variable order fractional differential operators in anomalous diffusion modeling abstract the purpose of this paper is to offer a unified discussion of variable order differential operators in anomalous diffusion modeling the characteristics of the new models in contrast to constant order fractional diffusion models change with time space concentration or other independent quantities we introduced a classification of variable order fractional diffusion models based on the possible physical origins which prompt the variable order some potential applications of the variable order fractional diffusion models are also discussed

equivalent continuous and discrete realizations of levy flights a model of one dimensional motion of an inertial particle abstract the paper is devoted to the relationship between the continuous markovian description of levy flights developed previously see e g i a lubashevsky truncated levy flights and generalized cauchy processes eur phys j b 82 2011 189 195 and references therein and their equivalent representation in terms of discrete steps of a wandering particle a certain generalization of continuous time random walks to simplify understanding the key points of the technique to be created our consideration is confined to the one dimensional model for continuous random motion of a particle with inertia its dynamics governed by stochastic self acceleration is described as motion on the phase plane x v comprising the position x and velocity v d x d t of the given particle a notion of random walks inside a certain neighborhood l of the line v 0 the x axis and outside it is developed it enables us to represent a continuous trajectory of particle motion on the plane x v as a collection of the corresponding discrete steps each of these steps matches one complete fragment of the velocity fluctuations originating and terminating at the boundary of l as demonstrated the characteristic length of particle spatial displacement is mainly determined by velocity fluctuations with large amplitude which endows the derived random walks along the x axis with the characteristic properties of levy flights using the developed classification of random trajectories a certain parameter free core stochastic process is constructed its peculiarity is that all the characteristics of levy flights similar to the exponent of the levy scaling law are no more than the parameters of the corresponding transformation from the particle velocity v to the related variable of the core process in this way the previously found validity of the continuous markovian model for all the regimes of levy flights is explained based on the obtained results an efficient single peak approximation is constructed in particular it enables us to calculate the basic characteristics of levy flights using the probabilistic properties of extreme velocity fluctuations and the shape of the most probable trajectory of particle motion within such extreme fluctuations

nonlinear evolution equations associated with a riemann hilbert scattering problem in an earlier paper nonlinear evolution equations associated with a riemann hilbert scattering problem which reduces in an appropriate limit to the zakharov shabat akns scattering problem were considered here we discuss certain necessary constraints associated with the scattering problem and their impact upon the associated evolution equations moreover the direct linearization of the nonlinear evolution equations and an algorithm to construct ann soliton solution are given

on the initial value problem for the modified benjamin ono equation the initial value problem for the modified benjamin ono equation is solved by exploiting the fokas ablowitz inverse scattering transform for a certain complex extension of the benjamin ono equation the asymptotics of the jost functions for small positive values of the spectral parameter are sharpened and lead to a previously overlooked nongeneric case

time dependent schrodinger like equation with nonlocal term we investigate a time dependent schrodinger like equation in presence of a nonlocal term by using the method of variable separation and the green function approach we analyze the green function for different forms of the memory kernel and the nonlocal term results for delta potential energy function are presented distributed order memory kernels are also considered and the asymptotic behaviors of the green function are derived by using tauberian theorem the obtained results for the green function for the considered schrodinger like equation may be transformed to those for the probability distribution function of a diffusion like equation with memory kernel and can be used to explain various anomalous diffusive behaviors

periodic ilw equation with discrete laplacian we study an integro differential equation which generalizes the periodic intermediate long wave ilw equation the kernel of the singular integral involved is an elliptic function written as a second order difference of the weierstrass function using sato s formulation we show the integrability and construct some special solutions an elliptic solution is also obtained we present a conjecture based on a poisson structure that gives an alternative description of this integrable hierarchy we note that this poisson algebra in turn is related to a quantum algebra related to the family of macdonald difference operators

