Chaotic transport in deterministic sine-Gordon soliton ratchets.

We investigate homogeneous and inhomogeneous sine-Gordon ratchet systems in which a temporal symmetry and the spatial symmetry, respectively, are broken. We demonstrate that in the inhomogeneous systems with ac driving the soliton dynamics is chaotic in certain parameter regions, although the soliton motion is unidirectional. This is qualitatively explained by a one-collective-coordinate theory which yields an equation of motion for the soliton that is identical to the equation of motion for a single particle ratchet which is known to exhibit chaotic transport in its underdamped regime. For a quantitative comparison with our simulations we use a two-collective-coordinate (2CC) theory. In contrast to this, homogeneous sine-Gordon ratchets with biharmonic driving, which breaks a temporal shift symmetry, do not exhibit chaos. This is explained by a 2CC theory which yields two ODEs: one is linear, the other one describes a parametrically driven oscillator which does not exhibit chaos. The latter ODE can be solved by a perturbation theory which yields a hierarchy of linear equations that can be solved exactly order by order. The results agree very well with the simulations.

Soliton ratchets in sine-Gordon systems with additive inhomogeneities.

We investigate the ratchet dynamics of solitons of a sine-Gordon system with additive inhomogeneities. We show by means of a collective coordinate approach that the soliton moves like a particle in an effective potential which is a result of the inhomogeneities. Different degrees of freedom of the soliton are used as collective coordinates in order to study their influence on the motion of the soliton. The collective coordinates considered are the soliton position, its width and offset, and the height of the spikes that appear on the soliton. The results of the theory are compared with numerical simulations of the full system.

Thermal diffusion of solitons on anharmonic chains with long-range coupling.

We extend our studies of thermal diffusion of nontopological solitons to anharmonic Fermi-Pasta-Ulam-type chains with additional long-range couplings. The observed superdiffusive behavior in the case of nearest-neighbor interaction turns out to be the dominating mechanism for the soliton diffusion on chains with long-range interactions. Using a collective variable technique in the framework of a variational analysis for the continuum approximation of the chain, we derive a set of stochastic integrodifferential equations for the collective variables (CVs) soliton position and the inverse soliton width. This set can be reduced to a statistically equivalent set of Langevin-type equations for the CV, which shares the same Fokker-Planck equation. The solution of the Langevin set and the Langevin dynamics simulations of the discrete system agree well and demonstrate that the variance of the soliton increases stronger than linearly with time (superdiffusion). This result for the soliton diffusion on anharmonic chains with long-range interactions reinforces the conjecture that superdiffusion is a generic feature of nontopological solitons.

Envelope solitons on anharmonic damped atomic chains.

We investigate the influence of dissipation on envelope solitons on anharmonic chains. We consider both Stokes and hydrodynamical damping and derive the evolution equations for the envelope in both the continuum and the quasi-continuum approximation of the chain. We introduce an appropriate collective variable ansatz for the envelope in order to describe the effect of damping on the soliton shape. We derive ordinary differential equations for the evolution of the three collective variables amplitude, width, and chirp which describe the spatial modulation of the envelope. The analytical results are in good agreement with the simulations of the discrete system for high-energy excitations on the chain. Our results derived from the quasi-continuum approximation show significant improvements compared to the continuum approximation.

Persistent breathers in long-ranged discrete nonlinear Schrödinger models.

The DNLS model including Kac-Baker long-range interactions and nonlinear damping exhibits prominent effects in computer simulations. The combination of long-range forces and damping yields a periodic pattern of stationary breathers from an originally uniformly distributed background. The inverse interaction radius determines the periodicity which can be understood in the quasicontinuum approximation of the system. For the undamped system, we investigate the impact of the long-range interactions on the transition to the persistent-breather phase, which only depends on the energy and the norm of the DNLS. Using Monte Carlo techniques, we can monitor the localization strength as a function of the the long-range radius and the system temperature, which is formally negative in the persistent-breather phase.

Optimization of soliton ratchets in inhomogeneous sine-Gordon systems.

