Hamiltonian Lattices Research Articles

Darboux’s Theorem, Lie series and the standardization of the Salerno and Ablowitz–Ladik models

In the framework of nonlinear Hamiltonian lattices, we revisit the proof of Moser-Darboux’s Theorem, in order to present a general scheme for its constructive applicability to Hamiltonian models with non-standard symplectic structures. We take as a guiding example the Salerno and Ablowitz–Ladik (AL) models: we justify the form of a well-known change of coordinates which is adapted to the Gauge symmetry, by showing that it comes out in a natural way within the general strategy outlined in the proof. Moreover, the full or truncated Lie-series technique in the extended phase-space is used to transform the Salerno model, at leading orders in the Darboux coordinates: thus the dNLS Hamiltonian turns out to be a normal form of the Salerno and AL models; as a byproduct we also get estimates of the dynamics of these models by means of dNLS one. We also stress that, once it is cast into the perturbative approach, the method allows to deal with the cases where the explicit trasformation is not known, or even worse it is not writable in terms of elementary functions.

Room temperature polaritonic soft-spin XY Hamiltonian in organic–inorganic halide perovskites

Abstract Exciton–polariton condensates, due to their nonlinear and coherent characteristics, have been employed to construct spin Hamiltonian lattices for potentially studying spin glass, critical dephasing, and even solving optimization problems. Here, we report the room-temperature polariton condensation and polaritonic soft-spin XY Hamiltonian lattices in an organic–inorganic halide perovskite microcavity. This is achieved through the direct integration of high-quality single-crystal samples within the cavity. The ferromagnetic and antiferromagnetic couplings in both one- and two-dimensional condensate lattices have been observed clearly. Our work shows a nonlinear organic–inorganic hybrid perovskite platform for future investigations as polariton simulators.

The Effect of On-Site Potentials on Supratransmission in One-Dimensional Hamiltonian Lattices.

We investigated a class of one-dimensional (1D) Hamiltonian N-particle lattices whose binary interactions are quadratic and/or quartic in the potential. We also included on-site potential terms, frequently considered in connection with localization phenomena, in this class. Applying a sinusoidal perturbation at one end of the lattice and an absorbing boundary on the other, we studied the phenomenon of supratransmission and its dependence on two ranges of interactions, 0<α<∞ and 0<β<∞, as the effect of the on-site potential terms of the Hamiltonian varied. In previous works, we studied the critical amplitude As(α,Ω) at which supratransmission occurs, for one range parameter α, and showed that there was a sharp threshold above which energy was transmitted in the form of large-amplitude nonlinear modes, as long as the driving frequency Ω lay in the forbidden band-gap of the system. In the absence of on-site potentials, it is known that As(α,Ω) increases monotonically the longer the range of interactions is (i.e., as α⟶0). However, when on-site potential terms are taken into account, As(α,Ω) reaches a maximum at a low value of α that depends on Ω, below which supratransmission thresholds decrease sharply to lower values. In this work, we studied this phenomenon further, as the contribution of the on-site potential terms varied, and we explored in detail their effect on the supratransmission thresholds.

Halide perovskites enable polaritonic XY spin Hamiltonian at room temperature.

Exciton polaritons, the part-light and part-matter quasiparticles in semiconductor optical cavities, are promising for exploring Bose-Einstein condensation, non-equilibrium many-body physics and analogue simulation at elevated temperatures. However, a room-temperature polaritonic platform on par with the GaAs quantum wells grown by molecular beam epitaxy at low temperatures remains elusive. The operation of such a platform calls for long-lifetime, strongly interacting excitons in a stringent material system with large yet nanoscale-thin geometry and homogeneous properties. Here, we address this challenge by adopting a method based on the solution synthesis of excitonic halide perovskites grown under nanoconfinement. Such nanoconfinement growth facilitates the synthesis of smooth and homogeneous single-crystalline large crystals enabling the demonstration of XY Hamiltonian lattices with sizes up to 10 × 10. With this demonstration, we further establish perovskites as a promising platform for room temperature polaritonic physics and pave the way for the realization of robust mode-disorder-free polaritonic devices at room temperature.

