General integrable problems of classical mechanics

Several classical problems of mechanics are shown to be integrable for the special systems of coupled rigid bodies, introduced in this work and calledC ^k -central configurations. It is proven that dynamics of an arbitraryC ^k -central configuration in a Newtonian gravitational field with an arbitrary quadratic potential is integrable in the Liouville sense and in the theta-functions of Riemann surfaces. Hidden symmetry of the inertial dynamics of these configurations is disclosed and reductions of the Lagrange equations to the Euler equations on Lie coalgebras are obtained. Reductions and integrable cases of a heavyC ^k -central configuration rotation around a fixed point are indicated. Separation of rotations of a space station type orbiting system, being aC ^k -central configuration of rigid bodies, is proven. This result leads to the possibility of the independent stabilization of rotations of the rigid bodies in such orbiting configurations.Supported by the Ministry of Colleges and Universities of Ontario and the Natural Sciences and Engineering Research Council of Canada     

Third and fourth order invariants for one-dimensional time-dependent classical systems

The construction of invariants up to fourth order in velocities has been carried out for one-dimensional, time-dependent classical dynamical systems. While the exact results are recovered for the first and second order integrable systems, the results for the third and fourth order invariants are expressed in terms of nonlinearpotential equations. Noticing the separability of the potential in space and time variables these nonlinear equations are reduced to a tractable form. A possible solution for the third order case suggests a new integrable systemV(q, t) ∼t ^−4/3 q ^1/2. Alexander von Humboldt-Stiftung Fellow, on leave from the Department of Physics, Ramjas College (University of Delhi), Delhi 110 007, India.     

A one-particle potential integrable on a family of quadrics

It is shown that one particle in the potential ( - ∑ _k-1 ⁿ q _k ⁿ /a _k )^-1 is completely integrable and n independent rational integrals in involution are found. The restriction of this system to any quadric ∑ _k=1 ⁿ q _k ² /(a _k - z)=1 is integrable too. The system is separable in generalized elliptic coordinates. Supported in part by the Polish Ministry of Science, Technology and Higher Education, Project MR-1-7.     

Construction of exact dynamical invariants of two-dimensional classical system

A general method is used for the construction of second constant of motion of fourth order in momenta using the complex coordinates (z, _z ^- ). A fourth-order potential equation is obtained whose solutions directly provide a large class of integrable systems. The potential equation is tested with an interesting example which admits second constants of motion.     

Point-contact tunneling spectroscopy between a Nb tip and an ideal topological insulator Sn-doped Bi1.1Sb0.9Te2S

We find that the quantum-classical correspondence in integrable systems is characterized by two time scales. One is the Ehrenfest time below which the system is classical; the other is the quantum revival time beyond which the system is fully quantum. In between, the quantum system can be well approximated by classical ensemble distribution in phase space. These results can be summarized in a diagram which we call Ehrenfest diagram. We derive an analytical expression for Ehrenfest time, which is proportional to h~(-1/2). According to our formula, the Ehrenfest time for the solar-earth system is about 10~(26) times of the age of the solar system. We also find an analytical expression for the quantum revival time, which is proportional to h~(-1). Both time scales involve ω(I), the classical frequency as a function of classical action. Our results are numerically illustrated with two simple integrable models. In addition, we show that similar results exist for Bose gases, where 1/N serves as an effective Planck constant.     

Exact decay of correlations for mixtures and rotators

We give the exact asymptotic form, at low activity, of the correlations of a classical fluid consisting of several species of particles interacting by means of integrable two-body potentials. Our results also extend to classical dipoles withr ^–3 potential in two dimensions.     

Two new integrable couplings of the soliton hierarchies with self-consistent sources

A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with sl(4) has been presented (Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613). Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl(4). In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.     

The Camassa-Holm Hierarchy, N-Dimensional Integrable Systems, and Algebro-Geometric Solution on a Symplectic Submanifold

This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. Furthermore, under two constraint conditions between the potentials and the eigenfunctions, the CH spectral problem is cast in: 1. a new Neumann-like N-dimensional system when it is restricted into a symplectic submanifold of ^2N which is proven to be integrable by using the Dirac-Poisson bracket and the r-matrix process; and 2. a new Bargmann-like N-dimensional system when it is considered in the whole ^2N which is proven to be integrable by using the standard Poisson bracket and the r-matrix process.     

Painlevé analysis and integrability of the damped anharmonic oscillator equation

The Painlevé analysis is applied to the anharmonic oscillator equation . The following three integrable cases are identified: (i)C=0,d ²=25A/6,A>0,B arbitrary, (ii)d ²=9A/2,B=0,A>0,C arbitrary and (iii)d ²=−9A/4,C=2B ²/(9A),A<0,C<0,B arbitrary. The first two integrable choices are already reported in the literature. For the third integrable case the general solution is found involving elliptic function with exponential amplitude and argument.     

