transient aging in fractional brownian and langevin equation motion stochastic processes driven by stationary fractional gaussian noise that is fractional brownian motion and fractional langevin equation motion are usually considered to be ergodic in the sense that after an algebraic relaxation time and ensemble averages of physical observables coincide recently it was demonstrated that fractional brownian motion and fractional langevin equation motion under external confinement are transiently nonergodic char22 time and ensemble averages behave differently char22 from the moment when the particle starts to sense the confinement here we show that these processes also exhibit transient aging that is physical observables such as the time averaged mean squared displacement depend on the time lag between the initiation of the system at time t 0 and the start of the measurement at the aging time t a in particular it turns out that for fractional langevin equation motion the aging dependence on t a is different between the cases of free and confined motion we obtain explicit analytical expressions for the aged moments of the particle position as well as the time averaged mean squared displacement and present a numerical analysis of this transient aging phenomenon

solitons in a nonlinear model medium the periodic stationary solutions of a model nonlinear evolution equation simulating the propagation of short wave perturbations in a relaxing medium are studied solutions expressed by a multiple valued function are shown to exist a method for determining the nonlinear interaction between solitary waves is suggested an example of a collision of solitons is given

accelerated growth in outgoing links in evolving networks deterministic versus stochastic picture in several real world networks such as the internet world wide web etc the number of links grow in time in a nonlinear fashion we consider growing networks in which the number of outgoing links is a nonlinear function of time but new links between older nodes are forbidden the attachments are made using a preferential attachment scheme in the deterministic picture the number of outgoing links m t at any time t is taken as n t ensuremath theta where n t is the number of nodes present at that time the continuum theory predicts a power law decay of the degree distribution p k ensuremath propto k ensuremath 1 ensuremath 2 1 ensuremath ensuremath theta while the degree of the node introduced at time t i is given by k t i t t i ensuremath theta phantom rule 0 2em 0ex t t i 1 ensuremath theta 2 when the network is evolved till time t numerical results show a growth in the degree distribution for small k values at any nonzero ensuremath theta in the stochastic picture m t is a random variable as long as m t is independent of time the network shows a behavior similar to the barab asi albert ba model different results are obtained when m t is time dependent e g when m t follows a distribution p m ensuremath propto m ensuremath ensuremath lambda the behavior of p k changes significantly as ensuremath lambda is varied for ensuremath lambda g3 the network has a scale free distribution belonging to the ba class as predicted by the mean field theory for smaller values of ensuremath lambda it shows different behavior characteristic features of the clustering coefficients in both models have also been discussed

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

bimodality of the interspike interval distributions for subordinated diffusion models of integrate and fire neurons abstract a subordinated langevin process with a random operational time in the form of an inverse strictly increasing levy type subordinator is considered as a generalization of the conventional perfect and leaky integrate and fire neuron models the parent process is given by standard brownian motion the effect of the random activity of synaptic inputs which arises from other neurons forming local and distant networks is modeled via a levy exponent of the subordinator using a first passage time formulation in an external force field we find exact expressions for the laplace transform of the output interspike interval isi density more detailed analysis is presented on the properties of the isi distribution in the case of the levy exponent which corresponds to the truncated double order time fractional diffusion equation for the probability density of the membrane potential particularly it is shown that at some parameter regimes the isi density exhibits a bimodal structure moreover it is demonstrated that the isis regularity is maximized at an intermediate value of the mean input current

synchronizability in complex ad hoc dynamical networks with accelerated growth recent research works have been pursued in connection with network synchronizability under various constraints for different topological structures and evolving mechanisms however the fundamental question of how the synchronizability of the networks relates to accelerated growth and ad hoc property in the evolving processes remains underexplored here we study the ad hoc dynamical models with accelerated growth where the total number of edges increases faster than linearly with network size by adopting three attachment mechanisms random rewired and preferential attachment we investigate the second largest eigenvalues and the ratios of the extremal eigenvalues of coupled matrices in the accelerated models with different evolving parameters for the new models we demonstrate the robustness and fragility of synchronization against random and specific attacks by numerical simulations we find that accelerated growth represents a convenient tool for improving the synchronizability of an evolving network furthermore we show that not only network synchronization but also topological periodicity has robustness and fragility against random failures and specifical removal of the nodes respectively in particular when ad hoc property is suggested in the evolving networks we find that the deletion of nodes is easier to change network synchronizability and robustness compared to the addition of nodes

