltonian systems and transseries expansions

This paper studies analytic Liouville-nonintegrable and C ∞-Liouville-integrable Hamiltonian systems with two degrees of freedom. We prove the property for a class of Hamiltonians more general than the one studied in Gorni and Zampieri (Differ Geom Appl 22:287–296, 2005). We also show that a certain monodromy property of an ordinary differential equation obtained as a subsystem of a given Hamiltonian and the transseries expansion of a first integral play an important role in the analysis (cf. (4)). In the former half the analytic Liouville-nonintegrability for a class of Hamiltonians satisfying the above condition is shown. For these analytic nonintegrable Hamiltonians one cannot construct nonanalytic first integrals concretely as in Gorni and Zampieri (Differ Geom Appl 22:287–296, 2005). In the latter half, the nonanalytic integrability from the viewpoint of a transseries expansion of a first integral is discussed. More precisely, we construct a first integral in a formal transseries expansion in a general situation. Then we show convergence of transseries or existence of the first integral being asymptotically equal to a given formal transseries solution.

An integrable coupling hierarchy of the Mkdv_integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy

A four-by-four matrix spectral problem is introduced, locality of solution of the related stationary zero curvature equation is proved. An integrable coupling hierarchy of the Mkdv_integrable systems is presented. The Hamiltonian structure of the resulting integrable coupling hierarchy is established by means of the variational identity. It is shown that the resulting integrable couplings are all Liouville integrable Hamiltonian systems. Ultimately, through the nonisospectral zero curvature representation, a nonisospectral integrable hierarchy associated with the resulting integrable couplings is constructed.

Spinning gas clouds: Liouville integrable cases

We consider the class of ellipsoidal gas clouds expanding into a vacuum [1, 2] which has been shown to be a Liouville integrable Hamiltonian system [3]. This system presents several interesting features, such as the Painleve property [4, 5], the existence of Backlund transformations and the separability of variables, all shown to be present in at least several sub-cases.

Role of Singular Tori in the Dynamics of Spatiotemporal Nonlinear Wave Systems

We show that the peculiar topological properties inherent to singular tori play a major role in the spatiotemporal dynamics of counterpropagating nonlinear waves. Under rather general conditions, these Hamiltonian wave systems exhibit a relaxation process towards a stationary state. We show that this stationary state converges exponentially towards the singular torus of the associated Liouville-integrable Hamiltonian system in the limit of an infinite medium. The singular torus then appears as an attractor for the infinite dimensional dynamical system, a feature which is illustrated by several key models of spatiotemporal wave interactions.

The Liouville integrability of integrable couplings of Volterra lattice equation

Abstract A hierarchy of integrable couplings of Volterra lattice equations with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebra, the two Hamiltonian forms are deduced for each lattice equation in the resulting hierarchy. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations are all Liouville integrable discrete Hamiltonian systems.

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-four matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems.