Manifestation of Hamiltonian monodromy in nonlinear wave systems

We show that the concept of dynamical monodromy plays a natural fundamental role in the spatiotemporal dynamics of counterpropagating nonlinear wave systems. By means of an adiabatic change of the boundary conditions imposed to the wave system, we show that Hamiltonian monodromy manifests itself through the spontaneous formation of a topological phase singularity (2π- or π-phase defect) in the nonlinear waves. This manifestation of dynamical Hamiltonian monodromy is illustrated by generic nonlinear wave models. In particular, we predict that its measurement can be realized in a direct way in the framework of a nonlinear optics experiment.     

Uncovering Fractional Monodromy

The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n ₁:(?n ₂) resonant Hamiltonian systems with n ₁, n ₂ coprime natural numbers. We consider, in particular, systems that for n ₁, n ₂ > 1 contain one-parameter families of singular fibres which are ‘curled tori’. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n ₁, n ₂. Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.     

Hamiltonian Monodromy via Picard-Lefschetz Theory

 In this paper, we investigate the ``Hamiltonian' monodromy of the fibration in Liouville tori of certain integrable systems via (real) algebraic geometry. Using Picard-Lefschetz theory in a relative Prym variety, we determine the Hamiltonian monodromy of the ``geodesic flow on SO(4)'. Using a relative generalized Jacobian, we prove that the Hamiltonian monodromy of the spherical pendulum can also be obtained by the Picard-Lefschetz formula. Received: 28 September 2001 / Accepted: 12 April 2002 Published online: 12 August 2002     

Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra?

The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.     

On geometric phases for soliton equations

This paper develops a new complex Hamiltonian structure forn-soliton solutions for a class of integrable equations such as the nonlinear Schrödinger, sine-Gordon and Korteweg-de Vries hierarchies of equations that yields, amongst other things, geometric phases in the sense of Hannay and Berry. For example, one of the possible soliton geometric phases is manifested by the well known phase shift that occurs for interacting solitons. The main new tools are complex angle representations that linearize the corresponding Hamiltonian flows on associated noncompact Jacobi varieties. This new structure is obtained by taking appropriate limits of the differential equations describing the class of quasi-periodic solutions. A method of asymptotic reduction of the angle representations is introduced for investigating soliton geometric phases that are related to the presence of monodromy at singularities in the space of parameters. In particular, the phase shift of interacting solitons can be expressed as an integral over a cycle on an associated Riemann surface. In this setting, soliton geometric asymptotics are constructed for studying geometric phases in the quantum case. The general approach is worked out in detail for the three specific hierarchies of equations mentioned. Some links with -functions, the braid group and geometric quantization are pointed out as well.Communicated by A. Jaffe     

Another Note on Focus-Focus Singularities

We show a natural relation between the monodromy formula for focus-focus singularities of integrable Hamiltonian systems and a formula of Duistermaat–Heckman, and extend the main results of our previous note ( ¹-action, monodromy, and topological classification) to the case of degenerate focus-focus singularities. We also consider the non-Hamiltonian case, local normal forms, etc.     

Dynamical manifestations of Hamiltonian monodromy

Monodromy is the simplest obstruction to the existence of global action–angle variables in integrable Hamiltonian dynamical systems. We consider one of the simplest possible systems with monodromy: a particle in a circular box containing a cylindrically symmetric potential-energy barrier. Systems with monodromy have nontrivial smooth connections between their regular Liouville tori. We consider a dynamical connection produced by an appropriate time-dependent perturbation of our system. This turns studying monodromy into studying a physical process. We explain what aspects of this process are to be looked upon in order to uncover the interesting and somewhat unexpected dynamical behavior resulting from the nontrivial properties of the connection. We compute and analyze this behavior.     

Dual isomonodromic deformations and moment maps to loop algebras

The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to dual pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painlevé transcendentsP _V and _VI .Research supported in part by the Natural Sciences and Engineering Research Council of Canada.     

推广的多分量费米型量子可导非线性Schrdinger模型的可积性

导出了推广的多分量费米型量子可导非线性Schr dinger模型的哈密顿量 .利用代数Betheansatz方法 ,找到了此模型的量子monodromy矩阵所满足的量子Yang Baxter方程 ,并证明了其可积性 .     

Hamiltonian systems with detuned 1:1:2 resonance: Manifestation of bidromy

We consider a generalization of the 1:1:2 resonant swing-spring [see H. Dullin, A. Giacobbe, R.H. Cushman, Physica D 190 (2004) 15] which is suggested both by the symmetries of this system and by its physical and in particular molecular realizations [see R.H. Cushman, H.R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovskií, B.I. Zhilinskií, Phys. Rev. Lett. 93 (2004) 024302-1-024302-4]. Our generic integrable system is detuned off the exact Fermi resonance 1:2. The three-dimensional (3D) image of its energy-momentum map EM consists either of two or three qualitatively different non-intersecting 3D regions: a regular region at low vibrational excitation, a region with monodromy similar to that studied for the exact resonance, and in some cases—an intermediate region in which the 3D set of regular values of EM is partially self-overlapping while remaining connected. In the presence of this latter region, the system has an interesting property which we called bidromy. We analyze monodromy and bidromy for a concrete integrable classical Hamiltonian system of three coupled oscillators and for its quantum analog. We also show that the bifurcation involved in the transition from the regular region to the region with monodromy can be regarded as a special resonant equivariant analog of the Hamiltonian Hopf bifurcation.     

