Hamiltonian Case Research Articles

Witten Non Abelian Localization for Equivariant K-Theory, and the [𝑄,𝑅]=0 Theorem

The purpose of the present paper is two-fold. First, we obtain a non-abelian localization theorem when M is any even dimensional compact manifold : following an idea of E. Witten, we deform an elliptic symbol associated to a Clifford bundle on M with a vector field associated to a moment map. Second, we use this general approach to reprove the [Q,R] = 0 theorem of Meinrenken-Sjamaar in the Hamiltonian case, and we obtain mild generalizations to almost complex manifolds. This non-abelian localization theorem can be used to obtain a geometric description of the multiplicities of the index of general spin^c Dirac operators (see the preprint arXiv:1411.7772).

Multi-pulse Orbits and Homoclinic Trees in a Non-autonomous Resonant Hamiltonian System

In this study, we develop the energy-phase method to deal with the high-dimensional non-autonomous nonlinear dynamical systems. Our generalized energy-phase method applies to integrable, two-degree-of freedom non-autonomous resonant Hamiltonian systems. As an example, we investigate the multi-pulse orbits and homoclinic trees for a parametrically excited, simply supported rectangular thin plate of two-mode approximation. In both the Hamiltonian and dissipative case we find homoclinic trees, which describe the repeated bifurcations of multi-pulse solutions, and we present visualizations of these complicated structures.

Second Order Analysis for Strong Solutions in the Optimal Control of Parabolic Equations

In this paper we provide a second order analysis for strong solutions in the optimal control of parabolic equations. We consider first the case of box constraints on the control in a general setting and then, in the case of a quadratic Hamiltonian, we also impose final integral constraints on the state. For the first problem, in a rather general framework, we prove a characterization of the quadratic growth property in the strong sense, i.e., for admissible controls whose associated states are uniformly near to the state of the nominal control. Assuming a quadratic Hamiltonian in the state constrained case, we provide a sufficient second order optimality condition for the aforementioned quadratic growth property, which does not impose the uniqueness of the Lagrange multiplier set. As a consequence of our results, in the quadratic Hamiltonian case, under some continuity assumptions on the data we prove that the notions of quadratic growth in the strong sense coincide with the more standard notion of quadratic growth in the weak sense, i.e., with respect to controls which are uniformly near to the nominal one.

Directly solving the Hamilton–Jacobi equations by Hermite WENO Schemes

In this paper, we present a class of new Hermite weighted essentially non-oscillatory (HWENO) schemes based on finite volume framework to directly solve the Hamilton-Jacobi (HJ) equations. For HWENO reconstruction, both the cell average and the first moment of the solution are evolved, and for two dimensional case, HWENO reconstruction is based on a dimension-by-dimension strategy which is the first used in HWENO reconstruction. For spatial discretization, one of key points for directly solving HJ equation is the reconstruction of numerical fluxes. We follow the idea put forward by Cheng and Wang (2014) 3 to reconstruct the values of solution at Gauss-Lobatto quadrature points and numerical fluxes at the interfaces of cells, and for neither the convex nor concave Hamiltonian case, the monotone modification of numerical fluxes is added, which can guarantee the precision in the smooth region and converge to the entropy solution when derivative discontinuities come up. The third order TVD Runge-Kutta method is used for the time discretization. Extensive numerical experiments in one dimensional and two dimensional cases are performed to verify the efficiency of the methods.

Non-equilibrium thermo-field dynamics for anti-ferromagnetic spin system

The non-equilibrium thermo-field dynamics for an anti-ferromagnetic spin system interacting with a phonon reservoir is proposed for the case of a non-bilinear unperturbed Hamiltonian, which includes not only a bilinear part but also a non-bilinear part, in the spin–wave approximation. The two kinds of quasi-particle operators are introduced, and their forms are derived for the semi-free boson fields. It is shown that the two quasi-particles decay exponentially with the frequencies and life-times which are different from each other. It is also shown that each quasi-particle changes to the other tilde quasi-particle through the spin–phonon interaction. The spin–spin correlation functions and longitudinal magnetization for the anti-ferromagnetic spin system under an external static magnetic field are derived in the forms convenient for the perturbation expansions. The expectation values of the spin–wave energy and longitudinal magnetization and the spin–spin correlation functions are investigated numerically for an anti-ferromagnetic system of one-dimensional infinite spins interacting with a damped phonon-reservoir, in the region valid for the lowest spin–wave approximation and the narrowing-limit approximation in which the relaxation times of the spin system are much larger than the correlation time of the phonon reservoir. The two-point Green’s function of the semi-free spin–wave for the anti-ferromagnetic spin system is derived by introducing the thermal quartet notation, and it is given in a form of 4×4 matrix.

Vortex nucleation in a dissipative variant of the nonlinear Schrödinger equation under rotation

In the present work, we motivate and explore the dynamics of a dissipative variant of the nonlinear Schrödinger equation under the impact of external rotation. As in the well established Hamiltonian case, the rotation gives rise to the formation of vortices. We show, however, that the most unstable mode leading to this instability scales with an appropriate power of the chemical potential μ of the system, increasing proportionally to μ2/3. The precise form of the relevant formula, obtained through our asymptotic analysis, provides the most unstable mode as a function of the atomic density and the trap strength. We show how these unstable modes typically nucleate a large number of vortices in the periphery of the atomic cloud. However, through a pattern selection mechanism, prompted by symmetry-breaking, only few isolated vortices are pulled in sequentially from the periphery towards the bulk of the cloud resulting in highly symmetric stable vortex configurations with far fewer vortices than the original unstable mode. These results may be of relevance to the experimentally tractable realm of finite temperature atomic condensates.

Melnikov Functions in Quadratic Perturbations of Generalized Lotka–Volterra Systems

We present a detailed analysis of Melnikov functions which arise in quadratic perturbations of generalized Lotka---Volterra vector fields with the first integral x?ys(1 ? x ? y). That analysis was sketched in ?ola?dek (J Differ Equ 109:223---273, 1994). In particular, we prove that the maximal number of limit cycles in the generic case equals 2 and in the Hamiltonian triangle case, this number is 3.

Interaction-free evolution in the presence of time-dependent Hamiltonians

The generalization of the concept of interaction-free evolutions (IFE) [Napoli et al., Phys. Rev. A 89, 062104 (2014)] to the case of time-dependent Hamiltonians is discussed. It turns out that the time-dependent case allows for much richer structures of interaction-free states and interaction-free subspaces. The general condition for the occurrence of IFE is found and exploited to analyze specific situations. Several examples are presented, each one associated to a class of Hamiltonians with specific features.

The Thermal Statistics of Quasi-Probabilities’ Analogs in Phase Space

We focus attention upon the thermal statistics of the classical analogs of quasi-probabilities (QP) in phase space for the important case of quadratic Hamiltonians. We consider the three more important OPs: Wigner’s,P-, and Husimi’s. We show that, for all of them, the ensuing semiclassical entropy is a function only of the fluctuation productΔxΔp. We ascertain that the semiclassical analog ofP-distribution seems to become unphysical at very low temperatures. The behavior of several other information quantifiers reconfirms such an assertion in manifold ways. We also examine the behavior of the statistical complexity and of thermal quantities like the specific heat.

