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arXiv1996

Mobility and diffusivity in a generalized Frenkel-Kontorova model

Molecular dynamics simulations are used to investigate the atomic mobility and diffusivity of a generalized Frenkel-Kontorova model which takes into account anharmonic (exponential) interaction of atoms subjected to a three-dimensional substrate potential periodic in two dimensions and nonconvex (Morse) in the third dimension. The numerical results are explained by a phenomenological theory which treats a system of strongly interacting atoms as a system of weakly interacting quasiparticles (kinks). Model parameters are chosen close to those for {\bf K}-{\bf W}(112) adsorption system.

patt-sol↗

arXiv1996

Transport and conservation laws

We study the lowest order conservation laws in one-dimensional (1D) integrable quantum many-body models (IQM) as the Heisenberg spin 1/2 chain, the Hubbard and t-J model. We show that the energy current is closely related to the first conservation law in these models and therefore the thermal transport coefficients are anomalous. Using an inequality on the time decay of current correlations we show how the existence of conserved quantities implies a finite charge stiffness (weight of the zero frequency component of the conductivity) and so ideal conductivity at finite temperatures.

cond-mat.str-el↗

arXiv1997

Resonances of dynamical checkerboard states in Josephson arrays with self-inductance

We study the dynamics of fully frustrated, underdamped Josephson arrays. Experiments reveal remarkable similarities among the dc current-voltage characteristics of several kinds of square and triangular arrays, where two resonant voltages are observed. Simulations indicate that a dynamical checkerboard solution underlies these similarities. By assuming such a solution, we reduce the governing equations to three coupled pendulum equations, and thereby calculate the voltages of the intrinsic resonances analytically.

patt-sol↗

arXiv1997

Solitonic approach to the dimerization problem in correlated one-dimensional systems

Using exact diagonalizations we consider self-consistently the lattice distortions in odd Peierls-Hubbard and spin-Peierls periodic rings in the adiabatic harmonic approximation. From the tails of the inherent spin soliton the dimerization d_\infty of regular even rings is found by extrapolations to infinite ring lengths. Considering a wide region of electron-electron onsite interaction values U>0 compared with the band width 4t_0 at intermediately strong electron-phonon interaction g, known relationships obtained by other methods are reproduced and/or refined within one unified approach: such as the maximum of d_\infty at U \simeq 3 t_0 for g \simeq 0.5 and its shift to zero for g \to g_c \approx 0.7. The hyperbolic tangent shape of the spin soliton is retained for any U and g <~ 0.6. In the spin-Peierls limit the d_\infty are found to be in agreement with results of DMRG computations.

cond-mat.str-el↗

arXiv1998

Parametrically forced sine-Gordon equation and domain walls dynamics in ferromagnets

A parametrically forced sine-Gordon equation with a fast periodic {\em mean-zero} forcing is considered. It is shown that $π$-kinks represent a class of solitary-wave solutions of the equation. This result is applied to quasi-one-dimensional ferromagnets with an easy plane anisotropy, in a rapidly oscillating magnetic field. In this case the $π$-kink solution we have introduced corresponds to the uniform ``true'' domain wall motion, since the magnetization directions on opposite sides of the wall are anti-parallel. In contrast to previous work, no additional anisotropy is required to obtain a true domain wall. Numerical simulations showed good qualitative agreement with the theory.

patt-sol↗

arXiv1999

Mechanisms of spatial current-density instabilities in $p^+ - p^- - n - p^+ -n^{++}$ structures

Semiconductor $p^+ - p^- - n - p^+ - n^{++}$ structures with large junction and contact areas are treated as 1 \times 2-dimensional active media, in which self-organized pattern formation can be expected. The local bistable behavior of the structures may emanate from two different mechanisms both governed by a nonlinear current feedback-loop between the electrons and holes injected from the outer layers. By considering the device to be composed of an active subsystem with negative differential resistance and a passive resistive layer with positive differential resistance an analytical approach is suggested to understand and describe the corresponding physical mechanisms in a self-consistent way. Analytical solutions of the derived model equations allow a description of homogeneous stationary states and yield explicit expressions of the current-density vs. voltage characteristics of the whole structure and its subsystems. A stability analysis of the homogeneous states with respect to two-dimensional transversal harmonic fluctuations is performed for the two cases under study. The resulting dispersion relations allow two different types of instability. While the first one is of Ridley's type which is characteristic for any spatially extended electrical system with negative differential resistance, the second type can be considered as a solid-state analogue of Turing's instability known as a generic instability mechanism which may lead, e. g., to the formation of periodic patterns.

