Hamiltonian Perturbation Research Articles

Inheritance properties of Krylov subspace methods for continuous-time algebraic Riccati equations

We investigate the theory behind the Krylov subspace methods for large-scale continuous-time algebraic Riccati equations. We show that the solvability of the projected algebraic Riccati equation need not be assumed but can be inherited. This study of inheritance properties is the first of its kind. We study the stabilizability and detectability of the control system, the stability of the associated Hamiltonian matrix and perturbation in terms of residuals. Special attention is paid to the stabilizing and positive semi-definite properties of approximate solutions. Illustrative numerical examples for the inheritance properties are presented.

Krylov subspace methods for discrete-time algebraic Riccati equations

We apply the Krylov subspace methods to large-scale discrete-time algebraic Riccati equations. The solvability of the projected algebraic Riccati equation is not assumed but is shown to be inherited from the original equation. Solvability in terms of stabilizability, detectability, stability radius of the associated Hamiltonian matrix and perturbation theory are considered. We pay particular attention to the stabilizing and the positive semi-definite properties of approximate solutions. Illustrative numerical examples are presented.

A large probability averaging theorem for the defocusing NLS

We consider the nonlinear Schrödinger equation on the one dimensional torus, with a defocusing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measures with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant for times of order , being the inverse of the temperature and a positive number (we prove ). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory.

Multi-pulse jumping orbits and chaotic dynamics of cantilevered pipes conveying time-varying fluid

Global bifurcations and multi-pulse chaotic motions of cantilevered pipes conveying time-varying fluid under external excitation are investigated. The method of multiple scales and Galerkin’s approach are utilized on the partial differential governing equation to yield the four-dimensional averaged equation with 1:2 internal resonance and primary parameter resonance. Based on the averaged equations, the normal form theory is adopted to derive the explicit expressions of normal form associated with a double zeroes and a pair of purely imaginary eigenvalues. Then the energy-phase method is employed to analyze the chaotic dynamics by identifying the existence of the multi-pulse Silnikov-type orbits in the perturbed phase space. The homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated in both Hamiltonian and dissipative perturbation. The diagrams indicate a gradual breakup of homoclinic tree in the system as the dissipative factor grows. Numerical simulations are performed to show the multi-pulse jumping orbits and Silnikov-type chaotic behaviors may occur. The influence of the external excitation and the flow velocity of the cantilevered pipe on the dynamics of the system is discussed simultaneously in numerical results. The global dynamics also exhibits the existence of the chaos in the sense of Smale horseshoes for the cantilevered pipe conveying time-varying fluid under external excitation.

A Hamiltonian Pertubation Approach to Construction of Geometric Integrators for Optimal Control Problems

In this paper, we discuss computational methods for optimal control problems which preserve some geometric properties of the system. A Hamiltonian perturbation method is developed, which does not involve any discretization of either the cost function, or the dynamic equations. Instead, the approach relies mainly on an iterative successive approximation of the value-function, which converges uniformly to the solution. The approach is most effective when the Hamiltonian of the system is a polynomial function of the phase coordinates, and an example is presented to demonstrate this.

Measure and Capacity of Wandering Domains in Gevrey Near-integrable Exact Symplectic Systems

A wandering domain for a diffeomorphism is an open connected set whose iterates are pairwise disjoint. We endow A^n = T^n x R^n with its usual exact symplectic structure. An integrable diffeomorphism Φ^h, i.e. the time-one map of a Hamiltonian h which depends only on the action variables, has no nonempty wandering domains. The aim of this paper is to estimate the size (measure and Gromov capacity) of wandering domains in the case of an exact symplectic perturbation of Φ^h , in the analytic or Gevrey category. Upper estimates are related to Nekhoroshev theory, lower estimates are related to examples of Arnold diffusion. This is a contribution to the quantitative Hamiltonian perturbation initiated in previous works on the optimality of long term stability estimates and diffusion times; our emphasis here is on discrete systems because this is the natural setting to study wandering domains. We first prove that the measure (or the capacity) of these wandering domains is exponentially small, with an upper bound of the form exp(-c e^(-1/2nα)), where e is the size of the perturbation, α is the Gevrey exponent (α is at least 1, α=1 for analytic systems) and c is some positive constant depending mildly on h. This is obtained as a consequence of an exponential stability theorem for near-integrable exact symplectic maps, in the analytic or Gevrey category, for which we give a complete proof based on the most recent improvements of Nekhoroshev theory for Hamiltonian flows, and which requires the development of specific Gevrey suspension techniques. The second part of the paper is devoted to the construction of near-integrable Gevrey systems possessing wandering domains, for which the capacity (and thus the measure) can be estimated from below. We suppose that n is at least 2, essentially because KAM theory precludes Arnold diffusion in too low a dimension. For any α>1, we produce examples with lower bounds of the form exp(-c e^(-1/2(n-1)(α-1))). This is done by means of a coupling technique, involving rescaled standard maps possessing wandering discs in A and near-integrable systems possessing periodic domains of arbitrarily large periods in A^(n-1). The most difficult part of the construction consists in obtaining a perturbed pendulum-like system on A with periodic islands of arbitrarily large periods, whose areas are explicitly estimated from below. Our proof is based on a version due to Herman of the translated curve theorem.

