One-dimensional Hamiltonian Systems Research Articles

Abundant optical soliton solutions for the stochastic fractional fokas system using bifurcation analysis

In this study, the stochastic fractional Fokas system (SFFS) with M-truncated derivatives is considered. A certain wave transformation is applied to convert this system to a one-dimensional conservative Hamiltonian system. Based on the qualitative theory of dynamical systems, the bifurcation and phase portrait are examined. Utilizing the conserved quantity, we construct some new traveling wave solutions for the SFFS. Due to the fact that the Fokas system is used to explain nonlinear pulse transmission in mono-mode optical fibers, the given solutions may be applied to analyze an extensive variety of crucial physical phenomena. To clarify the effects of the M-truncated derivative and Wiener process, the dynamic behaviors of the various obtained solutions are depicted with 3-D and 2-D curves.

Bifurcation and Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation

This paper focuses on the construction of traveling wave solutions to the nonlinear Klein--Gordon equation by employing the qualitative theory of planar dynamical systems. Based on this theory, we analytically study the existence of periodic, kink (anti-kink), and solitary wave solutions. We then attempt to construct such solutions. For this purpose, we apply a well-known traveling wave solution to convert the nonlinear Klein--Gordon equation into an ordinary differential equation that can be written as a one-dimensional Hamiltonian system. The qualitative theory is applied to investigate and describe phase portraits of the Hamiltonian system. Based on the bifurcation constraints on the system parameters, we integrate the conserved quantities to build new wave solutions that can be classified into periodic, kink (anti-kink), and solitary wave solutions. Some of the obtained solutions are clarified graphically and their connection with the phase orbits is derived.

Distorted stability pattern and chaotic features for quantized prey-predator-like dynamics.

Nonequilibrium and instability features of prey-predator-like systems associated to topological quantum domains emerging from a quantum phase-space description are investigated in the framework of the Weyl-Wigner quantum mechanics. Reporting about the generalized Wigner flow for one-dimensional Hamiltonian systems, H(x,k), constrained by ∂^{2}H/∂x∂k=0, the prey-predator dynamics driven by Lotka-Volterra (LV) equationsis mapped onto the Heisenberg-Weyl noncommutative algebra, [x,k]=i, where the canonical variables x and k are related to the two-dimensional LV parameters, y=e^{-x} and z=e^{-k}. From the non-Liouvillian pattern driven by the associated Wigner currents, hyperbolic equilibrium and stability parameters for the prey-predator-like dynamics are then shown to be affected by quantum distortions over the classical background, in correspondence with nonstationarity and non-Liouvillianity properties quantified in terms of Wigner currents and Gaussian ensemble parameters. As an extension, considering the hypothesis of discretizing the time parameter, nonhyperbolic bifurcation regimes are identified and quantified in terms of z-y anisotropy and Gaussian parameters. The bifurcation diagrams exhibit, for quantum regimes, chaotic patterns highly dependent on Gaussian localization. Besides exemplifying a broad range of applications of the generalized Wigner information flow framework, our results extend, from the continuous (hyperbolic regime) to discrete (chaotic regime) domains, the procedure for quantifying the influence of quantum fluctuations over equilibrium and stability scenarios of LV driven systems.

Relaxation Dynamics in a Long-Range System with Mixed Hamiltonian and Non-Hamiltonian Interactions