relaxation time distributions for an anomalously diffusing particle abstract as well known the generalized langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle here we focus attention on some new aspects of the dynamics successively considering the memory kernel the particle s mean velocity and the scattering function all these quantities are studied from a unique angle namely the discussion of the possible existence of a distribution of relaxation times characterizing their time decay although a very popular concept a relaxation time distribution cannot be associated with any time decreasing quantity from a mathematical point of view the decay has to be described by a completely monotonic function technically we use a memory kernel decaying as a mittag leffler function the mittag leffler functions interpolate between stretched or compressed exponential behaviour at short times and inverse power law behaviour at large times we show that in the case of a subdiffusive motion relaxation time distributions can be defined for the memory kernel and for the scattering function but not for the particle s mean velocity the situation is opposite in the superdiffusive case

exact solitary and periodic wave solutions of the k 2 2 equation defocusing branch an auxiliary elliptic equation method is presented for constructing exact solitary and periodic travelling wave solutions of the k 2 2 equation defocusing branch some known results in the literature are recovered more efficiently and some new exact travelling wave solutions are obtained also new stationary wave solutions are obtained

compacton peakon cuspons loop solutions and smooth solitons for the generalized kp mew equation in this paper we investigate all possible single traveling solitary wave solutions of the generalized kp mew equation u t u 2 x u 2 x x t x u y y 0 under boundary condition lim u a regular compacton solution of the generalized kp mew equation correspond to the case of a 0 in the case of a 0 we find new exact soliton solutions including loop solution cusped soliton peaked soliton and smooth soliton solutions the parametric conditions of existence of the loop solution cusped soliton peaked soliton smooth soliton and compacton solutions are given by using the phase portrait analytical technique mathematical analysis and numerical simulations are provided for these soliton solutions of the generalized kp mew equation we show some graphs to explain these solutions

single peak solitary wave solutions for the osmosis k 2 2 equation under inhomogeneous boundary condition abstract the qualitative theory of differential equations is applied to the k 2 2 equation with osmosis dispersion smooth peaked and cusped solitary wave solutions of the osmosis k 2 2 equation under inhomogeneous boundary condition are obtained the parametric conditions of existence of the smooth peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique asymptotic analysis and numerical simulations are provided for smooth peaked and cusped solitary wave solutions of the osmosis k 2 2 equation

correlation functions for the fractional generalized langevin equation in the presence of internal and external noise we study generalized fractional langevin equations in the presence of a harmonic potential general expressions for the mean velocity and particle displacement the mean squared displacement position and velocity correlation functions as well as normalized displacement correlation function are derived we report exact results for the cases of internal and external friction that is when the driving noise is either internal and thus the fluctuation dissipation relation is fulfilled or when the noise is external the asymptotic behavior of the generalized stochastic oscillator is investigated and the case of high viscous damping overdamped limit is considered additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed in addition the cases of a constant external force and the force free case are obtained the validity of the generalized einstein relation for this process is discussed the considered fractional generalized langevin equation may be used to model anomalous diffusive processes including single file type diffusion

single peak solitary wave solutions for the ch kp 2 1 equation under boundary condition abstract the qualitative theory of differential equations is applied to the ch kp 2 1 equation u t 2 u x u 2 x x t u u x x u y y 0 our procedure shows that the ch kp 2 1 equation either has the regular peakon soliton cuspon soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x u a 0 or possesses compacton solutions only when lim x u a 0 in particular mathematical analysis and numerical simulations of the ch kp 2 1 equation are provided for those peakon cuspon compacton loop soliton and smooth soliton solutions

aging effects in free quantum brownian motion the two time correlation function cxx t t 0 of the displacement x t x t0 of a free quantum brownian particle with respect to its position at a given time t0 is calculated analytically in the framework of the caldeira and leggett ohmic dissipation model as a result at any temperature t cxx t t 0 exhibits aging i e it depends explicitly on both times t and t 0 and not only on the time dierence t t 0 even in the limit of large age t 0 t06t 0 6t in contrast with a dynamic variable in equilibrium such as the particle velocity the equilibrium quantum uctuation dissipation theorem qfdt has to be modied in order to relate the response function xx t t 0 t ocxx t t 0 since this latter quantity takes into account even those uctuations of the displacement which take place during the waiting time tw t 0 t0 we describe the deviation from qfdt in terms of an eective inverse temperature e tw the behaviour of this quantity as a function of for given values of t and tw is analysed in the classical limit it is shown that e tw d d d tw where d t denotes the time dependent diusion coecient c 2000 elsevier science b v all rights reserved