patt-sol↗

arXiv1998

Higher order Shapiro steps in ac-driven Josephson junctions

We demonstrate that the well known phase-locking mechanism leading to Shapiro steps in ac-driven Josephson junctions is always accompanied by a higher order phase-locking mechanism similar to that of the parametrically driven pendulum. This effect, resulting in a $π$-periodic effective potential for the phase, manifests itself clearly in the parameter regions where the usual Shapiro steps are expected to vanish.

patt-sol↗

arXiv1999

On modulational instability and energy localization in anharmonic lattices at finite energy density

The localization of vibrational energy, induced by the modulational instability of the Brillouin-zone-boundary mode in a chain of classical anharmonic oscillators with finite initial energy density, is studied within a continuum theory. We describe the initial localization stage as a gas of envelope solitons and explain their merging, eventually leading to a single localized object containing a macroscopic fraction of the total energy of the lattice. The initial-energy-density dependences of all characteristic time scales of the soliton formation and merging are described analytically. Spatial power spectra are computed and used for the quantitative explanation of the numerical results.

cond-mat↗

arXiv1993

Rotating Rayleigh-Bénard Convection: Aspect Ratio Dependence of the Initial Bifurcations

The initial bifurcations in rotating Rayleigh-Bénard convection are studied in the range of dimensionless rotation rate $0 < Ω< 2150$ for an aspect-ratio-2.5 cylindrical cell. We used simultaneous optical shadowgraph, heat transport and local temperature measurements to determine the stability and characteristics of the azimuthally-periodic wall convection state. We also show that the second transition corresponds to the onset of bulk convection. Our results for critical Rayleigh numbers, precession frequencies and critical mode numbers agree well with theoretical results. The dynamics of the wall convection state can be described by a complex Ginzburg-Landau amplitude equation.

patt-sol↗

arXiv1993

Multidimensional Pattern Formation Has an Infinite Number of Constants of Motion

Extending our previous work on 2D growth for the Laplace equation we study here {\it multidimensional} growth for {\it arbitrary elliptic} equations, describing inhomogeneous and anisotropic pattern formations processes. We find that these nonlinear processes are governed by an infinite number of conservation laws. Moreover, in many cases {\it all dynamics of the interface can be reduced to the linear time--dependence of only one ``moment" $M_0$} which corresponds to the changing volume while {\it all higher moments, $M_l$, are constant in time. These moments have a purely geometrical nature}, and thus carry information about the moving shape. These conserved quantities (eqs.~(7) and (8) of this article) are interpreted as coefficients of the multipole expansion of the Newtonian potential created by the mass uniformly occupying the domain enclosing the moving interface. Thus the question of how to recover the moving shape using these conserved quantities is reduced to the classical inverse potential problem of reconstructing the shape of a body from its exterior gravitational potential. Our results also suggest the possibility of controlling a moving interface by appropriate varying the location and strength of sources and sinks.

patt-sol↗

arXiv1993

Domain Walls in Non-Equilibrium Systems and the Emergence of Persistent Patterns

Domain walls in equilibrium phase transitions propagate in a preferred direction so as to minimize the free energy of the system. As a result, initial spatio-temporal patterns ultimately decay toward uniform states. The absence of a variational principle far from equilibrium allows the coexistence of domain walls propagating in any direction. As a consequence, *persistent* patterns may emerge. We study this mechanism of pattern formation using a non-variational extension of Landau's model for second order phase transitions. PACS numbers: 05.70.Fh, 42.65.Pc, 47.20.Ky, 82.20Mj

patt-sol↗

arXiv1993

Dissipative Boussinesq System of Equations in the Bénard-Marangoni Phenomenon

By using the long-wave approximation, a system of coupled evolution equations for the bulk velocity and the surface perturbations of a Bénard-Marangoni system is obtained. It includes nonlinearity, dispersion and dissipation, and it can be interpreted as a dissipative generalization of the usual Boussinesq system of equations. As a particular case, a strictly dissipative version of the Boussinesq system is obtained. Finnaly, some speculations are made on the nature of the physical phenomena described by this system of equations.