Nilpotency and the Hamiltonian property for cancellative residuated lattices

This paper studies nilpotent and Hamiltonian cancellative residuated lattices and their relationship with nilpotent and Hamiltonian lattice-ordered groups. In particular, results about lattice-ordered groups are extended to the domain of residuated lattices. The two key ingredients that underlie the considerations of this paper are the categorical equivalence between Ore residuated lattices and lattice-ordered groups endowed with a suitable modal operator; and Malcev’s description of nilpotent groups of a given nilpotency class c in terms of a semigroup equation.

Stability Properties of 1-Dimensional Hamiltonian Lattices with Nonanalytic Potentials

We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from nonanalytic potentials. In particular, we study the dynamics of a model governed by a “graphene-type” force law and one inspired by Hollomon’s law describing “work-hardening” effects in certain elastic materials. Our main aim is to show that, although similarities with the analytic case exist, some of the local and global stability properties of nonanalytic potentials are very different than those encountered in systems with polynomial interactions, as in the case of 1D Fermi–Pasta–Ulam–Tsingou (FPUT) lattices. Our approach is to study the motion in the neighborhood of simple periodic orbits representing continuations of normal modes of the corresponding linear system, as the number of particles [Formula: see text] and the total energy [Formula: see text] are increased. We find that the graphene-type model is remarkably stable up to escape energy levels where breakdown is expected, while the Hollomon lattice never breaks, yet is unstable at low energies and only attains stability at energies where the harmonic force becomes dominant. We suggest that, since our results hold for large [Formula: see text], it would be interesting to study analogous phenomena in the continuum limit where 1D lattices become strings.

Nonlinear Supratransmission in Quartic Hamiltonian Lattices With Globally Interacting Particles and On-Site Potentials

Abstract We investigate a family of one-dimensional (1D) Hamiltonian semi-infinite particle lattices whose interactions involve exclusively terms of fourth order in the potential. Our aim is to examine their distinct role in the dynamics, in the absence of quadratic (harmonic) interactions, which are typically included in most studies, as they are known to play an important role in many physical phenomena. We also include in our potentials on-site terms of the sine-Gordon type, which are also considered in many studies in connection with localization effects. Our 1D lattices are subjected to sinusoidal perturbation on one end and an absorbing boundary on the other. To simulate a semi-infinite chain, we will consider a relatively long chain with string coupling. Using reliable finite difference discretization schemes, we establish the existence of nonlinear supratransmission for both short-range and long-range interactions, and demonstrate that the presence of quadratic interactions is not necessary for a system to show nonlinear supratransmission. Additionally, we provide diagrams depicting novel relations between the critical amplitude at which supratransmission is triggered versus driving frequency and a parameter measuring the length of the interactions. Our investigation also shows that the presence of on-site potentials is also not crucial for the system to present supratransmission.

Low dimensional completely resonant tori in Hamiltonian Lattices and a Theorem of Poincaré

We present an extension of a classical result of Poincare (1892) about continuation of periodic orbits and breaking of completely resonant tori in a class of nearly integrable Hamiltonian systems, which covers most Hamiltonian Lattice models. The result is based on the fixed point method of the period map and exploits a standard perturbation expansion of the solution with respect to a small parameter. Two different statements are given, about existence and linear stability: a first one, in the so called non-degenerate case, and a second one, in the completely degenerate case. A pair of examples inspired to the existence of localized solutions in the discrete NLS lattice is provided.

Complex Dynamics and Statistics of 1-D Hamiltonian Lattices: Long Range Interactions and Supratransmission

In this paper, I review a number of results that my co-workers and I have obtained in the field of 1-Dimensional (1D) Hamiltonian lattices. This field has grown in recent years, due to its importance in revealing many phenomena that concern the occurrence of chaotic behavior in conservative physical systems with a high number of degrees of freedom. After the establishment of the Kolomogorov-Arnol'd-Moser (KAM) theory in the 1960s, a wealth of results were obtained about such systems as small perturbations of completely integrable Ndegree- of-freedom Hamiltonians, where ordered motion is dominant in the form of invariant tori. Since the 1980s, however, and particularly in the last two decades, there has been great progress in understanding the properties of Hamiltonian 1D lattices far from the KAM regime, where "weak" and "strong" forms of chaos begin to play an increasingly significant role. It is the purpose of this review to address and highlight some of these advances, in which the author has made several contributions concerning the dynamics and statistics of these lattices.