Well-posedness for the fifth-order KdV equation in the energy space

We prove that the initial value problem (IVP) associated to the fifth-order KdV equation (0.1) ∂ t u − ∂ x 5 u = c 1 ∂ x u ∂ x 2 u + c 2 ∂ x ( u ∂ x 2 u ) + c 3 ∂ x ( u 3 ) , \begin{equation*} \tag {0.1} \partial _tu-\partial ^5_x u=c_1\partial _xu\partial _x^2u+c_2\partial _x(u\partial _x^2u)+c_3\partial _x(u^3), \end{equation*} where x ∈ R x \in \mathbb R , t ∈ R t \in \mathbb R , u = u ( x , t ) u=u(x,t) is a real-valued function and α , c 1 , c 2 , c 3 \alpha , \ c_1, \ c_2, \ c_3 are real constants with α ≠ 0 \alpha \neq 0 , is locally well-posed in H s ( R ) H^s(\mathbb R) for s ≥ 2 s \ge 2 . In the Hamiltonian case (i.e. when c 1 = c 2 c_1=c_2 ), the IVP associated to (0.1) is then globally well-posed in the energy space H 2 ( R ) H^2(\mathbb R) .

A center-stable manifold for the energy-critical wave equation in ℝ3 in the symmetric setting

Consider the focusing semilinear wave equation in ℝ3 with energy-critical nonlinearity [Formula: see text] This equation admits stationary solutions of the form [Formula: see text] called solitons, which solve the elliptic equation [Formula: see text] Restricting ourselves to the space of symmetric solutions ψ for which ψ(x) = ψ(-x), we find a local center-stable manifold, in a neighborhood of ϕ(x, 1), for this wave equation in the weighted Sobolev space [Formula: see text] Solutions with initial data on the manifold exist globally in time for t ≥ 0, depend continuously on initial data, preserve energy, and can be written as the sum of a rescaled soliton and a dispersive radiation term. The proof is based on a new class of reverse Strichartz estimates, recently introduced by Beceanu and Goldberg and adapted here to the case of Hamiltonians with a resonance.

Dirac equation in low dimensions: The factorization method

We present a general approach to solve the (1+1) and (2+1)-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials.

Validity of the GGE for quantum quenches from interacting to noninteracting models

.In the majority of analytical verifications of the conjecture that the Generalized Gibbs Ensemble (GGE) describes the large time asymptotics of local observables in quantum quench problems, both post- and pre-quench Hamiltonians are essentially noninteracting. We test this conjecture studying field correlations in the more general case of an arbitrary pre-quench Hamiltonian, while keeping the post-quench one noninteracting. We first show that, in the previously studied special case of a noninteracting pre-quench Hamiltonian, the validity of the conjecture is a consequence of Wick’s theorem. We then show that this conjecture is more generally valid for an arbitrary interacting pre-quench Hamiltonian, but this time as a consequence of the cluster decomposition property of the initial state, which is a fundamental principle for generic physical states. For arbitrary initial states that do not satisfy the cluster decomposition property, the above conjecture is not generally true. As a byproduct of our investigation we obtain an analytical derivation of earlier numerical results for the large time evolution of correlations after a quantum quench of the interaction in the Lieb–Liniger model from a nonzero value to zero.

Normal form and Nekhoroshev stability for nearly integrable hamiltonian systems with unconditionally slow aperiodic time dependence

The aim of this paper is to extend the results of Giorgilli and Zehnder for aperiodic time dependent systems to a case of general nearly-integrable convex analytic Hamiltonians. The existence of a normal form and then a stability result are shown in the case of a slow aperiodic time dependence that, under some smallness conditions, is independent on the size of the perturbation.

Absence of thermospin current response of a spin-orbit-coupled two-dimensional electron gas

We consider the spin current flowing in a two-dimensional electron gas with Rashba and linear Dresselhaus spin-orbit interaction as a linear response to a temperature gradient, taking into account the contribution due to the thermoelectric effect. We derive a relation connecting the electrically and thermally driven spin and charge conductivities, for the Hamiltonians corresponding to samples grown in the main crystallographic directions. Based on this connection, it is shown that the transverse and longitudinal spin currents generated by the temperature gradient vanish exactly for each spin-orbit Hamiltonian case. This result is in contrast to the recently predicted finite thermospin Hall effect for such class of systems.

