In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, the standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non-self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framework that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed “inverse cubic law” of the return pdf tails.

The effect of a time-dependent flow in an oscillatory chemical system supporting front propagation is studied. Resonant target patterns depend on the strength and frequency of the time-dependent flow. The flow time scale needed to entrain the system to the resonant target period of oscillation depends on the closeness to the natural oscillation frequency of the medium. The flow strength needed to obtain these patterns is interpreted in terms of mixing optimization, and we give conditions for the flow that guarantee the best mixing with the Bernoulli property.

The interaction of nonlinear Schrödinger solitons with extended inhomogeneities with modified group-velocity (GV) and group-velocity dispersion (GVD) coefficients is investigated numerically. Increased GVD coefficients act as potential barriers and yield reflection or transmission of the incoming soliton. Decreased GVD coefficients act as potential wells, and for a given range of parameters the scattering results exhibit periodically repeating windows of trapping and transmission as a function of the length of the segment. It is shown that the escape of the soliton is due to a resonance between the period of the shape oscillations of the soliton inside the segment and the length of the latter. Segments with modified GV coefficients act as potential wells for both positive and negative values of the GV mismatch and can also lead to periodic capture-transmission scattering patterns.

Large amplitude standing waves of spatially periodic sine-Gordon equations are excited and controlled by sweeping the frequency of a small, spatially modulated driving oscillation through resonances in the system. The approach is based on capturing the system into resonances and subsequent adiabatic, persistent phase locking (autoresonance) yielding control via a single external parameter (the driving frequency). Plasma oscillations in the system are excited by using a small amplitude drive in the form of a chirped frequency standing wave, while emergence of autoresonant breather oscillations requires driving by a combination of small amplitude oscillation and standing waves.

Anisotropic crystals containing impurity atoms, when the principal optical axis of the uniaxial crystal and the vector of electrical dipole moment of the impurity atoms are perpendicular to each other and are directed along different crystallographic axes, are shown to have three different mechanisms of the formation of breathers depending on the direction of the wave propagation and on the symmetry of the medium. Explicit analytic expressions for the parameters of breathers and the effective nonlinear susceptibilities for extraordinary waves are obtained. All uniaxial crystals with quadratic susceptibilities can be divided into three different groups, according to the crystal classes. Each group is characterized by a universal structure of the breather zones. The structure of the breather zones of the trigonal, tetragonal, and hexagonal crystals with cubic susceptibilities depends neither on the crystallographic system nor on the crystal classes and coincides with the structure of the breather zones of the trigonal and hexagonal crystals with quadratic susceptibilities and crystal classes 3, $3m$, $\overline{6}$, $\overline{6}m2$. It is shown that structure of breather zones of the different groups of crystals for the cases when the principal optical axis of the uniaxial crystal and the vector of electrical dipole moment of the impurity atoms are mutual parallel or perpendicular to each other are significantly different.

We show analytically that the heat conductivity of oscillator chains diverges with system size $N$ as $N^{1∕3}$, which is the same as for one-dimensional fluids. For long cylinders, we use the hydrodynamic equations for a crystal in one dimension. This is appropriate for stiff systems such as nanotubes, where the eventual crossover to a fluid only sets in at unrealistically large $N$. Despite the extra equation compared to a fluid, the scaling of the heat conductivity is unchanged. For strictly one-dimensional chains, we show that the dynamic equations are those of a fluid at all length scales even if the static order extends to very large $N$. The discrepancy between our results and numerical simulations on Fermi-Pasta-Ulam chains is discussed.

We present a study of the isotropic-nematic interface in a system of hard spherocylinders. First we compare results from Monte Carlo simulations and Onsager density functional theory for the interfacial profiles of the orientational order parameter and the density. Those interfacial properties that are not affected by capillary waves are in good agreement, despite the fact that Onsager theory overestimates the coexistence densities. Then we show results of a Monte Carlo study of the capillary waves of the interface. In agreement with recent theoretical investigations [Elgeti and Schmid, Eur. Phys. J. E 18, 407 (2005)] we find a strongly anisotropic capillary wave spectrum. For the wave numbers accessed in our simulations, the spectrum is quadratic, i.e., elasticity does not play a role. We conjecture that this effect is due to the strong bending rigidity of the director field in suspensions of spherocylinders.