Sometimes the dynamics of a physical system is described by non-Hamiltonian equations of motion, and additionally, the system is characterized by long-range interactions. A concrete example is that of particles interacting with light as encountered in free-electron laser and cold-atom experiments. In this work, we study the relaxation dynamics to non-Hamiltonian systems, more precisely, to systems with interactions of both Hamiltonian and non-Hamiltonian origin. Our model consists of $N$ globally-coupled particles moving on a circle of unit radius; the model is one-dimensional. We show that in the infinite-size limit, the dynamics, similarly to the Hamiltonian case, is described by the Vlasov equation. In the Hamiltonian case, the system eventually reaches an equilibrium state, even though one has to wait for a long time diverging with $N$ for this to happen. By contrast, in the non-Hamiltonian case, there is no equilibrium state that the system is expected to reach eventually. We characterize this state with its average magnetization. We find that the relaxation dynamics depends strongly on the relative weight of the Hamiltonian and non-Hamiltonian contributions to the interaction. When the non-Hamiltonian part is predominant, the magnetization attains a vanishing value, suggesting that the system does not sustain states with constant magnetization, either stationary or rotating. On the other hand, when the Hamiltonian part is predominant, the magnetization presents long-lived strong oscillations, for which we provide a heuristic explanation. Furthermore, we find that the finite-size corrections are much more pronounced than those in the Hamiltonian case; we justify this by showing that the Lenard-Balescu equation, which gives leading-order corrections to the Vlasov equation, does not vanish, contrary to what occurs in one-dimensional Hamiltonian long-range systems.

Noncommutative phase-space Lotka-Volterra dynamics: The quantum analog.

The Lotka-Volterra (LV) dynamics is investigated in the framework of the Weyl-Wigner (WW) quantum mechanics extended to one-dimensional Hamiltonian systems, H(x,k) constrained by the ∂^{2}H/∂x∂k=0 condition. Supported by the Heisenberg-Weyl noncommutative algebra, where [x,k]=i, the canonical variables x and k are interpreted in terms of the LV variables, y=e^{-x} and z=e^{-k}, eventually associated with the number of individuals in a closed competitive dynamics: the so-called prey-predator system. The WW framework provides the ground for identifying how classical and quantum evolution coexist at different scales and for quantifying quantum analog effects. Through the results from the associated Wigner currents, (non-)Liouvillian and stationary properties are described for thermodynamic and Gaussian quantum ensembles in order to account for the corrections due to quantum features over the classical phase-space pattern yielded by the Hamiltonian description of the LV dynamics. In particular, for Gaussian statistical ensembles, the Wigner flow framework provides the exact profile for the quantum modifications over the classical LV phase-space trajectories so that Gaussian quantum ensembles can be interpreted as an adequate Hilbert space state configuration for comparing quantum and classical regimes. The generality of the framework developed here extends the boundaries of the understanding of quantumlike effects on competitive microscopical biosystems.

A lattice Gas Model for Generic One-Dimensional Hamiltonian Systems

We present a three-lane exclusion process that exhibits the same universal fluctuation pattern as generic one-dimensional Hamiltonian dynamics with short-range interactions, viz., with two sound modes in the Kardar-Parisi-Zhang (KPZ) universality class (with dynamical exponent z=3/2 and symmetric Prähofer-Spohn scaling function) and a superdiffusive heat mode with dynamical exponent z=5/3 and symmetric Lévy scaling function. The lattice gas model is amenable to efficient numerical simulation. Our main findings, obtained from dynamical Monte-Carlo simulation, are: (i) The frequently observed numerical asymmetry of the sound modes is a finite time effect. (ii) The mode-coupling calculation of the scale factor for the 5/3-Lévy-mode gives at least the right order of magnitude. (iii) There are significant diffusive corrections which are non-universal.

Asymptotic dynamics for the small data weakly dispersive one-dimensional Hamiltonian ABCD system