Left invertibility of formal Hamiltonian operators

In this paper, we investigate the left invertible completion of the formal Hamiltonian operators with unbounded entries. In particular, the left invertible completion of Hamiltonian cases is also given. Based on the space decomposition technique, the sufficient and necessary conditions are given according to whether the dimension of is finite or infinite.

Convergence of a semi-discretization scheme for the Hamilton–Jacobi equation: A new approach with the adjoint method

We consider a numerical scheme for the one dimensional time dependent Hamilton–Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L.C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the O(h) convergence rate in terms of the L∞ norm and O(h) in terms of the L1 norm, where h is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper.

Families of moment matching based, structure preserving approximations for linear port Hamiltonian systems

In this paper we propose a solution to the problem of moment matching with preservation of the port Hamiltonian structure, in the framework of time-domain moment matching. We characterize several families of parameterized port Hamiltonian models that match the moments of a given port Hamiltonian system, at a set of finite interpolation points. We also discuss the problem of Markov parameters matching for linear systems as a moment matching problem for descriptor representations associated with the given system, at zero interpolation points. Solving this problem yields families of parameterized reduced order models that achieve Markov parameter matching. Finally, we apply these results to the port Hamiltonian case, resulting in families of parameterized reduced order port Hamiltonian approximations.

Quadratic invariants for discrete clusters of weakly interacting waves

We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M × N matrix with entries 1, −1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J ≡ N − M* ⩾ N − M, where M* is the number of linearly independent rows in Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including the basic structures leading to M* < M. We illustrate our findings by presenting examples from the Charney–Hasegawa–Mima wave model, and by showing a classification of small (up to three-triad) clusters.

POINCARÉ CODE: A package of open-source implements for normalization and computer algebra reduction near equilibria of coupled ordinary differential equations

The Poincaré code is a Maple project package that aims to gather significant computer algebra normal form (and subsequent reduction) methods for handling nonlinear ordinary differential equations. As a first version, a set of fourteen easy-to-use Maple commands is introduced for symbolic creation of (improved variants of Poincaré’s) normal forms as well as their associated normalizing transformations. The software is the implementation by the authors of carefully studied and followed up selected normal form procedures from the literature, including some authors’ contributions to the subject. As can be seen, joint-normal-form programs involving Lie-point symmetries are of special interest and are published in CPC Program Library for the first time, Hamiltonian variants being also very useful as they lead to encouraging results when applied, for example, to models from computational physics like Hénon–Heiles. Program summaryProgram title: POINCARÉCatalogue identifier: AEPJ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 7694No. of bytes in distributed program, including test data, etc.: 597943Distribution format: tar.gzProgramming language: Maple V11.Computer: See specifications for running Maple V11 or above.Operating system: MS Windows.Classification: 4.3, 5, 16.9.Nature of problem:Computing structure-preserving normal forms near the origin for nonlinear vector fields.Solution method:14 Maple commands are designed to compute various normal forms as well as their generating functions and associated normalizing transformations for nonlinear systems of ordinary differential equations, including Hamiltonian ones. Further reduction of these normal forms–in the presence of Lie-point symmetries–is also considered. All algorithms are based on Lie transform, thus leading to the structure-preserving normal forms in the sense that all properties that can be formulated in terms of graded Lie algebras are preserved.Restrictions:The semisimple part of the leading matrix about the origin (resp. the quadratic part in the Hamiltonian case) is assumed to be already taken into diagonal form (resp. canonical form) via a linear change of variables. In other words, the eigenvalues must be explicitly known.Unusual features:Further reduction of Poincaré–Dulac normal form via the joint normal form commands, taking profitably Lie-point symmetries, is perhaps the main feature of the software. Besides, to the best of our knowledge, and except the Hamiltonian case, it seems that there is no CPC programs for computation of (structure-preserving) normal forms in the general case.Running time:Depends on the input data, mainly on system size, type of the leading matrix (semisimple or not), type of resonances, order of normalization and required options. Instantaneous for the provided examples.