A detailed ${}^2\mathrm{H}$ NMR study of a discotic monomer is reported in its columnar phase at $61.4\mathrm{MHz}$. A simulation of angular-dependent spectral patterns and/or signal intensities of the aromatic deuterons is shown to provide information on the small step (or jump) diffusion rates in the low temperature region of the ${\mathrm{D}}_{ho}$ phase, where the motion is in the intermediate motion regime. In the high end of this phase, the motion falls into the fast motion regime and spectral patterns become insensitive to the diffusion rate. In such a case, angular-dependent NMR signal intensities were used to give the required dynamic information. The diffusive motion about the columnar axis is found to be thermally activated with an activation energy of $69\mathrm{kJ}∕\mathrm{mol}$. Deuteron spin-lattice relaxation rates were measured at $61.4\mathrm{MHz}$ and interpreted by a conventional rotational diffusion model in order to complement and compare with the above study when possible.

The decay of stripe patterns in planarly aligned nematic liquid crystals has been studied experimentally and theoretically. The initial patterns have been generated by the electrohydrodynamic instability and a light diffraction technique has been used to monitor their decay. In our experiments different decay rates have been observed as a function of the pattern wave number. According to our theoretical analysis they belong to a spectrum of decay modes and are individually selected in dependence on the initial conditions. Additional insight has emerged from a refined physical optical description of the diffraction intensity. The results compare well with experiments, which include also controlled modifications of the initial conditions to assess different decay modes.

We studied the kinetic roughening dynamics of drying wet paper. The configurations of dry paper sheets are found to be self-similar, rater than self-affine. Accordingly, the paper roughening dynamics corresponds to the new class of anomalous kinetic roughening [J. J. Ramasco, J. M. López, and M. A. Rodríguez, Phys. Rev. Lett. 84, 2199 (2000)], characterized by the equal local and global roughness exponents $\zeta =\alpha =1$ and the dynamic exponent $z=1.0\pm 0.2$, whereas the spectral roughness exponent ${\alpha }_S>1$ is determined by the long-range correlations characterized by the fractal dimension $D$ of crumpled sheet.

Let the interface between two immiscible fluids in a Hele-Shaw cell have, at $t=0$, a wedge shape. As a wedge is scale-free, the fluid relaxation dynamics are self-similar. We find the dynamic exponent of this self-similar flow and show that the interface shape is given by the solution of an unusual inverse problem of potential theory. We solve this problem analytically for an almost flat wedge, and numerically otherwise. The wedge solution is useful for analysis of pinch-off singularities.

A complete surface acoustic wave band gap is found experimentally in a two-dimensional square-lattice piezoelectric phononic crystal etched in lithium niobate. Propagation in the phononic crystal is studied by direct generation and detection of surface waves using interdigital transducers. The complete band gap extends from $203\text{to}226\mathrm{MHz}$, in good agreement with theoretical predictions. Near the upper edge of the complete band gap, it is observed that radiation to the bulk of the substrate dominates. This observation is explained by introducing the concept of the sound line.

The transport properties of photons traveling through random media are of great fundamental and applied importance. For instance the dwell time due to resonant Mie scattering can lead to a significant reduction in transport velocity. Here, we have measured directly the energy-transport velocity of photons in strongly scattering media using a combination of time resolved transmission, measuring the diffusion coefficient, and angular resolved backscattering, yielding the transport mean free path. We find that the transport velocity is strongly reduced when the effective diameter of the scattering particles is a multiple of half the wavelength. This is consistent with the occurrence of resonant Mie scattering in the samples. We compare our data to previous theoretical calculations of the transport velocity as a function of scatterer size.