Consider the Hamiltonian a b c d abcd system in one dimension, with data posed in the energy space H 1 × H 1 H^1\times H^1 . This model, introduced by Bona, Chen, and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where a , c > 0 a,c>0 and b = d > 0 b=d>0 . Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this 2 × 2 2\times 2 system was given in [J. Math. Pure Appl. (9) 127 (2019), 121–159] in a strongly dispersive regime, i.e., under essentially the conditions \[ b = d > 2 9 , a , c > − 1 18 . b=d > \frac 29, \quad a,c>-\frac 1{18}. \] Additionally, decay was obtained inside a proper subset of the light cone ( − | t | , | t | ) (-|t|,|t|) . In this paper, we improve [J. Math. Pure Appl. (9) 127 (2019), 121–159] in three directions. First, we enlarge the set of parameters ( a , b , c , d ) (a,b,c,d) for which decay to zero is the only available option, considering now the so-called weakly dispersive regime a , c ∼ 0 a,c\sim 0 : we prove decay if now \[ b = d > 3 16 , a , c > − 1 48 . b=d > \frac 3{16}, \quad a,c>-\frac 1{48}. \] This result is sharp in the case where a = c a=c , since for a , c a,c bigger, some a b c d abcd linear waves of nonzero frequency do have zero group velocity. Second, we sharply enlarge the interval of decay to consider the whole light cone, that is to say, any interval of the form | x | ∼ | v | t |x|\sim |v|t for any | v | > 1 |v|>1 . This result rules out, among other things, the existence of nonzero speed solitary waves in the regime where decay is present. Finally, we prove decay to zero of small a b c d abcd solutions in exterior regions | x | ≫ | t | |x|\gg |t| , also discarding super-luminical small solitary waves. These three results are obtained by performing new improved virial estimates for which better decay properties are deduced.

Solutions of a Two-Particle Interacting Quantum Walk.

We study the solutions of an interacting Fermionic cellular automaton which is the analogue of the Thirring model with both space and time discrete. We present a derivation of the two-particle solutions of the automaton recently in the literature, which exploits the symmetries of the evolution operator. In the two-particle sector, the evolution operator is given by the sequence of two steps, the first one corresponding to a unitary interaction activated by two-particle excitation at the same site, and the second one to two independent one-dimensional Dirac quantum walks. The interaction step can be regarded as the discrete-time version of the interacting term of some Hamiltonian integrable system, such as the Hubbard or the Thirring model. The present automaton exhibits scattering solutions with nontrivial momentum transfer, jumping between different regions of the Brillouin zone that can be interpreted as Fermion-doubled particles, in stark contrast with the customary momentum-exchange of the one-dimensional Hamiltonian systems. A further difference compared to the Hamiltonian model is that there exist bound states for every value of the total momentum and of the coupling constant. Even in the special case of vanishing coupling, the walk manifests bound states, for finitely many isolated values of the total momentum. As a complement to the analytical derivations we show numerical simulations of the interacting evolution.

Parabolic matching of hyperbolic system using Control by Interconnection

Abstract The structural difference between one-dimensional (1D) hyperbolic and parabolic port Hamiltonian system (PHS) is discussed. Then, using a Control by Interconnection (CbI) approach, a distributed state feedback is designed in order to transform an hyperbolic PHS into a parabolic one, the latter being asymptotically stable and even purely dissipative (with no oscillating modes). Distributed wave damping in 1D vibro-acoustic pipes, using piezo actuators, is considered as an illustration example for the proposed control design.

Step Density Profiles in Localized Chains

We consider two types of strongly disordered one-dimensional Hamiltonian systems coupled to baths (energy or particle reservoirs) at the boundaries: strongly disordered quantum spin chains and disordered classical harmonic oscillators. These systems are believed to exhibit localization, implying in particular that the conductivity decays exponentially in the chain length $L$. We ask however for the profile of the (very slowly) transported quantity in the steady state. We find that this profile is a step-function, jumping in the middle of the chain from the value set by the left bath to the value set by the right bath. The width of the step grows not faster than $\sqrt{L}$. This is confirmed by numerics on a disordered quantum spin chain of 9 spins and on much longer chains of harmonic oscillators. In the case of harmonic oscillators, we also observe a drastic breakdown of local equilibrium at the step, resulting in a chaotic temperature profile.

