Searcharxiv⌕ Search

SEARCH · Searcharxiv

Results for “patt-sol”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 307 records · Page 17Linked to original sources
arXiv1995

Absolutely stable solitons in two-component active systems

As is known, a solitary pulse in the complex cubic Ginzburg-Landau (GL) equation is unstable. We demonstrate that a system of two linearly coupled GL equations with gain and dissipation in one subsystem and pure dissipation in another produces absolutely stable solitons and their bound states. The problem is solved in a fully analytical form by means of the perturbation theory. The soliton coexists with a stable trivial state; there is also an unstable soliton with a smaller amplitude, which is a separatrix between the two stable states. This model has a direct application in nonlinear fiber optics, describing an Erbium-doped laser based on a dual-core fiber.

patt-sol↗
arXiv1996

Convective Fingering of an Autocatalytic Reaction Front

We report experimental observations of the convection-driven fingering instability of an iodate-arsenous acid chemical reaction front. The front propagated upward in a vertical slab; the thickness of the slab was varied to control the degree of instability. We observed the onset and subsequent nonlinear evolution of the fingers, which were made visible by a {\it p}H indicator. We measured the spacing of the fingers during their initial stages and compared this to the wavelength of the fastest growing linear mode predicted by the stability analysis of Huang {\it et. al.} [{\it Phys. Rev. E}, {\bf 48}, 4378 (1993), and unpublished]. We find agreement with the thickness dependence predicted by the theory.

patt-sol↗
arXiv1995

Front Stability in Mean Field Models of Diffusion Limited Growth

We present calculations of the stability of planar fronts in two mean field models of diffusion limited growth. The steady state solution for the front can exist for a continuous family of velocities, we show that the selected velocity is given by marginal stability theory. We find that naive mean field theory has no instability to transverse perturbations, while a threshold mean field theory has such a Mullins-Sekerka instability. These results place on firm theoretical ground the observed lack of the dendritic morphology in naive mean field theory and its presence in threshold models. The existence of a Mullins-Sekerka instability is related to the behavior of the mean field theories in the zero-undercooling limit.

patt-sol↗
arXiv1996

Influence of through-flow on linear pattern formation properties in binary mixture convection

We investigate how a horizontal plane Poiseuille shear flow changes linear convection properties in binary fluid layers heated from below. The full linear field equations are solved with a shooting method for realistic top and bottom boundary conditions. Through-flow induced changes of the bifurcation thresholds (stability boundaries) for different types of convective solutions are deter- mined in the control parameter space spanned by Rayleigh number, Soret coupling (positive as well as negative), and through-flow Reynolds number. We elucidate the through-flow induced lifting of the Hopf symmetry degeneracy of left and right traveling waves in mixtures with negative Soret coupling. Finally we determine with a saddle point analysis of the complex dispersion relation of the field equations over the complex wave number plane the borders between absolute and convective instabilities for different types of perturbations in comparison with the appropriate Ginzburg-Landau amplitude equation approximation. PACS:47.20.-k,47.20.Bp, 47.15.-x,47.54.+r

patt-sol↗
arXiv1996

Multiple-Time Higher-Order Perturbation Analysis of the Regularized Long-Wavelength Equation

By considering the long-wave limit of the regularized long wave (RLW) equation, we study its multiple-time higher-order evolution equations. As a first result, the equations of the Korteweg-de Vries hierarchy are shown to play a crucial role in providing a secularity-free perturbation theory in the specific case of a solitary-wave solution. Then, as a consequence, we show that the related perturbative series can be summed and gives exactly the solitary-wave solution of the RLW equation. Finally, some comments and considerations are made on the N-soliton solution, as well as on the limitations of applicability of the multiple scale method in obtaining uniform perturbative series.

patt-sol↗
arXiv1996

Self-replication and splitting of domain patterns in reaction-diffusion systems with fast inhibitor

An asymptotic equation of motion for the pattern interface in the domain-forming reaction-diffusion systems is derived. The free boundary problem is reduced to the universal equation of non-local contour dynamics in two dimensions in the parameter region where a pattern is not far from the points of the transverse instabilities of its walls. The contour dynamics is studied numerically for the reaction-diffusion system of the FitzHugh-Nagumo type. It is shown that in the asymptotic limit the transverse instability of the localized domains leads to their splitting and formation of the multidomain pattern rather than fingering and formation of the labyrinthine pattern.

patt-sol↗
arXiv1996

Scenarios of domain pattern formation in a reaction-diffusion system

We performed an extensive numerical study of a two-dimensional reaction-diffusion system of the activator-inhibitor type in which domain patterns can form. We showed that both multidomain and labyrinthine patterns may form spontaneously as a result of Turing instability. In the stable homogeneous system with the fast inhibitor one can excite both localized and extended patterns by applying a localized stimulus. Depending on the parameters and the excitation level of the system stripes, spots, wriggled stripes, or labyrinthine patterns form. The labyrinthine patterns may be both connected and disconnected. In the the stable homogeneous system with the slow inhibitor one can excite self-replicating spots, breathing patterns, autowaves and turbulence. The parameter regions in which different types of patterns are realized are explained on the basis of the asymptotic theory of instabilities for patterns with sharp interfaces developed by us in Phys. Rev. E. 53, 3101 (1996). The dynamics of the patterns observed in our simulations is very similar to that of the patterns forming in the ferrocyanide-iodate-sulfite reaction.

patt-sol↗
arXiv1996

Temporal Modulation of Traveling Waves in the Flow Between Rotating Cylinders With Broken Azimuthal Symmetry

The effect of temporal modulation on traveling waves in the flows in two distinct systems of rotating cylinders, both with broken azimuthal symmetry, has been investigated. It is shown that by modulating the control parameter at twice the critical frequency one can excite phase-locked standing waves and standing-wave-like states which are not allowed when the system is rotationally symmetric. We also show how previous theoretical results can be extended to handle patterns such as these, that are periodic in two spatial direction.

patt-sol↗
arXiv1996

Structure of Complex-Periodic and Chaotic Media with Spiral Waves

The spatiotemporal structure of reactive media supporting a solitary spiral wave is studied for systems where the local reaction law exhibits a period-doubling cascade to chaos. This structure is considerably more complex than that of simple period-1 oscillatory media. As one moves from the core of the spiral wave the local dynamics takes the form of perturbed period-doubled orbits whose character varies with spatial location relative to the core. An important feature of these media is the existence of a curve where the local dynamics is effectively period-1. This curve arises as a consequence of the necessity to reconcile the conflict between the global topological organization of the medium induced by the presence of a spiral wave and the topological phase space structure of local orbits determined by the reaction rate law. Due to their general topological nature, the phenomena described here should be observable in a broad class of systems with complex periodic behavior.

patt-sol↗
arXiv1996

Dynamical Properties of Multi-Armed Global Spirals in Rayleigh-Benard Convection

Explicit formulas for the rotation frequency and the long-wavenumber diffusion coefficients of global spirals with $m$ arms in Rayleigh-Benard convection are obtained. Global spirals and parallel rolls share exactly the same Eckhaus, zigzag and skewed-varicose instability boundaries. Global spirals seem not to have a characteristic frequency $ω_m$ or a typical size $R_m$, but their product $ω_m R_m$ is a constant under given experimental conditions. The ratio $R_i/R_j$ of the radii of any two dislocations ($R_i$, $R_j$) inside a multi-armed spiral is also predicted to be constant. Some of these results have been tested by our numerical work.

patt-sol↗
arXiv1996

Mutually Penetrating Motion of Self-Organized 2D Patterns of Soliton-Like Structures

Results of numerical simulations of a recently derived most general dissipative-dispersive PDE describing evolution of a film flowing down an inclined plane are presented. They indicate that a novel complex type of spatiotemporal patterns can exist for strange attractors of nonequilibrium systems. It is suggested that real-life experiments satisfying the validity conditions of the theory are possible: the required sufficiently viscous liquids are readily available.

patt-sol↗
arXiv1996

Synchronization, chaos, and breakdown of collective domain oscillations in reaction-diffusion systems

The universal equations describing collective oscillations of the multidomain patterns of small period in an arbitrary $d$-dimensional reaction-diffusion system of the activator-inhibitor type are asymptotically derived. It is shown that not far from the instability leading to the formation of the pulsating multidomain pattern the oscillations of different domains synchronize. In one dimension standing and traveling waves of the oscillation phase are realized. In addition to these, in two dimensions target and spiral waves of the oscillation phase, as well as spatio-temporal chaos of domain oscillations, are feasible. Further inside the unstable region the collective oscillations break down, so the pulsating multidomain pattern transforms into an irregular pulsating pattern, the uniform self-oscillations, or turbulence. The parameter regions where these effects occur are analyzed. The effects of the pattern's disorder are also studied. The conclusions of the analysis are supported by the numerical simulations of a concrete model. The obtained results explain the dynamics of Turing patterns observed in the experiments on chlorite-iodide-malonic acid reaction.

