on equations of kp type we discuss the local cauchy problem for the generalised kadomtsev petviashvili equation namely in the periodic and nonperiodic settings

functional topology of evolving urban drainage networks we investigated the scaling and topology of engineered urban drainage networks udns in two cities and further examined udn evolution over decades udn scaling was analyzed using two power law scaling characteristics widely employed for river networks 1 hack s law of length l area a l ah and 2 exceedance probability distribution of upstream contributing area p a a for the smallest udns 2 km2 length area scales linearly h 1 but power law scaling h 0 6 emerges as the udns grow while p a plots for river networks are abruptly truncated those for udns display exponential tempering p a a exp c the tempering parameter c decreases as the udns grow implying that the distribution evolves in time to resemble those for river networks however the power law exponent for large udns tends to be greater than the range reported for river networks differences in generative processes and engineering design constraints contribute to observed differences in the evolution of udns and river networks including subnet heterogeneity and nonrandom branching

power law behavior in geometric characteristics of full binary trees natural river networks exhibit regular scaling laws in their topological organization here we investigate whether these scaling laws are unique characteristics of river networks or can be applicable to general binary tree networks we generate numerous binary trees ranging from purely ordered trees to completely random trees for each generated binary tree we analyze whether the tree exhibits any scaling property found in river networks i e the power laws in the size distribution the length distribution the distance load relationship and the power spectrum of width function we found that partially random trees generated on the basis of two distinct types of deterministic trees i e deterministic critical and supercritical trees show contrasting characteristics partially random trees generated on the basis of deterministic critical trees exhibit all power law characteristics investigated in this study with their fitted exponents close to the values observed in natural river networks over a wide range of random degree on the other hand partially random trees generated on the basis of deterministic supercritical trees rarely follow scaling laws of river networks

on strong anomalous diffusion superdiffusive behavior i e x2 t t2 with 1 2 is in general not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusive regime i e q const 1 2 occurring in random shear flows strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this is due to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling p x t t f x t of the probability density this implies that the effective equation at large scale and long time for p x t obeys neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

on the fractal interpretation of the mainstream length drainage area relationship the exponent of the mainstream length drainage area relationship is interpreted as a fractal quantity superseding the previous allometric interpretation the fractal interpretation is based on the assumption that cartographic generalization is applied evenly to all map scales twenty three drainage basins of the eaton river quebec canada were delineated from topographic maps at three different scales 1 20 000 1 50 000 and 1 125 000 the exponent of the length area relation is much lower 0 546 at the largest scale than at the smallest scales 0 65 and its value corresponds to that obtained from a richardson analysis of 10 interior stream segments at the 1 20 000 map scale the exponent is entirely fractal the larger exponent values obtained at the smallest scales exceed the fractal value and incorporate an allometric component this component is not functional however and it merely reflects the generalization process of cartographic abstraction of stream heads as scale is reduced the fractal dimension of streams should not be inferred from the exponent of the length area relation because its value is likely to be scale dependent

periodic solutions of kadomtsev petviashvili on etablit l existence de solutions globales pour l equation non lineaire de kadomtsev petviashvili u t uu x u xxx d 1 u yy periodiques en x et en y et d 1 denote la primitive en x

scaling of swimming sequences in copepod behavior data analysis and simulation abstract we consider symbolic sequences of copepod behavior classified as break and slow swimming states successive residence times for each state have been recorded for 327 symbolic times series obtained from 52 different trajectories we performed a symbolic dynamics analysis of the succession of states and showed that the probability densities of the duration of each state obey power laws for large times we analyzed the transition between states and showed as expected the non markovian properties of this dynamics we further simulate a copepod trajectory using an alternate renewal process for the generation of the sequence time series and using a brownian motion for the copepod motion during swimming periods

fractional langevin equations of distributed order distributed order fractional langevin like equations are introduced and applied to describe anomalous diffusion without unique diffusion or scaling exponent it is shown that these fractional langevin equations of distributed order can be used to model the kinetics of retarding subdiffusion whose scaling exponent decreases with time and the strongly anomalous ultraslow diffusion with mean square displacement which varies asymptotically as a power of logarithm of time

multifractal description of wind power fluctuations using arbitrary order hilbert spectral analysis a b s t r a c t the objectives are to study and model the aggregate wind power fluctuations dynamics in the multifractal framework we present here the analysis of aggregate power output sampled at 1 hz during three years we decompose the data into several intrinsic mode functions imfs using empirical mode decomposition emd we use a new approach ar bitrary order hilbert spectral analysis a combination of the emd approach with hilbert spectral analysis or hilbert huang transform and the classical structure function anal ysis to extract the scaling exponents or multifractal spectrum q this function provides a full characterization of a process at all intensities and all scales the application of both methods i e structure function and arbitrary order hilbert spectral analyses gives similar results indicating that the aggregate power output from a wind farm possesses intermit tent and multifractal properties in order to check this result we generate stochastic simu lations of a multifractal random walk mrw using a log normal stochastic equation we show that the simulation results are fully compatible with the experimental results

on hack s law hack s law is reviewed emphasizing its implications for the elongation of river basins as well as its connections with their fractal characteristics the relation between hack s law and the internal structure of river basins is investigated experimentally through digital elevation models it is found that hack s exponent elongation and some relevant fractal characters are closely related the self affine character of basin boundaries is shown to be connected to the power law decay of the probability of total contributing areas at any link and to hack s law an explanation for hack s law is derived from scaling arguments from the results we suggest that a statistical framework referring to the scaling invariance of the entire basin structure should be used in the interpretation of hack s law