Persistent currents by reservoir engineering

We demonstrate that persistent currents can be induced in a quantum system in contact with a structured reservoir, without the need of any applied gauge field. The working principle of the mechanism leading to their presence is based on the extension to the many-body scenario of non-reciprocal Lindblad dynamics, recently put forward by Metelmann and Clerk in Phys. Rev. X 5, 021025 (2015): Non-reciprocity can be generated by suitably balancing coherent interactions with their corresponding dissipative version, induced by the coupling to a common structured environment, such to make total interactions directional. Specifically, we consider an interacting spin/boson model in a ring-shaped one-dimensional lattice coupled to an external bath. By employing a combination of cluster mean-field, exact diagonalization and matrix-product-operator techniques, we show that solely dissipative effects suffice to engineer steady states with a persistent current that survives in the limit of large systems. We also verify the robustness of such current in the presence of additional dissipative or Hamiltonian perturbation terms.

On solving the nonlinear Schrödinger equation with an anti-cubic nonlinearity in presence of Hamiltonian perturbation terms

The soliton ansatz method, the tanh-coth method, the modified simple equation method, the new extended auxiliary equation method, the new mapping method, the rational (G′/G)-expansion method and the generalized Kudryashov method are applied in this paper. Solitons and other solutions for nonlinear Schrodinger equation with an anti-cubic nonlinearity in presence of Hamiltonian perturbation terms have been found. The used methods present a wider applicability for handling the nonlinear partial differential equations. A comparison of our new results with the well-known results is made.

Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control

The local invariant (stable, unstable, and center) manifolds of a saddle $$\times $$ center $$\times $$ center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller.

Field theoretical derivation of L\xfcscher\u2019s formula and calculation of finite volume form factors

We initiate a systematic method to calculate both the finite volume energy levels and form factors from the momentum space finite volume two-point function. By expanding the two point function in the volume we extracted the leading exponential volume correction both to the energy of a moving particle state and to the simplest non-diagonal form factor. The form factor corrections are given in terms of a regularized infinite volume 3-particle form factor and terms related to the Lüsher correction of the momentum quantization. We tested these results against second order Lagrangian and Hamiltonian perturbation theory in the sinh-Gordon theory and we obtained perfect agreement.

Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory

Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and, in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems has opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the frequency corrections for the dissipative oscillatory systems. As a further application, we use the perturbation theory to analytically calculate the Hannay angle for the van der Pol oscillator's limit cycle trajectory when its parameters-the strength of the nonlinearity and the frequency of the linear part-evolve cyclically and adiabatically. For this van der Pol oscillator, we also numerically calculate the corresponding geometric phase and establish its equivalence with the Hannay angle.

Three Dimensional Imaging of the Nucleon

We study the Wigner distributions of quarks and gluons in light-front dressed quark model using the overlap of light front wave functions (LFWFs). We take the target to be a dressed quark, this is a composite spin $-1/2$ state of quark dressed with a gluon. This state allows us to calculate the quark and gluon Wigner distributions analytically in terms of LFWFs using Hamiltonian perturbation theory. We analyze numerically the Wigner distributions of quark and gluon and report their nature in the contour plots. We use an improved numerical technique to remove the cutoff dependence of the Fourier transformed integral over ${\bf \Delta}_\perp$.

Hamiltonian perturbation of optical solitons with parabolic law nonlinearity using three integration methodologies

This work secures optical soliton solution to the nonlinear Schrodinger’s equation that appears with parabolic law nonlinearity and a couple of Hamiltonian type perturbation terms that are strong. Three integration schemes retrieve soliton solutions to the model. They are csch-function method, tanh-coth algorithm and modified simple equation technique.