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

codifference as a practical tool to measure interdependence abstract correlation and spectral analysis represent the standard tools to study interdependence in statistical data however for the stochastic processes with heavy tailed distributions such that the variance diverges these tools are inadequate the heavy tailed processes are ubiquitous in nature and finance we here discuss codifference as a convenient measure to study statistical interdependence and we aim to give a short introductory review of its properties by taking different known stochastic processes as generic examples we present explicit formulas for their codifferences we show that for the gaussian processes codifference is equivalent to covariance for processes with finite variance these two measures behave similarly with time for the processes with infinite variance the covariance does not exist however the codifference is relevant we demonstrate the practical importance of the codifference by extracting this function from simulated as well as real data taken from turbulent plasma of fusion device and financial market we conclude that the codifference serves as a convenient practical tool to study interdependence for stochastic processes with both infinite and finite variances as well

fluctuations in multiplicative systems with jumps fluctuation properties of the langevin equation including a multiplicative power law noise and a quadratic potential are discussed the noise has the l evy stable distribution if this distribution is truncated the covariance can be derived in the limit of large time it falls exponentially covariance in the stable case studied for the cauchy distribution exhibits a weakly stretched exponential shape and can be approximated by the simple exponential the dependence of that function on system parameters is determined then we consider a dynamics which involves the above process and obeys the generalized langevin equation the same as for the gaussian case the resulting distributions possess power law tails which fall similarly to those for the driving noise whereas central parts can assume the gaussian shape moreover a process with the covariance 1 t at large time is constructed and the corresponding dynamical equation is solved diffusion properties of systems for both covariances are discussed

a galerkin finite element method for the modified distributed order anomalous sub diffusion equation abstract in this paper we consider a modified time distributed order anomalous sub diffusion equation for the description of processes becoming less anomalous over the course of time first we discretize the integral term using a mid point quadrature rule and transform the time distributed order fractional diffusion equation into a multi term equation then applying the backward difference method in time and galerkin finite element method in space we present a fully discrete scheme to solve the time distributed order fractional diffusion equation moreover the stability and convergence of the proposed method are also discussed numerical examples are given to verify the accuracy and stability of our scheme

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

eigenfunctions and eigenvalues for a scalar riemann hilbert problem associated to inverse scattering a complete set of eigenfunctions is introduced within the riemann hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations in particular we consider the time independent and time dependent schrodinger problems which are related to the kdv and kpi equations possessing solitons and lumps respectively non standard scalar products orthogonality and completeness relations are derived for these problems the complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations we classify two different types of bifurcations of new eigenvalues and analyze their characteristic features one type corresponds to thresholdless generation of solitons in the kdv equation while the other predicts a threshold for generation of lumps in the kpi equation

singularity confinement analysis of integrodifferential equations of benjamin ono type we analyse the integrability of various types of intregrodifferential equations belonging to the benjamin ono family the method used is a combination of two well known integrability detectors the painleve method for continuous systems and the singularity confinement for discrete systems we confirm the results of hietarinta based on the study of multisoliton solutions in particular we show that the third order extension to benjamin ono that he proposed does pass the test and is thus an excellent candidate for integrability

multidimensional nonlinear evolution equations and inverse scattering abstract in this paper we will review some recent work done in the field of integrable nonlinear evolution equations and inverse scattering we will concentrate on the basic underlying areas and refer interested readers to suitable references for complete details specifically background material can be found in various texts on this subject e g 1 by ablowitz and segur more recent references will be given as necessary the outline of the paper is as follows 1 1 introductory remarks 2 2 a discussion of two separate but related issues namely a solving certain nonlinear evolutions equations in infinite space and b inverse scattering these are important problems having many physical applications moreover they are related to each other by what we refer to as the inverse scattering transform ist 3 3 at the end of the paper we will make some remarks on the possibility of solving nonlinear evolution equations in high dimensions i e equations with more than two spatial and one time variable by using the ist method as we now understand it

an application of the inverse scattering transform to the modified intermediate long wave equation the modified intermediate long wave milw equation is a 1 1 dimensional nonlinear singular integro differential equation that possesses soliton solutions in an appropriate limit the milw equation reduces to the well known modified korteweg de vries equation in this paper we solve the initial value problem for the milw equation through a suitable implementation of the inverse scattering transform and use of the miura type transformation that maps solutions of the milw equation into solutions of a complexified version of the standard intermediate long wave ilw equation the initial value used for the milw equation is assumed to be real valued sufficiently smooth and decaying to zero as the absolute value of the spatial variable approaches large values an interesting feature of the procedure we develop is that soliton solutions for the ilw and milw equations can be derived by appropriate specializations of a master set of equations