The effect of long-range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on-site potential

We examine the role of long--range interactions on the dynamical and statistical properties of two 1D lattices with on--site potentials that are known to support discrete breathers: the Klein--Gordon (KG) lattice which includes linear dispersion and the Gorbach--Flach (GF) lattice, which shares the same on--site potential but its dispersion is purely nonlinear. In both models under the implementation of long--range interactions (LRI) we find that single--site excitations lead to special low--dimensional solutions, which are well described by the undamped Duffing oscillator. For random initial conditions we observe that the maximal Lyapunov exponent $\lambda $ %: (a) tends to a positive value for KG and grows like $\varepsilon^(.25)$ for GF as the energy density $\varepsilon=E/N$ increases; (b) saturates to a positive value as the number of particles $N$ increase, scales as $N^{-0.12}$ in the KG model and as $N^{-0.27}$ in the GF with LRI, suggesting in that case an approach to integrable behavior towards the thermodynamic limit. Furthermore, under LRI, their non-Gaussian momentum distributions are distinctly different from those of the FPU model.

Existence of breathing patterns in globally coupled finite-size nonlinear lattices

ABSTRACTWe prove the existence of time-periodic solutions representing breathing patterns in general nonlinear Hamiltonian finite-size lattices with global coupling. As a first step the existence of localised solutions of a two site segment, where one oscillator performs larger-amplitude motion compared to the other one, is established. To this end the existence problem is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder’s Fixed Point Theorem. For isoenergetic states the degree of localisation can be tuned in a wide range by changing the initial data and the coupling strength. Subsequently, it is proven that a related localised state can be excited in the extended nonlinear lattice forming a breathing pattern with a single site of large amplitude against a background of uniform small-amplitude states. Furthermore, it is demonstrated that also spatial patterns are possible that are built up from any combination of the small-amplitude state and the large-amplitude state.

An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices.

In this work, we revisit a criterion, originally proposed in Friesecke & Pego (Friesecke & Pego 2004 Nonlinearity17, 207-227. (doi:10.1088/0951715/17/1/013)), for the stability of solitary travelling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-travelling frame, as well as at that of the Floquet problem arising when considering the travelling wave as a periodic orbit modulo shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the travelling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Complex Dynamics and Statistics in Hamiltonian 1-Dimensional Lattices

In this paper I review in a brief and introductory way some important developments in the analysis of the dynamics and statistics of N – dimensional Hamiltonian systems, in which my research team and I have played an important role over the last two decades. The results I describe here have helped us understand the surprising importance of simple periodic orbits and their local stability properties in revealing crucial dynamical and statistical properties of the systems as a whole. This has led us to introduce the concepts of “strong” and “weak” chaos that are expected to play a significant role in better understanding the complexity of these multi-dimensional systems, which have important applications in solid state, field theory, superconductivity and more recently nonlinear optics. undefined undefined Звуковая функция ограничена 200 символами Настройки : История : Обратная связь : Donate Закрыть

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability.

In this work, we provide two complementary perspectives for the (spectral) stability of solitary traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. One is as an eigenvalue problem for a stationary solution in a cotraveling frame, while the other is as a periodic orbit modulo shifts. We connect the eigenvalues of the former with the Floquet multipliers of the latter and using this formulation derive an energy-based spectral stability criterion. It states that a sufficient (but not necessary) condition for a change in the wave stability occurs when the functional dependence of the energy (Hamiltonian) H of the model on the wave velocity c changes its monotonicity. Moreover, near the critical velocity where the change of stability occurs, we provide an explicit leading-order computation of the unstable eigenvalues, based on the second derivative of the Hamiltonian H^{''}(c_{0}) evaluated at the critical velocity c_{0}. We corroborate this conclusion with a series of analytically and numerically tractable examples and discuss its parallels with a recent energy-based criterion for the stability of discrete breathers.