scale free property of complex ad hoc networks with accelerated growth a complex ad hoc network model with accelerated growth is proposed in this paper in the evolving process of the model at each time step with the addition probability c1c1 a new vertex with m t m t edges is added into the model which new adding edges are connected with the old vertices according to the preferential attachment rule and with the deletion probability c2 c1 c2 c1 a random chosen vertex is deleted and its associated edges are disappeared simultaneously the accelerated growth means that at each time step the number of new adding edges is the increasing function of the time tt we consider mainly m t t m t t where 0 10 1 to analyze the degree distribution of the model using the continuous approach we prove theoretically and numerically that the model follows the scale free degree distribution and generates both the barabasi and albert ba model and the model introduced by sarshar and roychowdhury sr in addition we present the appropriate numerical range of c1c1 c2c2 and according to theoretical analyzes and numerical simulation the model takes on a certain universality to complex evolving network and has the guiding significance to the application of the peer to peer networks and complex ad hoc networks

new solitary wave solutions with compact support for the kdv like k m n equations with fully nonlinear dispersion in this paper a new method different from the adomian decomposition method namely the extend sine cosine method is proposed to investigate the compacton solutions of the kdv like equations with fully nonlinear dispersion k m n equations u t u m x u n x x x 0 the new exact solitary wave solutions with compact support of the equations are found by our new method the two special cases k 2 2 and k 3 3 are chosen to illustrate the concrete scheme of our approach in k m n equations an entirely new general formulas for the solutions of k m n equations for all types where m n are established our results include not only some known results in literature as special cases but also some new exact special solutions the method presented by this paper is suitable for studying compacton solutions of some other equations

fokker planck description for a linear delayed langevin equation with additive gaussian noise we construct an equivalent probability description of linear multi delay langevin equations subject to additive gaussian white noise by exploiting the time convolutionless transform and a time variable transformation we are able to write a fokker planck equation fpe for the 1 time and for the 2 time probability distributions valid irrespective of the regime of stability of the langevin equations we solve exactly the derived fpes and analyze the aging dynamics by studying analytically the conditional probability distribution we discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non markov process as both preparation and observation times have bearing on its dynamics as our analytic procedure can also be applied to linear langevin equations with memory kernels we compare the non markov dynamics of a one delay system with that of a generalized langevin equation with an exponential as well as a power law memory application to a generalization of the green kubo formula is also presented

single peak solitary wave solutions for the generalized kp mew 2 2 equation under boundary condition the qualitative theory of differential equations is applied to the kp mew 2 2 equation q t q 2 x q 2 x x t x q y y 0 our procedure shows that the kp mew 2 2 equation either has the regular peakon soliton cuspon soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x q x a 0 or possesses compacton solutions only when lim x q x a 0 in particular we present mathematical analysis and numerical simulations are provided for those peakon cuspon compacton loop soliton and smooth soliton solutions

classification of the travelling waves in the nonlinear dispersive kdv equation abstract in this paper we classify the travelling wave solutions to the nonlinear dispersive kdv equation called k 2 2 equation the parameter region is specified and the parameter dependence of its solitary waves is described besides the previously known compacton solutions the equation is shown to admit more new solutions such as cuspons peakons loopons stumpons and fractal like waves furthermore by the qualitative results we give some new explicit travelling wave solutions

special exact soliton solutions for the k 2 2 equation with non zero constant pedestal abstract special exact solutions of the k 2 2 equation u t u 2 x u 2 xxx 0 are investigated by employing the qualitative theory of differential equations our procedure shows that the k 2 2 equation either has loop soliton cusped soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x u a 0 or possesses compacton solutions only when lim x u 0 mathematical analysis and numerical simulations are provided for these soliton solutions of the k 2 2 equation

the stability of solutions of vakhnenko s equation vakhnenko s equation has two families of travelling wave solutions the method of rowlands and infeld 1990 is used to investigate whether these solutions are stable to long wavelength perturbations of small amplitude the method predicts stability for both families of solutions some comments on the validity of the method are given