Ergodicity of One-dimensional Systems Coupled to the Logistic Thermostat

We analyze the ergodicity of three one-dimensional Hamiltonian systems, with harmonic, quartic and Mexican-hat potentials, coupled to the logistic thermostat. As criteria for ergodicity we employ: the independence of the Lyapunov spectrum with respect to initial conditions; the absence of visual holes in two-dimensional Poincare sections; the agreement between the histograms in each variable and the theoretical marginal distributions; and the convergence of the global joint distribution to the theoretical one, as measured by the Hellinger distance. Taking a large number of random initial conditions, for certain parameter values of the thermostat we find no indication of regular trajectories and show that the time distribution converges to the ensemble one for an arbitrarily long trajectory for all the systems considered. Our results thus provide a robust numerical indication that the logistic thermostat can serve as a single one-parameter thermostat for stiff one-dimensional systems.

One-Dimensional Self-Organization and Nonequilibrium Phase Transition in a Hamiltonian System.

Self-organization and nonequilibrium phase transitions are well known to occur in two- and three-dimensional dissipative systems. Here, instead, we provide numerical evidence that these phenomena also occur in a one-dimensional Hamiltonian system. To this end, we calculate the heat conductivity by coupling the two ends of our system to two heat baths at different temperatures. It is found that when the temperature difference is smaller than a critical value, the heat conductivity increases with the system size in power law with an exponent considerably smaller than 1. However, as the temperature difference exceeds the critical value, the system's behavior undergoes a transition and the heat conductivity tends to diverge linearly with the system size. Correspondingly, an ordered structure emerges. These findings suggest a new direction for exploring the transport problems in one dimension.

Orbits with minimal period for a class of autonomous second order one-dimensional Hamiltonian systems

Abstract We show, under an iterative condition which is similar to but stronger than that of Ambrosetti and Rabinowitz and by using a variational method, the existence of a T-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential z ¨ + V ′ ⁢ ( z ) = 0 , z ∈ ℝ , \ddot{z}+V^{\prime}(z)=0,\quad z\in\mathbb{R}, for any T > 0 {T>0} . Moreover, such a solution has T / k {T/k} as a minimal period for some integer 1 ≤ k ≤ 3 {1\leq k\leq 3} .

Scaling of the dynamics of a homogeneous one-dimensional anisotropic classical Heisenberg model with long-range interactions.

The dynamics of quasistationary states of long-range interacting systems with N particles can be described by kinetic equations such as the Balescu-Lenard and Landau equations. In the case of one-dimensional homogeneous systems, two-body contributions vanish as two-body collisions in one dimension only exchange momentum and thus cannot change the one-particle distribution. Using a Kac factor in the interparticle potential implies a scaling of the dynamics proportional to N(δ) with δ=1 except for one-dimensional homogeneous systems. For the latter different values for δ were reported for a few models. Recently it was shown by Rocha Filho and collaborators [Phys. Rev. E 90, 032133 (2014)] for the Hamiltonian mean-field model that δ=2 provided that N is sufficiently large, while small N effects lead to δ≈1.7. More recently, Gupta and Mukamel [J. Stat. Mech. (2011) P03015] introduced a classical spin model with an anisotropic interaction with a scaling in the dynamics proportional to N(1.7) for a homogeneous state. We show here that this model reduces to a one-dimensional Hamiltonian system and that the scaling of the dynamics approaches N(2) with increasing N. We also explain from theoretical consideration why usual kinetic theory fails for small N values, which ultimately is the origin of noninteger exponents in the scaling.

Controlling the range of interactions in the classical inertial ferromagnetic Heisenberg model: analysis of metastable states

A numerical analysis of a one-dimensional Hamiltonian system, composed by N classical localized Heisenberg rotators on a ring, is presented. A distance rij between rotators at sites i and j is introduced, such that the corresponding two-body interaction decays with rij as a power-law, (α ⩾ 0). The index α controls the range of the interactions, in such a way that one recovers both the fully-coupled (i.e. mean-field limit) and nearest-neighbour-interaction models in the particular limits α = 0 and α → ∞, respectively. The dynamics of the model is investigated for energies U below its critical value (U < Uc), with initial conditions corresponding to zero magnetization. The presence of quasi-stationary states (QSSs), whose durations tQSS increase for increasing values of N, is verified for values of α in the range 0 ⩽ α < 1, like the ones found for the similar model of XY rotators. Moreover, for a given energy U, our numerical analysis indicates that tQSS ∼ Nγ, where the exponent γ decreases for increasing α in the range 0 ⩽ α < 1 and particularly, our results suggest that γ → 0 as α → 1. The growth of tQSS with N could be interpreted as a breakdown of ergodicity, which is shown herein to occur for any value of α in this interval.