on the initial value problem for a class of nonlinear integral evolution equations including the sine hilbert equation a method for solving a class of nonlinear singular integral evolution equations for decaying initial values on the line is presented the underlying scattering problem is a matrix riemann hilbert problem scattering analysis shows that the spectrum is purely discrete an application is to the so called sine hilbert equation h t c sin where c is a constant and h denotes the hilbert transform

distribution free estimation of local growth rates around interval censored anchoring events biological processes are usually defined on timelines that are anchored by specific events for example cancer growth is typically measured by the change in tumor size from the time of oncogenesis in the absence of such anchoring events longitudinal assessments of the outcome lose their temporal reference in this paper we considered the estimation of local change rates in the outcomes when the anchoring events are interval censored viewing the subject specific anchoring event times as random variables from an unspecified distribution we proposed a distribution free estimation method for the local growth rates around the unobserved anchoring events we expressed the rate parameters as stochastic functionals of the anchoring time distribution and showed that under mild regularity conditions consistent and asymptotically normal estimates of the rate parameters could be achieved with a n convergence rate we conducted a carefully designed simulation study to evaluate the finite sample performance of the method to motivate and illustrate the use of the proposed method we estimated the skeletal growth rates of male and female adolescents before and after the unobserved pubertal growth spurt pgs times

the scattering transform for the benjamin ono equation in the small dispersion limit abstract using exact formulae for the scattering data of the benjamin ono equation valid for general rational potentials recently obtained in miller and wetzel 17 we rigorously analyze the scattering data in the small dispersion limit in particular we deduce precise asymptotic formulae for the reflection coefficient the location of the eigenvalues and their density and the asymptotic dependence of the phase constant associated with each eigenvalue on the eigenvalue itself our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering and they also provide new details not previously reported in the literature

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

relaxation and diffusion models with non singular kernels abstract anomalous relaxation and diffusion processes have been widely quantified by fractional derivative models where the definition of the fractional order derivative remains a historical debate due to its limitation in describing different kinds of non exponential decays e g stretched exponential decay meanwhile many efforts by mathematicians and engineers have been made to overcome the singularity of power function kernel in its definition this study first explores physical properties of relaxation and diffusion models where the temporal derivative was defined recently using an exponential kernel analytical analysis shows that the caputo type derivative model with an exponential kernel cannot characterize non exponential dynamics well documented in anomalous relaxation and diffusion a legitimate extension of the previous derivative is then proposed by replacing the exponential kernel with a stretched exponential kernel numerical tests show that the caputo type derivative model with the stretched exponential kernel can describe a much wider range of anomalous diffusion than the exponential kernel implying the potential applicability of the new derivative in quantifying real world anomalous relaxation and diffusion processes

backlund transformation conservation laws and inverse scattering transform of a model integrodifferential equation for water waves the backlund transformation bt an infinite number of conservation laws and the inverse scattering transform ist of a model integrodifferential equation for water waves in fluids of finite depth y matsuno j math phys 29 49 1989 are constructed by employing the bilinear transformation method the model equation is also shown to pass the painleve test these facts prove the complete integrability of the equation both the deep and shallow water limits of various results thus obtained are then investigated in detail in addition a new method to evaluate conserved quantities for pure n soliton is developed by utilizing actively the time part of the bt it is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the benjamin ono and the korteweg de vries equations the most important problem left open in this paper is to solve various ist equations

asymptotically optimal estimation of smooth functionals for interval censoring case 2 for a version of the interval censoring model case 2 in which the observation intervals are allowed to be arbitrarily small we consider estimation of functionals that are differentiable along hellinger differentiable paths the asymptotic information lower bound for such functionals can be represented as the squared l2 norm of the canonical gradient in the observation space this canonical gradient has an implicit expression as a solution of an integral equation that does not belong to one of the standard types we study an extended version of the integral equation that can also be used for discrete distribution functions like the nonparametric maximum likelihood estimator npmle and derive the asymptotic normality and efficiency of the npmle from properties of the solutions of the integral equations

