On the spectra of periodic waves for infinite-dimensional Hamiltonian systems

We consider the problem of determining the spectrum for the linearization of an infinite-dimensional Hamiltonian system about a spatially periodic traveling wave. By using a Bloch-wave decomposition, we recast the problem as determining the point spectra for a family of operators J_γL_γ, where J_γ is skew-symmetric with bounded inverse and L_γ is symmetric with compact inverse. Our main result relates the number of unstable eigenvalues of the operator J_γL_γ to the number of negative eigenvalues of the symmetric operator L_γ. The compactness of the resolvent operators allows us to greatly simplify the proofs, as compared to those where similar results are obtained for linearizations about localized waves. The theoretical results are general, and apply to a larger class of problems than those considered herein. The theory is applied to a study of the spectra associated with periodic and quasi-periodic solutions to the nonlinear Schrödinger equation, as well as periodic solutions to the generalized Korteweg-de Vries equation with power nonlinearity.     

Some applications of stochastic averaging method for quasi Hamiltonian systems in physics

Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response statistics. In the present paper, the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced. The applications of the stochastic averaging method in studying the dynamics of active Brownian particles, the reaction rate theory, the dynamics of breathing and denaturation of DNA, and the Fermi resonance and its effect on the mean transition time are reviewed. Supported by the National Natural Science Foundation of China (Grant Nos. 10772159 and 10802074), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060335125), and the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y7080070)     

Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems

We develop canonical perturbation theory for a physically interesting class of infinite-dimensional systems. We prove stability up to exponentially large times for dynamical situations characterized by a finite number of frequencies. An application to two model problems is also made. For an arbitrarily large FPU-like system with alternate light and heavy masses we prove that the exchange of energy between the optical and the acoustical modes is frozen up to exponentially large times, provided the total energy is small enough. For an infinite chain of weakly coupled rotators we prove exponential stability for two kinds of initial data: (a) states with a finite number of excited rotators, and (b) states with the left part of the chain uniformly excited and the right part at rest.     

Periodic solutions of some infinite-dimensional Hamiltonian systems associated with non-linear partial difference equations I

We establish existence of a dense set of non-linear eigenvalues,E, and exponentially localized eigenfunctions,u _E , for some non-linear Schrödinger equations of the form $$Eu_E (x) = [( - \Delta + V(x))u_E ](x) + \lambda u_E (x)^3 ,$$ bifurcating off solutions of the linear equation with λ=0. The pointsx range over a lattice, ? ^d ,d=1,2,3,..., Δ is the finite difference Laplacian, andV(x) is a random potential. Such equations arise in localization theory and plasma physics. Our analysis is complicated by the circumstance that the linear operator ?Δ+V(x) has dense point spectrum near the edges of its spectrum which leads to small divisor problems. We solve these problems by developing some novel bifurcation techniques. Our methods extend to non-linear wave equations with random coefficients.     

Periodic solutions of some infinite-dimensional Hamiltonian systems associated with non-linear partial difference equations. II

This is our second paper devoted to the study of some non-linear Schrödinger equations with random potential. We study the non-linear eigenvalue problems corresponding to these equations. We exhibit a countable family of eigenfunctions corresponding to simple eigenvalues densely embedded in the band tails. Contrary to our results in the first paper, the results established in the present paper hold for an arbitrary strength of the non-linear (cubic) term in the non-linear Schrödinger equation.     

Finite de Finetti theorem for infinite-dimensional systems

We formulate and prove a de Finetti representation theorem for finitely exchangeable states of a quantum system consisting of k infinite-dimensional subsystems. The theorem is valid for states that can be written as the partial trace of a pure state |Psi/Psi| chosen from a family of subsets {Cn} of the full symmetric subspace for n subsystems. We show that such states become arbitrarily close to mixtures of pure power states as n increases. We give a second equivalent characterization of the family {Cn}.     

Some applications of pulse techniques in EPR

Overview of application of pulsed EPR is given.     

Reduction theory for symmetry breaking with applications to nematic systems

We formulate Euler–Poincaré and Lagrange–Poincaré equations for systems with broken symmetry. We specialize the general theory to present explicit equations of motion for nematic systems, ranging from single nematic molecules to biaxial liquid crystals. The geometric construction applies to order parameter spaces consisting of either unsigned unit vectors (directors) or symmetric matrices (alignment tensors). On the Hamiltonian side, we provide the corresponding Poisson brackets in both Lie–Poisson and Hamilton–Poincaré formulations. The explicit form of the helicity invariant for uniaxial nematics is also presented, together with a whole class of invariant quantities (Casimirs) for two-dimensional incompressible flows.     