Unidirectional motion of solitons can take place, although the applied force has zero average in time, when the spatial symmetry is broken by introducing a potential V(x) , which consists of periodically repeated cells with each cell containing an asymmetric array of strongly localized inhomogeneities at positions xi. A collective coordinate approach shows that the positions, heights, and widths of the inhomogeneities (in that order) are the crucial parameters so as to obtain an optimal effective potential Uopt that yields a maximal average soliton velocity. Uopt essentially exhibits two features: double peaks consisting of a positive and a negative peak, and long flat regions between the double peaks. Such a potential can be obtained by choosing inhomogeneities with opposite signs (e.g., microresistors and microshorts in the case of long Josephson junctions) that are positioned close to each other, while the distance between each peak pair is rather large. These results of the collective variable theory are confirmed by full simulations for the inhomogeneous sine-Gordon system.

Paramagnetic colloidal ribbons in a precessing magnetic field.

We investigate the dynamics of a kink in a damped parametrically driven nonlinear Klein-Gordon equation. We show by using a method of averaging that, in the high-frequency limit, the kink moves in an effective potential and is driven by an effective constant force. We demonstrate that the shape of the solitary wave can be controlled via the frequency and the eccentricity of the modulation. This is in accordance with the experimental results reported in a recent paper [Casic et al., Phys. Rev. Lett. 110, 168302 (2013)], where the dynamic self-assembly and propulsion of a ribbon formed from paramagnetic colloids in a time-dependent magnetic field has been studied.

Anomalies of ac driven solitary waves with internal modes: nonparametric resonances induced by parametric forces.

We study the dynamics of kinks in the straight phi(4) model subjected to a parametric ac force, both with and without damping, as a paradigm of solitary waves with internal modes. By using a collective coordinate approach, we find that the parametric force has a nonparametric effect on the kink motion. Specifically, we find that the internal mode leads to a resonance for frequencies of the parametric driving close to its own frequency, in which case the energy of the system grows as well as the width of the kink. These predictions of the collective coordinate theory are verified by numerical simulations of the full partial differential equation. We finally compare this kind of resonance with that obtained for nonparametric ac forces and conclude that the effect of ac drivings on solitary waves with internal modes is exactly the opposite of their character in the partial differential equation.

Kink topology control by high-frequency external forces in nonlinear Klein-Gordon models.

A method of averaging is applied to study the dynamics of a kink in the damped double sine-Gordon equation driven by both external (nonparametric) and parametric periodic forces at high frequencies. This theoretical approach leads to the study of a double sine-Gordon equation with an effective potential and an effective additive force. Direct numerical simulations show how the appearance of two connected π kinks and of an individual π kink can be controlled via the frequency. An anomalous negative mobility phenomenon is also predicted by theory and confirmed by simulations of the original equation.

Spin dynamics simulations of two-dimensional clusters with Heisenberg and dipole-dipole interactions.

Spin dynamics with the Landau-Lifshitz equation has provided topics for a wealth of research endeavors. We introduce here a numerical integration method which explicitly uses the precession motion of a spin about the local field, thus intrinsically conserving spin lengths, and therefore allowing for relatively quick results for a large number of situations with varying temperatures and couplings. This method is applied to the effect of long-range dipole-dipole interactions in two-dimensional clusters of spins with nearest-neighbor XY-Heisenberg exchange interactions on a square lattice at finite temperature. The structures thus obtained are analyzed through orientational correlations functions. Magnon dispersion curves, different from those of the standard Heisenberg model, are obtained and discussed. The number of vortices in the system is discussed as a function of temperature and typical examples of vortex dynamics are shown.

Overdamped sine-Gordon kink in a thermal bath.

We study sine-Gordon kink diffusion at finite temperature in the overdamped limit. By means of a general perturbative approach, we calculate the first- and second-order (in temperature) contributions to the diffusion coefficient. We compare our analytical predictions with numerical simulations. The good agreement allows us to conclude that, up to temperatures where kink-antikink nucleation processes cannot be neglected, a diffusion constant linear and quadratic in temperature gives a very accurate description of the diffusive motion of the kink. The quadratic temperature dependence is shown to stem from the interaction with the phonons. In addition, we calculate and compute the average value < phi(x,t)> of the wave function as a function of time, and show that its width grows with square root of t. We discuss the interpretation of this finding and show that it arises from the dispersion of the kink center positions of individual realizations which all keep their width.