patt-sol↗
arXiv1997

Propagation and Structure of Planar Streamer Fronts

Streamers often constitute the first stage of dielectric breakdown in strong electric fields: a nonlinear ionization wave transforms a non-ionized medium into a weakly ionized nonequilibrium plasma. New understanding of this old phenomenon can be gained through modern concepts of (interfacial) pattern formation. As a first step towards an effective interface description, we determine the front width, solve the selection problem for planar fronts and calculate their properties. Our results are in good agreement with many features of recent three-dimensional numerical simulations. In the present long paper, you find the physics of the model and the interfacial approach further explained. As a first ingredient of this approach, we here analyze planar fronts, their profile and velocity. We encounter a selection problem, recall some knowledge about such problems and apply it to planar streamer fronts. We make analytical predictions on the selected front profile and velocity and confirm them numerically. (abbreviated abstract)

patt-sol↗
arXiv1996

Electroconvection in a Suspended Fluid Film: A Linear Stability Analysis

A suspended fluid film with two free surfaces convects when a sufficiently large voltage is applied across it. We present a linear stability analysis for this system. The forces driving convection are due to the interaction of the applied electric field with space charge which develops near the free surfaces. Our analysis is similar to that for the two-dimensional Bénard problem, but with important differences due to coupling between the charge distribution and the field. We find the neutral stability boundary of a dimensionless control parameter ${\cal R}$ as a function of the dimensionless wave number $κ$. ${\cal R}$, which is proportional to the square of the applied voltage, is analogous to the Rayleigh number. The critical values ${{\cal R}_c}$ and ${κ_c}$ are found from the minimum of the stability boundary, and its curvature at the minimum gives the correlation length ${ξ_0}$. The characteristic time scale ${τ_0}$, which depends on a second dimensionless parameter ${\cal P}$, analogous to the Prandtl number, is determined from the linear growth rate near onset. ${ξ_0}$ and ${τ_0}$ are coefficients in the Ginzburg-Landau amplitude equation which describes the flow pattern near onset in this system. We compare our results to recent experiments.

patt-sol↗
arXiv1996

On a Conjecture of Goriely for the Speed of Fronts of the Reaction--Diffusion Equation

In a recent paper Goriely considers the one--dimensional scalar reaction--diffusion equation $u_t = u_{xx} + f(u)$ with a polynomial reaction term $f(u)$ and conjectures the existence of a relation between a global resonance of the hamiltonian system $ u_{xx} + f(u) = 0$ and the asymptotic speed of propagation of fronts of the reaction diffusion equation. Based on this conjecture an explicit expression for the speed of the front is given. We give a counterexample to this conjecture and conclude that additional restrictions should be placed on the reaction terms for which it may hold.

patt-sol↗
arXiv1997

Order Parameter Equations for Front Transitions: Planar and Circular Fronts

Near a parity breaking front bifurcation, small perturbations may reverse the propagation direction of fronts. Often this results in nonsteady asymptotic motion such as breathing and domain breakup. Exploiting the time scale differences of an activator-inhibitor model and the proximity to the front bifurcation, we derive equations of motion for planar and circular fronts. The equations involve a translational degree of freedom and an order parameter describing transitions between left and right propagating fronts. Perturbations, such as a space dependent advective field or uniform curvature (axisymmetric spots), couple these two degrees of freedom. In both cases this leads to a transition from stationary to oscillating fronts as the parity breaking bifurcation is approached. For axisymmetric spots, two additional dynamic behaviors are found: rebound and collapse.

patt-sol↗
arXiv1996

Parametric Forcing of Waves with Non-Monotonic Dispersion Relation: Domain Structures in Ferrofluids?

Surface waves on ferrofluids exposed to a dc-magnetic field exhibit a non-monotonic dispersion relation. The effect of a parametric driving on such waves is studied within suitable coupled Ginzburg-Landau equations. Due to the non-monotonicity the neutral curve for the excitation of standing waves can have up to three minima. The stability of the waves with respect to long-wave perturbations is determined $via$ a phase-diffusion equation. It shows that the band of stable wave numbers can split up into two or three sub-bands. The resulting competition between the wave numbers corresponding to the respective sub-bands leads quite naturally to patterns consisting of multiple domains of standing waves which differ in their wave number. The coarsening dynamics of such domain structures is addressed.

patt-sol↗