langevin formulation of a subdiffusive continuous time random walk in physical time systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power law scaling of the mean square displacement one of the most common models to describe such subdiffusive dynamics is the continuous time random walk ctrw stochastic trajectories of a ctrw can be described in terms of the subordination of a normal diffusive process by an inverse levy stable process here we propose an equivalent langevin formulation of a force free ctrw without subordination by introducing a different type of non gaussian noise we are able to express the ctrw dynamics in terms of a single langevin equation in physical time with additive noise we derive the full multipoint statistics of this noise and compare it with the scaled brownian motion sbm an alternative stochastic model describing subdiffusive dynamics interestingly these two noises are identical up to the second order correlation functions but different in the higher order statistics we extend our formalism to general waiting time distributions and force fields and compare our results with those of the sbm in the presence of external forces our proposed noise generates a different class of stochastic processes resembling a ctrw but with forces acting at all times

generalized fractional diffusion equations for accelerating subdiffusion and truncated levy flights fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time for processes lacking such scaling the corresponding description may be given by diffusion equations with fractional derivatives of distributed order such equations were introduced in a v chechkin r gorenflo and i sokolov phys rev e 66 046129 2002 for the description of the processes getting more anomalous in the course of time decelerating subdiffusion and accelerating superdiffusion here we discuss the properties of diffusion equations with fractional derivatives of the distributed order for the description of anomalous relaxation and diffusion phenomena getting less anomalous in the course of time which we call respectively accelerating subdiffusion and decelerating superdiffusion for the former process by taking a relatively simple particular example with two fixed anomalous diffusion exponents we show that the proposed equation effectively describes the subdiffusion phenomenon with diffusion exponent varying in time for the latter process we demonstrate by a particular example how the power law truncated levy stable distribution evolves in time to the distribution with power law asymptotics and gaussian shape in the central part the special case of two different orders is characteristic for the general situation in which the extreme orders dominate the asymptotics

generalized time dependent schrodinger equation in two dimensions under constraints we investigate a generalized two dimensional time dependent schrodinger equation on a comb with a memory kernel a dirac delta term is introduced in the schrodinger equation so that the quantum motion along the x direction is constrained at y 0 the wave function is analyzed by using green s function approach for several forms of the memory kernel which are of particular interest closed form solutions for the cases of dirac delta and power law memory kernels in terms of fox h function as well as for a distributed order memory kernel are obtained further a nonlocal term is also introduced and investigated analytically it is shown that the solution for such a case can be represented in terms of infinite series in fox h functions green s functions for each of the considered cases are analyzed and plotted for the most representative ones anomalous diffusion signatures are evident from the presence of the power law tails the normalized green s functions obtained in this work are of broader interest

exact solutions of triple order time fractional differential equations for anomalous relaxation and diffusion i the accelerating case a b s t r a c t in recent years the interest around the study of anomalous relaxation and diffusion processes is increased due to their importance in several natural phenomena moreover a further generalization has been developed by introducing time fractional differentiation of distributed order which ranges between 0 and 1 we refer to accelerating processes when the driving power law has a changing in time exponent whose modulus tends from less than 1 to 1 and to decelerating processes when such an exponent modulus decreases in time moving away from the linear behaviour accelerating processes are modelled by a time fractional derivative in the riemann liouville sense while decelerating processes by a time fractional derivative in the caputo sense here the focus is on the accelerating case while the decelerating one is considered in the companion paper after a short reminder about the derivation of the fundamental solution for a general distribution of time derivative orders we consider in detail the triple order case for both accelerating relaxation and accelerating diffusion processes and the exact results are derived in terms of an infinite series of h functions the method adopted is new and it makes use of certain properties of the generalized mittag leffler function and the h function moreover it provides an elegant generalization of the method introduced by langlands 2006 t a m langlands physica a 367 2006 136 to study the double order case of accelerating diffusion processes