patt-sol↗

arXiv1993

Temporal Forcing of Small-Amplitude Waves in Anisotropic Systems

We investigate the effect of resonant temporal forcing on an anisotropic system that exhibits a Hopf bifurcation to obliquely traveling waves in the absence of this forcing. We find that the forcing can excite various phase-locked standing-wave structures: rolls, rectangles and cross rolls. At onset, at most one of the two - rolls or rectangles - is stable. The cross rolls can arise in a secondary bifurcation and can be stable. Experimentally, they would appear as a periodic switching between a structure in which the `zig'-component dominates and one with dominating `zag'-structure. Since there are two symmetry-related states of this kind one may expect disordered structures to arise due to the break-up of the pattern into domains. The results are consistent with recent experiments on electro-convection in nematic liquid crystals by de la Torre and Rehberg. We also apply the general analysis to a model of the behavior near a Lifshitz point, where the angle of obliqueness vanishes. This analysis indicates that phase-locked standing rectangles are always unstable in this parameter regime.

patt-sol↗

arXiv1994

Kink stability, propagation, and length scale competition in the periodically modulated sine-Gordon equation

We have examined the dynamical behavior of the kink solutions of the one-dimensional sine-Gordon equation in the presence of a spatially periodic parametric perturbation. Our study clarifies and extends the currently available knowledge on this and related nonlinear problems in four directions. First, we present the results of a numerical simulation program which are not compatible with the existence of a radiative threshold, predicted by earlier calculations. Second, we carry out a perturbative calculation which helps interpret those previous predictions, enabling us to understand in depth our numerical results. Third, we apply the collective coordinate formalism to this system and demonstrate numerically that it accurately reproduces the observed kink dynamics. Fourth, we report on a novel occurrence of length scale competition in this system and show how it can be understood by means of linear stability analysis. Finally, we conclude by summarizing the general physical framework that arises from our study.

patt-sol↗

arXiv1994

New exact fronts for the nonlinear diffusion equation with quintic nonlinearities

We consider travelling wave solutions of the reaction diffusion equation with quintic nonlinearities $u_t = u_{xx} + μu (1 -u ) ( 1 +αu + βu^2 +γu^3)$. If the parameters $α, β$ and $γ$ obey a special relation, then the criterion for the existence of a strong heteroclinic connection can be expressed in terms of two of these parameters. If an additional restriction is imposed, explicit front solutions can be obtained. The approach used can be extended to polynomials whose highest degree is odd.

patt-sol↗

arXiv1993

Multiple Front Propagation Into Unstable States

The dynamics of transient patterns formed by front propagation in extended nonequilibrium systems is considered. Under certain circumstances, the state left behind a front propagating into an unstable homogeneous state can be an unstable periodic pattern. It is found by a numerical solution of a model of the Fréedericksz transition in nematic liquid crystals that the mechanism of decay of such periodic unstable states is the propagation of a second front which replaces the unstable pattern by a another unstable periodic state with larger wavelength. The speed of this second front and the periodicity of the new state are analytically calculated with a generalization of the marginal stability formalism suited to the study of front propagation into periodic unstable states. PACS: 47.20.Ky, 03.40.Kf, 47.54.+r

cond-mat↗

arXiv1994

Jam phases in two-dimensional cellular automata model of traffic flow

The jam phases in a two-dimensional cellular automata model of traffic flow are investigated by computer simulations. Two different types of the jam phases are found. The spatially diagonal long-range correlation obeys the power law at the low-density jam configurations. The diagonal correlation exponentially decays at the high-density jam. The exponent of the short-range correlation in the diagonal direction is introduced to define the transition between these two phases. We also discuss the stability of the jams.

patt-sol↗

arXiv1993

A New Class of Nonsingular Exact Solutions for Laplacian Pattern Formation

We present a new class of exact solutions for the so-called {\it Laplacian Growth Equation} describing the zero-surface-tension limit of a variety of 2D pattern formation problems. Contrary to common belief, we prove that these solutions are free of finite-time singularities (cusps) for quite general initial conditions and may well describe real fingering instabilities. At long times the interface consists of N separated moving Saffman-Taylor fingers, with ``stagnation points'' in between, in agreement with numerous observations. This evolution resembles the N-soliton solution of classical integrable PDE's.

patt-sol↗