An algorithm and its application for obtaining some kind of infinite-dimensional Hamiltonian canonical formulation

Using factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical evolution equation, infinite-dimensional Hamiltonian canonical system, factorization of differential operator, commutator,evolution equation, infinite-dimensional Hamiltonian canonical system, factorization of differential operator, commutatorProject supported by the National Natural Science Foundation of China (Grant No~10562002) and the Natural Science Foundation of Nei Mongol, China (Grant No~200508010103).2007-05-09{\partial}/{\partial x}Corresponding author．E-mail：alatanca@imu．edu.cn/qk/85823A/200711/25754042.html0200, 03405/9/2007 12:00:00 AMUsing factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical $`{\partial}/{\partial x}'$-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.http://cpb.iphy.ac.cn/CN/10.1088/1009-1963/16/11/002https://cpb.iphy.ac.cn/CN/article/downloadArticleFile.do?attachType=PDF&id=1088382007-11-20'-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.     

Canonical methods for Hamiltonian systems: numerical experiments

A Hamiltonian system possesses dynamics (e.g. preservation of volume in phase space and symplectic structure) that call for special numerical integrators, namely canonical methods. Recent research on this aspect have shown that canonical numerical integrators may be needed for Hamiltonian systems. In this paper, we focus on numerical experiments that compare canonical and non-canonical numerical integrators. Test problems are taken from different areas in physical sciences. These experiments help to buttress the claims that canonical numerical integrators give results that mimic the qualitative behavior of the original system and that canonical numerical integrators are suitable for long time integrations. Our experiments indicate that higher-order canonical methods allow for larger timestep than lower-order canonical methods.     

Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas

A method, called beatification, is presented for rapidly extracting weakly nonlinear Hamiltonian systems that describe the dynamics near equilibria of systems possessing Hamiltonian form in terms of noncanonical Poisson brackets. The procedure applies to systems like fluids and plasmas in terms of Eulerian variables that have such noncanonical Poisson brackets, i.e., brackets with nonstandard and possibly degenerate form. A collection of examples of both finite and infinite dimensions is presented.     

哈密顿Ermakov系统的形式不变性

把形式不变性的方法用于研究哈密顿Ermakov系统，从哈密顿Ermakov系统的形式不变性出发，运用比较系数法得到与形式不变性相应的点对称变换生成元的表达式及势能所满足的偏微分方程.结果表明，在点对称变换下，只有自治的哈密顿Ermakov系统才具有形式不变性. 关键词： 哈密顿Ermakov系统 拉格朗日函数 点对称变换 形式不变性     

Finite Hamiltonian systems: Linear transformations and aberrations

Finite Hamiltonian systems contain operators of position, momentum, and energy, having a finite number N of equally-spaced eigenvalues. Such systems are under the æis of the algebra su(2), and their phase space is a sphere. Rigid motions of this phase space form the group SU(2); overall phases complete this to U(2). But since N-point states can be subject to U(N) ?U(2) transformations, the rest of the generators will provide all N ² unitary transformations of the states, which appear as nonlinear transformations—aberrations—of the system phase space. They are built through the “finite quantization” of a classical optical system.     

Partial and complete observables for Hamiltonian constrained systems

We will pick up the concepts of partial and complete observables introduced by Rovelli in Conceptional Problems in Quantum Gravity, Birkhäuser, Boston (1991); Class Quant Grav, 8:1895 (1991); Phys Rev, D65:124013 (2002); Quantum Gravity, Cambridge University Press, Cambridge (2007) in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kucha?’s Bubble-Time Formalism (J Math Phys, 13:768, 1972). Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions. Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.     

Some properties of frustrated spin systems: extensions and applications of Lieb-Schupp approach

Lieb and Schupp have obtained a number of ground-state properties for frustrated Heisenberg models. The basic tool used was certain version of spin-reflection positivity method. One group of these results is related to singlet nature of ground state. It needs an assumption of reflection symmetry present in the system. In this paper, it is shown that analogous results hold also for other symmetries (inversion etc.). The second Lieb-Schupp result is matrix inequality, which imply inequalities between ground-state energies of certain systems. In the paper, the Lieb-Schupp inequality is applied to relate ground-state energies of various systems: spin chains, ladders and multidimensional lattices.     

A Hamiltonian approach for skew-product dynamical systems

The problem on the existence of Hamiltonian structures for (nonlinear) skew-product dynamical systems is studied via coupling Poisson structures. This research was partially supported by CONACYT under grant no. 55463.     

Variational principle for t-dependent classical Hamiltonian systems

For a non-conservative, classical system described by Lagrangian L(q?, q_i, t), the functional $\text{Z = ?}{}^{\mathrm{T}}{}_0\text{L}\text{d}\text{t + KT}$, where K is constant, is stationary with respect to variations in q_i(t) and T, given suitable boundary conditions. In the conservative case, for systems with a single periodic coordinate, this variational procedure reduces to that given by Luttinger and Thomas.