strongly and weakly self similar diffusion many dispersive processes have moments of displacements with large t behavior x p t p the study of p as a function of p provides a more complete characterization of the process than does the single number 2 also at long times the core of the concentration relaxes to a self similar profile while the large x tails consisting of particles which have experienced exceptional displacements are not self similar depending on the particular process the effect of the tails can be negligible and then p is a linear function of p strong self similarity but if the tails are important then p is a non trivial function of p weak self similarity in the weakly self similar case the low moments are determined by the self similar core while the high moments are determined by the non self similar tails the popular exponent 2 may be determined by either the core or the tails as representatives of a large class of dispersive processes for which p is a piecewise linear function of p we study two systems a stochastic model the generalized telegraph model and a deterministic area preserving map the

multifractal random walk in copepod behavior abstract a 3d copepod trajectory is recorded in the laboratory using two digital cameras the copepod undergoes a very structured type of trajectory with successive moves displaying intermittent amplitudes we perform a statistical analysis of this 3d trajectory using statistical tools developed in the field of turbulence and anomalous diffusion in natural sciences we show that the walk belongs to multifractal random walks characterized by a nonlinear moment scaling function for the distance versus time to our knowledge this is the first experimental study of multifractal anomalous diffusion in natural sciences we then propose a new type of stochastic process reproducing these multifractal scaling properties this can be directly used for stochastic numerical simulations and is thus of important potential applications in the field of animal movement study and more generally of anomalous diffusion studies

numerical study of strong anomalous diffusion the superdiffusion behavior i e x2 t t2 with 1 2 in general is not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusion regime i e q const 1 2 as in random shear flows the strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this corresponds to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling of the probability distribution i e p x t t f x t this implies that the effective equation at large scale and long time for p x t can obey neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

delay driven vegetation patterns of a plankton system on a network abstract a two component delayed reaction diffusion system is introduced to a complex network to describe the vegetation patterns in plankton system the positive equilibrium is shown by the linear stability analysis to be asymptotically stable in the absence of time delay but when the time delay increases beyond a threshold it loses its stability via the hopf bifurcation the stability and direction of the hopf bifurcation is investigated with the method of center manifold theory our result reveals that the stability of hopf bifurcation leads to the emergence of vegetation patterns numerical calculations are performed to confirm our theoretical results

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

turing pattern dynamics in an activator inhibitor system with superdiffusion the fractional operator is introduced to an activator inhibitor system to describe species anomalous superdiffusion the effects of the superdiffusive exponent on pattern formation and pattern selection are studied our linear stability analysis shows that the wave number of the turing pattern increases with the superdiffusive exponent a weakly nonlinear analysis yields a system of amplitude equations and the analysis of these amplitude equations predicts parameter regimes where hexagons stripes and their coexistence are expected numerical simulations of the activator inhibitor model near the stability boundaries confirm our analytical results since diffusion rate manifests in both diffusion constant and diffusion exponent we numerically explore their interactions on the emergence of turing patterns when the activator and inhibitor have different superdiffusive exponents we find that the critical ratio of the diffusion rate of the inhibitor to the activator required for the formation of the turing pattern increases monotonically with the superdiffusive exponent we conclude that small ratio than unity of anomalous diffusion exponent between the inhibitor and activator is more likely to promote the emergence of the turing pattern relative to the normal diffusion

a uniqueness theorem for the two dimensional generalized korteweg de vries equation abstract using elementary techniques we prove that the solution which is periodic in both x and y to the two dimensional generalized kdv equation is unique a theorem of continuous dependence of the solutions on the initial data is established some estimates of the solutions are also included

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

mimicking a turbulent signal sequential multiaffine processes an efficient method for the construction of a multiaffine process with prescribed scaling exponents is presented at variance with the previous proposals this method is sequential and therefore it is the natural candidate in numerical computations involving synthetic turbulence the application to the realization of a realistic turbulent like signal is discussed in detail the method represents a first step towards the realization of a realistic spatio temporal turbulent field

on a new hierarchy of symmetries for the kadomtsev petviashvili equation abstract we present a new hierarchy of symmetries for the kadomtsev petviashvili equation these new symmetries depend on the space and time variables explicitly together with the previously known classical symmetries they constitute an infinite dimensional lie algebra

multifractality of cerebral blood flow abstract scale invariance the property relating time series across multiple scales has provided a new perspective of physiological phenomena and their underlying control systems the traditional signal plus noise paradigm of the engineer was first replaced with a model in which biological time series have a fractal structure in time fractal physiology oxford university press oxford 1994 this new paradigm was subsequently shown to be overly restrictive when certain physiological signals were found to be characterized by more than one scaling parameter and therefore to belong to a class of more complex processes known as multifractals fractals plenum press new york 1988 here we demonstrate that in addition to heart rate nature 399 1999 461 and human gait phys rev e submitted for publication the nonlinear control system for cerebral blood flow cbf phys rev lett submitted for publication phys rev e 59 1999 3492 is multifractal we also find that this multifractality is greatly reduced for subjects with serious migraine and we present a simple model for the underlying control process to describe this effect