Information-theoretic differential geometry of quantum phase transitions

The manifold of coupling constants parametrizing a quantum Hamiltonian is equipped with a natural Riemannian metric with an operational distinguishability content. We argue that the singularities of this metric are in correspondence with the quantum phase transitions featured by the corresponding system. This approach provides a universal conceptual framework to study quantum critical phenomena which is differential geometric and information theoretic at the same time.     

Convergence of Hamiltonian systems to billiards

We examine in detail a physically natural and general scheme for gradually deforming a Hamiltonian to its corresponding billiard, as a certain parameter k varies from one to infinity. We apply this limiting process to a class of Hamiltonians with homogeneous potential-energy functions and further investigate the extent to which the limiting billiards inherit properties from the corresponding sequences of Hamiltonians. The results are mixed. Using theorems of Yoshida for the case of two degrees of freedom, we prove a general theorem establishing the "inheritability" of stability properties of certain orbits. This result follows naturally from the convergence of the traces of appropriate monodromy matrices to the billiard analog. However, in spite of the close analogy between the concepts of integrability for Hamiltonian systems and billiards, integrability properties of Hamiltonians in a sequence are not necessarily inherited by the limiting billiard, as we show by example. In addition to rigorous results, we include numerical examples of certain interesting cases, along with computer simulations. (c) 1998 American Institute of Physics.     

Geometric approach to Kac–Moody and Virasoro algebras

In this paper we show the existence of a group acting infinitesimally transitively on the moduli space of pointed-curves and vector bundles (with formal trivialization data) and whose Lie algebra is an algebra of differential operators. The central extension of this Lie algebra induced by the determinant bundle on the Sato Grassmannian is precisely a semidirect product of a Kac–Moody algebra and the Virasoro algebra. As an application of this geometric approach, we give a local Mumford-type formula in terms of the cocycle associated with this central extension. Finally, using the original Mumford formula we show that this local formula is an infinitesimal version of a general relation in the Picard group of the moduli of vector bundles on a family of curves (without any formal trivialization).     

Exact Solvability and Quantum Integrability of a Derivative Nonlinear Schrödinger Model

Using a variant of quantum inverse scattering method (QISM) which is directly applicable to field theoretical systems, we derive all possible commutation relations among the operator valued elements of the monodromy matrix associated with an integrable derivative nonlinear Schrödinger (DNLS) model. From these commutation relations we obtain the exact Bethe eigenstates for the quantum conserved quantities of DNLS model. We also explicitly construct the first few quantum conserved quantities including the Hamiltonian in terms of the basic field operators of this model. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This fact allows us to apply QISM to generate the spectrum of quantum DNLS Hamiltonian for the full range of its coupling constant.     

Stokes Matrices and Monodromy of the Quantum Cohomology of Projective Spaces

In this paper we compute Stokes matrices and monodromy of the quantum cohomology of projective spaces. This problem can be formulated in a “classical” framework, as the problem of computation of Stokes matrices and monodromy of differential equations with regular and irregular singularities. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups. Received: 24 October 1998 / Accepted: 27 April 1999     

螺旋光纤系统中非绝热条件几何相移

由于绝热条件几何相位量子逻辑门存在非绝热差错与退相干差错这一冲突，因此在拓扑量子计算中需要设计非绝热条件几何相门，以克服这一不足.证明螺旋光纤系统内光子有效哈密顿量恰好是一个Wang Matsumoto型哈密顿量，因此螺旋光纤系统能自动产生非绝热条件几何相移.同时还证明在螺旋光纤方案中，由极化光子与螺旋光纤相互作用哈密顿量所导致的动力学相位为零（这正是拓扑量子计算所要求的），以及在螺旋光纤系统中可以通过控制极化光子初始波矢，从而较容易获得条件初始态.总之，螺旋光纤系统方案能自动满足Wang与Matsumoto的核磁共振方案中为实现非绝热条件几何相移所提出的全部条件与要求. 关键词： 几何相位 螺旋光纤系统 Wang Matsumoto型哈密顿量 拓扑量子计算     

Hamiltonian systems on fibered manifolds

We offer a new geometric theory of Hamiltonian systems with an infinite number of degrees of freedom in which the Hamiltonian operators are nonlinear differential operators on fields. The Poisson bracket is carried into the vertical bracket by the mapping between functionals and Hamitonian operators which is established by a Hamiltonian structure.     

A Hamiltonian formulation of the Einstein-Cartan-Sciama-Kibble theory of gravity

We postulate the energy-momentum functionE for the ECSK theory of gravity and formulate the functional Hamiltonian equation in terms of the energy-momentum functionE and the symplectic 2-form . The system of partial differential equations which follows from the functional Hamilton equation is equivalent to the system of variational equations of the ECSK theory. The Hamiltonian method gives rise to a natural division of these equations into 10 constraint equations and the set of dynamical equations. We discuss the geometric sense of the constraint equations and their relations to the initial value problem.     

Is the Hamiltonian geometrical criterion for chaos always reliable?

It is found that the application of a newly developed geometrical criterion, in which negative eigenvalues of the associated matrix determined by the dynamical curvature of a conformal metric for a Hamiltonian system are used to identify the onset of local instability or chaos, is somewhat problematic in some circumstances. In fact, this criterion is neither necessary nor sufficient for the prediction of instability of orbits on a same energy hypersurface because it is not in good agreement with information on unstable or chaotic behavior given by the maximal Lyapunov exponent in general.