aging properties of an anomalously diffusing particule we report new results about the two time dynamics of an anomalously diffusing classical particle as described by the generalized langevin equation with a frequency dependent noise and the associated friction the noise is defined by its spectral density proportional to 1 at low frequencies with 0 1 subdiffusion or 1 2 superdiffusion using laplace analysis we derive analytic expressions in terms of mittag leffler functions for the correlation functions of the velocity and of the displacement while the velocity thermalizes at large times slowly in contrast to the standard brownian motion case 1 the displacement never attains equilibrium it ages we thus show that this feature of normal diffusion is shared by a subdiffusive or superdiffusive motion we provide a closed form analytic expression for the fluctuation dissipation ratio characterizing aging

symmetric and non symmetric waves in the osmosis k 2 2 equation we give an improved qualitative method to solve the osmosis k 2 2 equation this method combines several characteristics of other methods using this method the existence of symmetric and non symmetric wave solutions of the osmosis k 2 2 equation is studied besides abundant symmetric forms such as smooth wave solutions peaked waves cusped waves looped waves stumpons and fractal like waves this equation also admits non symmetric ones including breaking kink wave solutions breaking anti kink wave solutions and rampons as regards this equation most of those solutions either symmetric or non symmetric solutions have not appeared in the literature we also study the limiting behavior of all periodic solutions as the parameters tend to some special values

fractional diffusion equation with a generalized riemann liouville time fractional derivative in this paper the solution of a fractional diffusion equation with a hilfer generalized riemann liouville time fractional derivative is obtained in terms of mittag leffler type functions and fox s h function the considered equation represents a quite general extension of the classical diffusion heat conduction equation the methods of separation of variables laplace transform and analysis of the sturm liouville problem are used to solve the fractional diffusion equation defined in a bounded domain by using the fourier laplace transform method it is shown that the fundamental solution of the fractional diffusion equation with a generalized riemann liouville time fractional derivative defined in the infinite domain can be expressed via fox s h function it is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the caputo and riemann liouville sense are special cases of those diffusion equations with the hilfer generalized riemann liouville time fractional derivative the asymptotic behaviour of the solutions are found for large values of the spatial variable the fractional moments of the fundamental solution of the fractional diffusion equation are obtained the obtained results are relevant in the context of glass relaxation and aquifer problems

scale free and stable structures in complex ad hoc networks unlike the well studied models of growing networks where the dominant dynamics consist of insertions of new nodes and connections and rewiring of existing links we study ad hoc networks where one also has to contend with rapid and random deletions of existing nodes and hence the associated links we first show that dynamics based only on the well known preferential attachments of new nodes do not lead to a sufficiently heavy tailed degree distribution in ad hoc networks in particular the magnitude of the power law exponent increases rapidly from 3 with the deletion rate becoming ensuremath infty in the limit of equal insertion and deletion rates we then introduce a local and universal compensatory rewiring dynamic and show that even in the limit of equal insertion and deletion rates true scale free structures emerge where the degree distributions obey a power law with a tunable exponent which can be made arbitrarily close to 2 the dynamics reported in this paper can be used to craft protocols for designing highly dynamic peer to peer networks and also to account for the power law exponents observed in existing popular services

degree correlations in growing networks with deletion of nodes in this paper we study the degree distribution and the two node degree correlations in growing networks generated via a general linear preferential attachment of new nodes together with a uniformly random deletion of nodes by using a continuum approach we show that under some suitable combinations of parameters deletion rate and node attractiveness the degree distribution not only loses its scale free character but can even be supported on a small range of degrees moreover we obtain new results on two vertex degree correlations showing that for degree distributions with finite variance such correlations can change under a nonselective removal of nodes

fractional langevin equation overdamped underdamped and critical behaviors the dynamical phase diagram of the fractional langevin equation is investigated for a harmonically bound particle it is shown that critical exponents mark dynamical transitions in the behavior of the system four different critical exponents are found i ensuremath alpha c 0 402 ifmmode pm else textpm fi 0 002 marks a transition to a nonmonotonic underdamped phase ii ensuremath alpha r 0 441 dots marks a transition to a resonance phase when an external oscillating field drives the system and iii ensuremath alpha ensuremath chi 1 0 527 dots and iv ensuremath alpha ensuremath chi 2 0 707 dots mark transitions to a double peak phase of the loss when such an oscillating field present as a physical explanation we present a cage effect where the medium induces an elastic type of friction phase diagrams describing over and underdamped regimes with or without resonances show behaviors different from normal

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved