A theoretical derivation of cellular structures of flames

Using an equation for the formation of flame fronts derived by Sivashinsky and augmented by an additional term describing buoyancy effects we present an analytical treatment of the formation of cellular structures of flames formed by plane burners. In particular we find rectangular and square patterns. We first study the stability of the plane flame front by linear stability analysis and then transform the basic equation into a set of equations for the amplitudes of the stable and unstable modes. The amplitudes of the stable modes can be eliminated by the slaving principle so that generalized Ginzburg-Landau equations result which in a general frame were previously derived by one of us (H.H.). These equations are then solved explicitly and the stability of the resulting pattern is proven.     

Frequency locking in spatially extended systems

A variant of the complex Ginzburg-Landau equation is used to investigate the frequency-locking phenomena in spatially extended systems. With appropriate parameter values, a variety of frequency-locked patterns including flats, pi fronts, labyrinths, and 2pi/3 fronts emerge. We show that in spatially extended systems, frequency locking can be enhanced or suppressed by diffusive coupling. Novel patterns such as chaotically bursting domains and target patterns are also observed during the transition to locking.     

Retracting fronts induce spatiotemporal intermittency

The intermittent route to spatiotemporal complexity is analyzed in simple models which display a subcritical bifurcation without hysteresis. A new type of spatiotemporal complex behavior is found, induced by fronts which "clean" the perturbations around an unstable state. The mechanism which generates these "retracting fronts" through nonlinear dispersion is analyzed in the frame of the complex Ginzburg-Landau equation. For sufficiently strong nonlinear dispersion the effects also occur for a supercritical bifurcation.     

Fluctuations of one dimensional Ginzburg-Landau models in nonequilibrium

We study the fluctuation of one dimensional Ginzburg-Landau models in nonequilibrium along the hydrodynamic (diffusion) limit. The hydrodynamic limit has been proved to be a nonlinear diffusion equation by Fritz, Guo-Papanicolaou-Varadhan, etc. We proved that if the potential is uniformly convex then the fluctuation process is governed by an Ornstein-Uhlenbeck process whose drift term is obtained by formally linearizing the hydrodynamic equation.Work partially supported by the National Science Foundation under grant no. DMS 8806731 and DMS 9101196     

Resonantly forced inhomogeneous reaction-diffusion systems

The dynamics of spatiotemporal patterns in oscillatory reaction-diffusion systems subject to periodic forcing with a spatially random forcing amplitude field are investigated. Quenched disorder is studied using the resonantly forced complex Ginzburg-Landau equation in the 3:1 resonance regime. Front roughening and spontaneous nucleation of target patterns are observed and characterized. Time dependent spatially varying forcing fields are studied in the 3:1 forced FitzHugh-Nagumo system. The periodic variation of the spatially random forcing amplitude breaks the symmetry among the three quasi-homogeneous states of the system, making the three types of fronts separating phases inequivalent. The resulting inequality in the front velocities leads to the formation of "compound fronts" with velocities lying between those of the individual component fronts, and "pulses" which are analogous structures arising from the combination of three fronts. Spiral wave dynamics is studied in systems with compound fronts. (c) 2000 American Institute of Physics.     

Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation

It is shown that for smooth initial data solutions of the Robinson-Trautman equation (also known as the two-dimensional Calabi equation) exist for all positive “times,” and asymptotically converge to a constant curvature metric. Supported in part by NSF grant DMS-885773 to the Courant Institute and by the Polish Ministry of Science Research grant RPBP 01.3     

Exact Solutions of Discrete Complex Cubic Ginzburg-LandauEquation and Their Linear Stability

The discrete complex cubic Ginzburg-Landau equation is an important model to describe a number of physical systems such as Taylor and frustrated vortices in hydrodynamics and semiconductor laser arrays in optics. In this paper, the exact solutions of the discrete complex cubic Ginzburg-Landau equation are derived using homogeneous balance principle and the G' / G-expansion method, and the linear stability of exact solutions is discussed.     

Nonlinear evolution of two-magnetofluid instability

The nonlinear surface instability of a horizontal interface separating two magnetic fluids of different densities, magnetic permeabilities, and velocities, including surface tension effects, is investigated. The magnetic field is applied along the direction of streaming. It is shown that the evolution of the amplitude is governed by a nonlinear Ginzburg-Landau equation with the use of the multiple scale method. When the influence of streaming is neglected, the nonlinear diffusion equation is obtained. Further, it is shown that a nonlinear Schrödinger equation is obtained in the absence of gravity. The various stability criteria are discussed from these equations, of both Rayleigh-Taylor and Kelvin-Helmholtz problems, both analytically and numerically and the stability diagrams are obtained. Obtained also are the stability properties of solitary solutions to the Ginzburg-Landau equation in the case of constant surface tension.     

Chirped Lambert W-kink solitons of the complex cubic-quintic Ginzburg-Landau equation with intrapulse Raman scattering

In this paper, an exact explicit solution for the complex cubic-quintic Ginzburg-Landau equation is obtained, by using Lambert W function or omega function. More pertinently, we term them as Lambert W-kink-type solitons, begotten under the influence of intrapulse Raman scattering. Parameter domains are delineated in which these optical solitons exit in the ensuing model. We report the effect of model coefficients on the amplitude of Lambert W-kink solitons, which enables us to control efficiently the pulse intensity and hence their subsequent evolution. Also, moving fronts or optical shock-type solitons are obtained as a byproduct of this model. We explicate the mechanism to control the intensity of these fronts, by fine tuning the spectral filtering or gain parameter. It is exhibited that the frequency chirp associated with these optical solitons depends on the intensity of the wave and saturates to a constant value as the retarded time approaches its asymptotic value.     