Exact fourth order invariants for one-dimensional time-dependent Hamiltonian systems

We investigate exact fourth order invariants for one-dimensional time dependent dynamical systems by extending the approach of Struckmeier and Riedel (Phys Rev Lett 85:3833, 2000). The invariants contain a function of time \(f_4(t)\), which is a solution of a linear fourth order differential equation.

Statistical properties of one-dimensional parametrically kicked Hamilton systems.

We study the one-dimensional Hamiltonian systems and their statistical behavior, assuming the initial microcanonical distribution and describing its change under a parametric kick, which by definition means a discontinuous jump of a control parameter of the system. Following a previous work by Papamikos and Robnik [J. Phys. A: Math. Theor. 44, 315102 (2011)], we specifically analyze the change of the adiabatic invariant (the action) of the system under a parametric kick: A conjecture has been put forward that the change of the action at the mean energy always increases, which means, for the given statistical ensemble, that the Gibbs entropy in the mean increases. By means of a detailed analysis of a great number of case studies, we show that the conjecture largely is satisfied, except if either the potential is not smooth enough or if the energy is too close to a stationary point of the potential (separatrix in the phase space). Very fast changes in a time-dependent system quite generally can be well described by such a picture and by the approximation of a parametric kick, if the change of the parameter is sufficiently fast and takes place on the time scale of less than one oscillation period. We discuss our work in the context of the statistical mechanics in the sense of Gibbs.

Anomalous Fluctuations for a Perturbed Hamiltonian System with Exponential Interactions

A one-dimensional Hamiltonian system with exponential interactions perturbed by a conservative noise is considered. It is proved that energy superdiffuses and upper and lower bounds describing this anomalous diffusion are obtained

Dynamic Correlators of Fermi-Pasta-Ulam Chains and Nonlinear Fluctuating Hydrodynamics

We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics. In fact, our scheme is more general and would also cover other one-dimensional Hamiltonian systems, for example, classical and quantum fluids. Fluctuating hydrodynamics is a nonlinear system of conservation laws with noise. For a single mode, it is equivalent to the noisy Burgers equation, for which explicit solutions are available. Our focus is the case of several modes. No exact solution has been found so far, and we rely on a one-loop approximation. The resulting mode-coupling equations have a quadratic memory kernel and describe the time evolving 3×3 correlator matrix of the locally conserved fields. Long time asymptotics is computed analytically, and finite time properties are obtained through a numerical simulation of the mode-coupling equations.

Effects of complex time on periodic and nonperiodic classical trajectories of one-dimensional Hamiltonian systems

In recent years, much research has been carried out on extending both quantum mechanics and classical mechanics into the complex domain by making parameters of real hermitian Hamiltonians or total energy of the system complex. In this paper we investigate the effects of complex time on periodic and nonperiodic trajectories of both hermitian and nonhermitian one-dimensional classical Hamiltonian systems. Most of the periodic classical trajectories of real hermitian systems turn into nonperiodic and open when the energy or the parameters of the potential become complex. We show that when time is taken as a complex quantity with a specific fixed phase angle or as a specific complex function, nonperiodic trajectories become periodic and closed. Furthermore, we show that real hermitian systems, such as H = p2/2m + x4 + bx3 + cx2 + dx (b, c, and d are real quantities) possess classical periodic trajectories for real energies even when time is complex (i.e., t = treiτ). It was found that there is a discrete set of τ values for which the trajectories of the preceding system are closed and periodic and periods associated with them form a countably infinite set.