On a Class of Integrable Systems with a Cubic First Integral

A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained. We show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems. The relevant metrics are described by a finite number of parameters and lead to a large class of models mainly on the manifolds \mathbb S²{{\mathbb S}^2} and \mathbb H²{{\mathbb H}^2} . Many of these systems are globally defined and contain as special cases integrable systems due to Goryachev, Chaplygin, Dullin, Matveev and Tsiganov.     

Quantum geometry and quantum mechanics of integrable systems

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ^∞) and are invariant under quantum dynamics in the long-time range O(ħ ^−∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.     

Integrable Geodesic Flows on the (Super)extension of the Bott–Virasoro Group

In this Letter, we present an answer to the question posed by Marcel, Ovsienko and Roger in their paper (Lett. Math. Phys. 40 (1997), 31–39). The Itô equation, modified dispersive water wave equation and modified dispersionless long wave equation are shown to be the geodesic flows with respect to an L ² metric on the semidirect product space Diff ^s C (S ¹), where Diff ^s (S ¹) is the group of orientation-preserving Sobolev H ^s diffeomorphisms of the circle. We also study the geodesic flows with respect to H ¹ metric. The geodesic flows in this case yield different integrable systems admitting nonlinear dispersion terms. These systems exhibit more general wave phenomena than usual integrable systems. Finally, we study an integrable geodesic flow on the extended Neveu–Schwarz space.     

Bihamiltonian Structures and Quadratic Algebras in Hydrodynamics and on Non-Commutative Torus

We demonstrate the common bihamiltonian nature of several integrable systems. The first one is an elliptic rotator that is an integrable Euler-Arnold top on the complex group GL(N,) for any N, whose inertia ellipsiod is related to a choice of an elliptic curve. Its bihamiltonian structure is provided by the compatible linear and quadratic Poisson brackets, both of which are governed by the Belavin-Drinfeld classical elliptic r-matrix. We also generalize this bihamiltonian construction of integrable Euler-Arnold tops to several infinite-dimensional groups, appearing as certain large N limits of GL(N,). These are the group of a non-commutative torus (NCT) and the group of symplectomorphisms SDiff(T²) of the two-dimensional torus. The elliptic rotator on symplectomorphisms gives an elliptic version of an ideal 2D hydrodynamics, which turns out to be an integrable system. In particular, we define the quadratic Poisson algebra on the space of Hamiltonians on T² depending on two irrational numbers. In conclusion, we quantize the infinite-dimensional quadratic Poisson algebra in a fashion similar to the corresponding finite-dimensional case.     

Hilbert Schemes, Separated Variables, and D-Branes

We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on T ^*Σ for Σ=C, C ^* or elliptic curve, and on C ²/Γ and show that their complex deformations are integrable systems of Calogero–Sutherland–Moser type. We present the hyperk?hler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally we discuss the connections to physics of D-branes and string duality. Received: 2 November 2000 / Accepted: 7 May 2001     

The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation KM + MK = r s^T we introduce a scalar function S^{(i, j)} = s^T K^j (I + M)^?1Kⁱr which is defined as same as in discrete case. S^{(i, j)} satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on r and s, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of S^{(i, j)} defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property S^{(i, j)} = S^(j,i). The solution M provides t function by t = ∣I + M∣. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.     

Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM ⁿ. The central objects in this theory are supplementary invariant Poisson structuresP _c which are incompatable with the original Poisson structureP ₁ for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP ₁ andP ₂ in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP ₁ nor with respect toP ₂. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.Supported by NSERC grant OGPIN 337.     

Quasi-invariant measures on sets of piecewise smooth homeomorphisms of closed intervals and circles and representations of diffeomorphism groups

Families of measures are constructed that are quasi-invariant with respect to the action of C ¹-diffeomorphisms, for a closed interval and a circle, and have bounded Borel measurable second derivatives. A series of pairwise nonequivalent irreducible representations of the group of C ³-diffeomorphisms of the circle is introduced on the space of functions that are square integrable against these measures.     

Integrable discrete static Hamiltonian with a parametric double-well potential

A one-dimensional discrete conservative Hamiltonian with a generalized form of the Schmidt potential, is constructed with the help of a non-integrable discrete Hamiltonian whose parametrized double-well potential can be reduced to the ?⁴ potential. The new conservative Hamiltonian is completely integrable in the discrete static regime, and the associate exact nonlinear solution is shown to coincide with the continuum nonlinear periodic solution of the non-integrable Hamiltonian. Numerical simulations and nonlinear stability analysis suggest that the discrete mapping derived from the completely integrable Hamiltonian undergoes a bifurcation which does not leads to the chaotic phase with randomly pinned states, but instead to a phase where real solutions become rare forming a cluster of periodic points around an elliptic fixed point.     

Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons

In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified Camassa-Holm equation pointing out that in the peakon sector (a family of non-smooth solitons) only one of them, namely the one obtained from the distributional compatibility condition, supports the time invariance of the Sobolev H¹ norm.