Defect-mediated snaking: A new growth mechanism for localized structures

Stationary spatially localized patterns in parametrically driven systems are studied, focusing on the 2:1 and 1:1 resonance tongues as described by the forced complex Ginzburg-Landau equation. Homoclinic snaking is identified in both cases and the nature of the growth of the localized structures along the snaking branches is described. The structures grow from a central defect that inserts new rolls on either side, while pushing existing rolls outwards. This growth mechanism differs fundamentally from that found in other systems exhibiting homoclinic snaking in which new rolls are added at the fronts that connect the structure to the background homogeneous state.     

Analytic solutions of self-similar pulse based on Ginzburg-Landau equation with constant coefficients

Based on the constant coefficients of Ginzburg-Landau equation that considers the influence of the doped fiber retarded time on the evolution of self-similar pulse, the parabolic asymptotic self-similar solutions were obtained by the symmetry reduction algorithm. The parabolic asymptotic amplitude function, phase function, strict linear chirp function and the effective temporal pulse width of self-similar pulse are given in this paper. And these theoretical results are consistent with the numerical simulations. Supported by the Natural Science Foundation of Guangdong Province of China (Grant No. 04010397)     

Dissipative finite degrees of freedom dynamical system and description of optical systems with saturable amplification, saturable losses and filtering

Single-mode fiber optical system with saturable amplification, saturable losses and spectral filtering as proposed by Rozanov and Fedorov (1998) [10] is studied. The system of ordinary differential equations (ODE’s) that can help investigation of the original physical system is proposed. It allows calculation of linear and nonlinear fixed points as well as the study of their stability, so it can be used for analysis of coherent structures and their classification. Derived system of ODE’s extends the earlier one proposed by van Saarloos and Hohenberg (1992) [2], for the analysis of coherent structures of the qubic-quintic Ginzburg-Landau equation, by including additionally the temporal dependences of the gain and losses. In order to verify it, it was applied to the earlier considered cases of fast and slow changes in the amplification and losses. Earlier obtained localized structures namely pulses, have been observed via numerical solution of the proposed system. In addition, new families of fronts have been identified.     

Renormalization Group and the Ginzburg-Landau equation

We use Renormalization Group methods to prove detailed long time asymptotics for the solutions of the Ginzburg-Landau equations with initial data approaching, asx±, different spiraling stationary solutions. A universal pattern is formed, depending only on this asymptotics at spatial infinity.Supported by NSF grant DMS-8903041 and by EEC Grant SCI-CT91-0695TSTS     

Front reversals, wave traps, and twisted spirals in periodically forced oscillatory media

A new kind of nonlinear nonequilibrium patterns--twisted spiral waves--is predicted for periodically forced oscillatory reaction-diffusion media. We show, furthermore, that, in such media, spatial regions with modified local properties may act as traps where propagating waves can be stored and released in a controlled way. Underlying both phenomena is the effect of the wavelength-dependent propagation reversal of traveling phase fronts, always possible when homogeneous oscillations are modulationally stable without forcing. The analysis is performed using as a model the complex Ginzburg-Landau equation, applicable for reaction-diffusion systems in the vicinity of a supercritical Hopf bifurcation.     

Relation between Cauchy data for the scattering by a wedge

We justify an extension of the method of complex characteristics [6] for the Helmholtz equation in nonconvex angles. For convex angles, the method was introduced in [1] and developed in [6, 11]. On leave from the Institute for Information Transmission Problems, Russian Academy of Sciences. Supported partly by the Alexander von Humboldt Research Award, grant DFG 436 RUS 113/929/0-1, and by the FWF grant P19138-N13. Supported by Project SEP 2004-CO1-46769-F, CONACYT, México; SNI, México; Project MA-4.12, CIC of UMSNH, México.     

Modulation Equations: Stochastic Bifurcation in Large Domains

We consider the stochastic Swift-Hohenberg equation on a large domain near its change of stability. We show that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a stochastic Ginzburg-Landau equation. We then proceed to show that this approximation also extends to the invariant measures of these equations.     

Stability of convective patterns in reaction fronts: a comparison of three models

Autocatalytic reaction fronts generate density gradients that may lead to convection. Fronts propagating in vertical tubes can be flat, axisymmetric, or nonaxisymmetric, depending on the diameter of the tube. In this paper, we study the transitions to convection as well as the stability of different types of fronts. We analyze the stability of the convective reaction fronts using three different models for front propagation. We use a model based on a reaction-diffusion-advection equation coupled to the Navier-Stokes equations to account for fluid flow. A second model replaces the reaction-diffusion equation with a thin front approximation where the front speed depends on the front curvature. We also introduce a new low-dimensional model based on a finite mode truncation. This model allows a complete analysis of all stable and unstable fronts.     

Scattering for the wave equation on the Schwarzschild Metric

We study the wave equation for the Schwarzschild metric. Wave operators are constructed which yield solutions with given asymptotic behavior either at infinity or on the horizon. We prove asymptotic completeness for these wave operators.Supported by NSF grant No. PHY82-204399.     

Comments on the effect of disorder on transport with intermediate degree of coherence: Calculation of the mean square displacement

We use the generalized master equation with a simple exponential memory but with disorder in its spatial dependence to analyze the combined effect of coherence and randomness on the transport of quasiparticles. We calculate the mean-square-displacement and find that it retains well-known properties in the presence of randomness.Supported by NSF grant no. DMR-7919539     

Solution of the Ornstein-Zernike equation for a softcore Yukawa fluid. II. Numerical results

Numerical calculations are reported for the simplest case of the soft-core Yukawa fluid introduced in an earlier paper. Attention is given to the thermodynamic behavior, the correlation functions, and the interparticle potentials found by inverting the structural information using Percus-Yevick and hypernetted chain integration equation approximations.Supported by ARGC grant No. B7715646R.