fokker planck type equations associated with fractional brownian motion controlled by infinitely divisible processes abstract in this paper we study the anomalous diffusion process driven by fractional brownian motion delayed by general infinitely divisible subordinator we show the analyzed process is the stochastic representation of the fokker planck type equation that describes the probability density function of an introduced model moreover we study main characteristics of the examined process the first two moments that allow us in special cases for classification of it as a system with accelerating subdiffusion property

fractional differential equations and related exact mechanical models the aim of the paper is the description of fractional order differential equations in terms of exact mechanical models this result will be archived in the paper for the case of linear multiphase fractional hereditariness involving linear combinations of power laws in relaxation creep functions the mechanical model corresponding to fractional order differential equations is the extension of a recently introduced exact mechanical representation di paola and zingales 2012 33 and di paola et al 2012 34 of fractional order integrals and derivatives some numerical applications have been reported in the paper to assess the capabilities of the model in terms of a peculiar arrangement of linear springs and dashpots

levy mixing related to distributed order calculus subordinators and slow diffusions abstract the study of distributed order calculus usually concerns fractional derivatives of the form 0 1 u m d for some measure m eventually a probability measure in this paper an approach based on levy mixing is proposed non decreasing levy processes associated with levy triplets of the form a y b y d s y are considered and the parameter y is randomized by means of a probability measure the related subordinators are studied from different points of view some distributional properties are obtained and the interplay with inverse local times of markov processes is explored distributed order integro differential operators are introduced and adopted in order to write explicitly the governing equations of such processes an application to slow diffusions delayed brownian motion is discussed

stability of nonlinear periodic internal waves in a deep stratified fluid stability of the nonlinear periodic solutions of the benjamin ono equation with respect to one dimensional perturbations and taking into account transverse modulations is analytically studied the full set of eigenfrequencies and of small perturbations is found in the one dimensional case the dispersion equation for long wave oblique perturbations is obtained these perturbations are shown to be unstable under either sign of the dispersion of the linear dispersion relation the instability limits and the maximum growth rates are determined in the extreme cases of strongly and weakly nonlinear waves

explicit form of the first passage time density for accelerating subdiffusion the first passage time fpt statistics play a central role in different fields apart of description of the underlying system state and its evolution it is essential to investigate the time at which this state reaches a certain area for the first time in this paper we study the fpt problem for recently developed models of an accelerating subdiffusion which extends incorporating various short and long time scalings of the mean square displacement msd popular models of subdiffusion based on fractional order differential equations we derive the laplace transform of the fpt density function in the case of constant bias for a force depending linearly on the position we provide a full analytical formula for the fpt density in terms of generalized mittag leffler functions and the hermite polynomials we also provide a numerical evidence that fpt properties are strictly connected with msd scalings of the accelerating subdiffusion

moments for tempered fractional advection diffusion equations this paper develops moment formulas for exponentially tempered fractional advection diffusion equations tfades that transition from anomalous to asymptotic diffusion limits over time exact analytical expressions or series representations for spatial moments up to the fourth order are derived by integral transform or asymptotic expansion approach a fully lagrangian solver cross verified by an implicit eulerian approach is also developed to calculate numerically the complete evolution of moments for the tfades with complex initial and boundary conditions moment analysis identifies the diffusion equation that attracts the tempered anomalous diffusion in the long time limit fitting of moments measured at two end members of alluvial systems checks the applicability of moment analysis in understanding real diffusion

fractional differential models for anomalous diffusion abstract in this study we investigate three kinds of fractional differential models distributed order model variable order model and random order model for characterization of anomalous diffusion the characteristics physical advantages and potential applications of each model are highlighted the numerical simulations also validate our analytical and comparison results furthermore a generalized distributed variable order model and a more generalized distributed variable random order model are proposed to combine the advantages of distributed order model variable order model and random order model