Diverging Fluctuations of the Lyapunov Exponents.

We show that in generic one-dimensional Hamiltonian lattices the diffusion coefficient of the maximum Lyapunov exponent diverges in the thermodynamic limit. We trace this back to the long-range correlations associated with the evolution of the hydrodynamic modes. In the case of normal heat transport, the divergence is even stronger, leading to the breakdown of the usual single-function Family-Vicsek scaling ansatz. A similar scenario is expected to arise in the evolution of rough interfaces in the presence of suitably correlated background noise.

Stretch diffusion and heat conduction in one-dimensional nonlinear lattices.

For heat conduction in one-dimensional (1D) nonlinear Hamiltonian lattices, it has been known that conserved quantities play an important role in determining the actual heat conduction behavior. In closed or microcanonical Hamiltonian systems, the total energy and stretch are always conserved. Depending on the existence of external on-site potential, the total momentum can be conserved or not. All the momentum-conserving lattices have anomalous heat conduction except the 1D coupled rotator lattice. It was recently claimed that "whenever stretch (momentum) is not conserved in a 1D model, the momentum (stretch) and energy fields exhibit normal diffusion." The stretch in a coupled rotator lattice was also argued to be nonconserved due to the requirement of a finite partition function, which enables the coupled rotator lattice to fulfill this claim. In this work, we will systematically investigate stretch diffusion and heat conduction in terms of energy diffusion for typical 1D nonlinear lattices. Contrary to what was claimed, no clear connection between conserved quantities and heat conduction can be established. The actual situation might be more complicated than what was proposed.

Numerical integration of variational equations for Hamiltonian systems with long range interactions

We study numerically classical 1-dimensional Hamiltonian lattices involving inter-particle long range interactions that decay with distance like 1/rα, for α≥0. We demonstrate that although such systems are generally characterized by strong chaos, they exhibit an unexpectedly organized behavior when the exponent α<1. This is shown by computing dynamical quantities such as the maximal Lyapunov exponent, which decreases as the number of degrees of freedom increases. We also discuss our numerical methods of symplectic integration implemented for the solution of the equations of motion together with their associated variational equations. The validity of our numerical simulations is estimated by showing that the total energy of the system is conserved within an accuracy of 4 digits (with integration step τ=0.02), even for as many as N=8000 particles and integration times as long as 106 units.

Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices

We investigate dynamically and statistically diffusive motion in a chain of linearly coupled 2-dimensional symplectic McMillan maps and find evidence of subdiffusion in weakly and strongly chaotic regimes when all maps of the chain possess a saddle point at the origin and the central map is initially excited. In the case of weak coupling, there is either absence of diffusion or subdiffusion with q>1-Gaussian probability distributions, characterizing weak chaos. However, for large enough coupling and already moderate number of maps, the system exhibits strongly chaotic (q≈1) subdiffusive behavior, reminiscent of the subdiffusive energy spreading observed in a disordered Klein–Gordon Hamiltonian. Our results provide evidence that coupled symplectic maps can exhibit physical properties similar to those of disordered Hamiltonian systems, even though the local dynamics in the two cases is significantly different.

Energy spreading in strongly nonlinear disordered lattices

We study the scaling properties of energy spreading in disordered strongly nonlinear Hamiltonian lattices. Such lattices consist of nonlinearly coupled local linear or nonlinear oscillators, and demonstrate a rather slow, subdiffusive spreading of initially localized wave packets. We use a fractional nonlinear diffusion equation as a heuristic model of this process, and confirm that the scaling predictions resulting from a self-similar solution of this equation are indeed applicable to all studied cases. We show that the spreading in nonlinearly coupled linear oscillators slows down compared to a pure power law, while for nonlinear local oscillators a power law is valid in the whole studied range of parameters.

Scaling properties of energy spreading in nonlinear Hamiltonian two-dimensional lattices

In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find a nice agreement of the scaling predicted from the NDE with the spreading results from extensive numerical studies. Moreover, we show that the scaling works also for regular lattices with strongly nonlinear coupling, for which the scaling exponent is estimated analytically. This shows that the process of chaotic diffusion in such lattices does not require disorder.