The short paths between any two nodes and the robustness of connectivity are advanced properties of scale-free (SF) networks; however, they may be affected by geographical constraints in realistic situations. We consider geographical networks with the SF structure based on planar triangulation for online routings, and suggest scaling relations between the average distance or number of hops on the optimal paths and the network size. We also show that the tolerance to random failures and attacks on hubs is weakened in geographical networks, and that even then it is possible for the extremely vulnerable ones to be improved by adding with the local exchange of links.

By means of the Darboux transformation, we obtain analytical solutions for a soliton set on top of a plane-wave background in coupled Gross-Pitaevskii equations describing a binary Bose-Einstein condensate. We consider basic properties of the solutions with and without the cross interaction [cross phase modulation (XPM)] between the two components of the background. In the absence of the XPM, this solutions maintain properties of one-component condensates, such as the modulation instability (MI); in the presence of the cross interaction, the solutions exhibit different properties, such as restriction of the MI and soliton splitting.

A functional-differential equation equivalent to the infinitely hierarchic system of the coupled equations for many-point velocity distribution functions in turbulence established by Monin and Lundgren is presented, to be compared with the Hopf characteristic functional equation which governs the time evolution of a turbulent velocity field. It turns out that this functional entirely implies the Hopf functional, and vice versa. This functional allows us to view turbulence in a Lagrangian picture simulated by many-particle dynamics.

In Phys. Rev. B 42, 2290 (1990) we used a rigorous projection operator collective variable formalism for nonlinear Klein-Gordon equations to prove the continuum sine-Gordon (SG) equation has a long lived quasimode whose frequency ${\omega }_s=1.004{\Gamma }_0$ is in the continuum just above the lower phonon band edge with a lifetime $(1∕{\tau }_s)=0.0017{\Gamma }_0$. We confirmed the analytic calculations by simulations which agreed very closely with the analytic results. In Phys. Rev. E 62, R60 (2000) the authors performed two numerical investigation which they asserted “show that neither intrinsic internal modes nor quasimodes exist in contrast to previous results.” In this paper we prove their first numerical investigation could not possibly observe the quasimode in principle and their second numerical investigation actually demonstrates the existence of the SG quasimode. Our analytic calculations and verifying simulations were performed for a stationary sine-Gordon soliton fixed at the origin. Yet the authors in Phys. Rev. E 62, R60 (2000) state the explanation of our analytic simulations and confirming simulations are due to the Doppler shift of the phonons emitted by our stationary sine-Gordon soliton which thus has a zero Doppler shift.

In this reply to the comment by C. R. Willis, we show, by quoting his own statements, that the simulations reported in his original work with Boesch [Phys. Rev. B 42, 2290 (1990)] were done for kinks with nonzero initial velocity, in contrast to what Willis claims in his paper. We further show that his alleged proof, which assumes among other approximations that kinks are initially at rest, is not rigorous but an approximation. Moreover, there are other serious misconceptions which we discuss in our paper. As a consequence, our result that quasimodes do not exist in the sG equation [Phys. Rev. E 62, R60 (2000)] remains true.

In the system with two large spheres confined between two parallel plates, there are depletion interactions between the two large spheres and between one large sphere and the closely placed plate. Obviously, the depletion interactions exerted on one large sphere will be strongly affected by the presence of the closely placed plate or the other large sphere. This prediction is confirmed by the numerical results obtained through the acceptance ratio method (ARM) or density integration method (DIM), i.e., they are strengthened when two large spheres are contacted. Furthermore, it is found that the influences on the depletion forces are also sensitive to the angle of the centers’ connection line between the two large spheres and the confining walls. In addition, the numerical results show that the total depletion force exerted on one large sphere from both the other large sphere and the closely placed plate can be determined through ARM or DIM from the interactions between the two large spheres or between one large sphere and the corresponding closely placed plate.