Analysis of optical solitons in nonlinear negative-indexed materials with anti-cubic nonlinearity

This paper adopts the extended trial equation method to handle optical nonlinear negative-index materials that come with anti-cubic nonlinearity. In presence of Hamiltonian perturbation terms, soliton solutions are obtained. These are bright and singular solitons only. However, some additional wave solutions are also emerged.

Quasi-periodic solution of quasi-linear fifth-order KdV equation

We prove the existence of quasi-periodic small-amplitude solutions for quasi-linear Hamiltonian perturbation of the fifth order KdV equation on the torus in presence of a quasi-periodic forcing.

New Integration Methods for Perturbed ODEs Based on Symplectic Implicit Runge–Kutta Schemes with Application to Solar System Simulations

We propose a family of integrators, flow-composed implicit Runge–Kutta methods, for perturbations of nonlinear ordinary differential equations, consisting of the composition of flows of the unperturbed part alternated with one step of an implicit Runge–Kutta (IRK) method applied to a transformed system. The resulting integration schemes are symplectic when both the perturbation and the unperturbed part are Hamiltonian and the underlying IRK scheme is symplectic. In addition, they are symmetric in time (resp. have order of accuracy r) if the underlying IRK scheme is time-symmetric (resp. of order r). The proposed new methods admit mixed precision implementation that allows us to efficiently reduce the effect of round-off errors. We particularly focus on the potential application to long-term solar system simulations, with the equations of motion of the solar system rewritten as a Hamiltonian perturbation of a system of uncoupled Keplerian equations. We present some preliminary numerical experiments with a simple point mass Newtonian 10-body model of the solar system (with the sun, the eight planets, and Pluto) written in canonical heliocentric coordinates.

Resonant optical soliton perturbation with anti-cubic nonlinearity by extended trial function method

This paper studies the resonant optical solitons that comes with anti-cubic nonlinearity in presence of Hamiltonian perturbation terms. The extended trial function method is employed to retrieve cnoidal waves, bright and singular soliton solutions as well as periodic singular solutions to the model.

The proof of simultaneous conditional stabilization and destabilization of linear Hamiltonian systems

It is proved that each linear Hamiltonian system is simultaneously conditionally (with respect to a subspace of half dimension) stabilizable and destabilizable by infinitesimal Hamiltonian perturbation.

Ultra-strong nonlinear optical processes and trigonal warping in MoS2 layers

Nonlinear optical processes, such as harmonic generation, are of great interest for various applications, e.g., microscopy, therapy, and frequency conversion. However, high-order harmonic conversion is typically much less efficient than low-order, due to the weak intrinsic response of the higher-order nonlinear processes. Here we report ultra-strong optical nonlinearities in monolayer MoS2 (1L-MoS2): the third harmonic is 30 times stronger than the second, and the fourth is comparable to the second. The third harmonic generation efficiency for 1L-MoS2 is approximately three times higher than that for graphene, which was reported to have a large χ(3). We explain this by calculating the nonlinear response functions of 1L-MoS2 with a continuum-model Hamiltonian and quantum mechanical diagrammatic perturbation theory, highlighting the role of trigonal warping. A similar effect is expected in all other transition-metal dichalcogenides. Our results pave the way for efficient harmonic generation based on layered materials for applications such as microscopy and imaging.

Hamiltonian perturbation theory for ultra-differentiable functions

Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence M M bounding the growth of derivatives. In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BR M _M , and which generalizes the Bruno-Rüssmann condition; and Nekhoroshev’s theorem, where the stability time depends on the ultra-differentiable class of the pertubation, through the same sequence M M . Our proof uses periodic averaging, while a substitute for the analyticity width allows us to bypass analytic smoothing. We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and Marco-Sauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it matching), we manage to narrow the gap between stability hypotheses (e.g. the BR M _M condition) and instability hypotheses, thus circumbscribing the stability threshold. The formulas relating the growth M M of derivatives of the perturbation on the one hand, and the arithmetics of robust frequencies or the stability time on the other hand, bring light to the competition between stability properties of nearly integrable systems and the distance to integrability. Due to our method of proof using width of regularity as a regularizing parameter, these formulas are closer to optimal as the the regularity tends to analyticity.