distributed order diffusion equations and multifractality models and solutions we study distributed order time fractional diffusion equations characterized by multifractal memory kernels in contrast to the simple power law kernel of common time fractional diffusion equations based on the physical approach to anomalous diffusion provided by the seminal scher montroll weiss continuous time random walk we analyze both natural and modified form distributed order time fractional diffusion equations and compare the two approaches the mean squared displacement is obtained and its limiting behavior analyzed we derive the connection between the wiener process described by the conventional langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time a detailed analysis of the multifractal properties of distributed order diffusion equations is provided

generalized intermediate long wave hierarchy in zero curvature representation with noncommutative spectral parameter the simplest generalization of the intermediate long wave hierarchy ilw is considered to show how to extend the zakharov shabat dressing method to nonlocal i e integro partial differential equations the purpose is to give a procedure of constructing the zero curvature representation of this class of equations this result obtains by combining the drinfeld sokolov formalism together with the introduction of an operator valued spectral parameter namely a spectral parameter that does not commute with the space variable x this extension provides a connection between the ilwk hierarchy and the saveliev vershik continuum graded lie algebras in the case of ilw2 the fairlie zachos sinh algebra was found

robust nonparametric estimation of monotone regression functions with interval censored observations nonparametric estimation of monotone regression functions is a classical problem of practical importance robust estimation of monotone regression functions in situations involving interval censored data is a challenging yet unresolved problem herein we propose a nonparametric estimation method based on the principle of isotonic regression using empirical process theory we show that the proposed estimator is asymptotically consistent under a specific metric we further conduct a simulation study to evaluate the performance of the estimator in finite sample situations as an illustration we use the proposed method to estimate the mean body weight functions in a group of adolescents after they reach pubertal growth spurt

bifurcations of new eigenvalues for the benjamin ono equation a criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the benjamin ono bo equation this bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the bo equation the asymptotic approximation of an exponentially small new eigenvalue is derived the method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions explicit expressions are obtained for the soliton potentials

new integrable nonlinear integrodifferential equations and related solvable finite dimensional dynamical systems a new integrable nonlinear integrodifferential equation nide is proposed this equation may be interpreted as a model equation for deep water waves the n periodic and n soliton solutions for the equation are constructed by means of the bilinear transformation method these solutions have the same structure as that for the benjamin ono equation which describes internal waves in stratified fluids of great depth furthermore it is shown that the motion of the positions of the poles of solutions is related to certain solvable finite dimensional dynamical systems described by first order nonlinear ordinary differential equations the discussion is also made on a more general nide that may be interpreted as a model equation describing nonlinear waves in fluids of finite depth

on the zero dispersion limit of the benjamin ono cauchy problem for positive initial data we study the cauchy initial value problem for the benjamin ono equation in the zero dispersion limit and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by lax and levermore to analyze the corresponding limit for the korteweg de vries equation 2010 wiley periodicals inc

numerical method for solving diffusion wave phenomena we find solutions for the diffusion wave problem in 1d with n term time fractional derivatives whose orders belong to the intervals 0 1 1 2 and 0 2 respectively using the method of the approximation of the convolution by laguerre polynomials in the space of tempered distributions this method transfers the diffusion wave problem into the corresponding infinite system of linear algebraic equations through the coefficients which are uniquely solvable under some relations between the coefficients with index zero the method is applicable for nonlinear problems too

continuous time multidimensional markovian description of levy walks the paper presents a multidimensional model for nonlinear markovian random walks that generalizes the one we developed previously i lubashevsky r friedrich and a heuer phys rev e 79 011110 2009 in order to describe the levy type stochastic processes in terms of continuous trajectories of walker motion this approach may open a way to treat levy flights or levy random walks in inhomogeneous media or systems with boundaries in the future the proposed model assumes the velocity of a wandering particle to be affected by a linear friction and a nonlinear langevin force whose intensity is proportional to the magnitude of the velocity for its large values based on the singular perturbation technique the corresponding fokker planck equation is analyzed and the relationship between the system parameters and the levy exponent is found following actually the previous paper we demonstrate also that anomalously long displacements of the wandering particle are caused by extremely large fluctuations in the particle velocity whose duration is determined by the system parameters rather than the duration of the observation interval in this way we overcome the problem of ascribing to levy random walk non markov properties