is transverse macrodispersivity in three dimensional groundwater transport equal to zero a counterexample 1 in advective transport through weakly heterogeneous aquifers of random stationary and isotropic three dimensional permeability distribution transverse macrodispersivity t is found to be zero this was determined in the past by solving the transport equation at first order in the log conductivity variance 2 y however field findings indicate the presence of small but finite t the aim of the paper is to determine t for highly heterogeneous formations using a model that contains inclusions of conductivity k submerged in a matrix of conductivity k 0 for large k k 0 in the dilute medium approximation valid for small volume fraction n but arbitrary and for spherical inclusions it is found that t 0 because of the axisymmetry of flow past a sphere a medium made up of rotational ellipsoids of arbitrary random orientation macroscopically isotropic and of the same and n is devised as a counterexample it is found that because of the intertwining of streamlines t 0 being of order 1 4 for 1 these findings are confirmed by accurate numerical simulations of flow through a large number of interacting inclusions for 10 and n 0 2 jamming limit the large value t l 0 15 is attained the numerical simulations display the strong permanent deformation of stream tubes responsible for this phenomenon coined as advective mixing the two point covariance used in practice in order to characterize the aquifer structure is not able to detect the structures that produce advective mixing nevertheless the presence of high conductivity lenses inclined with respect to the mean flow may explain the occurrence of finite t in field applications

multipulses in discrete hamiltonian nonlinear systems in this work the behavior of multipulses in discrete hamiltonian nonlinear systems is investigated the discrete nonlinear schrodinger equation is used as the benchmark system for this study a singular perturbation methodology as well as a variational approach are implemented in order to identify the dominant factors in the discrete problem the results of the two methodologies are shown to coincide in assessing the interplay of discreteness and exponential tail tail pulse interaction they also allow one to understand why contrary to what is believed for their continuum siblings discrete systems can sustain static multipulse configurations a conclusion that is subsequently verified by numerical experiment

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

solute transport in highly heterogeneous aquifers the paper deals with the transport of nonreactive solute in heterogeneous formations with prescribed statistical properties of the hydraulic log conductivity y ln k available solutions obtained from perturbation methods are limited to first and second order solutions valid only for small values of the log conductivity variance 2y published numerical investigations give comparative results with finite values of 2y but some discrepancies among the results generate doubts about the capability of numerical methods to capture high order effects for media with large variance values when these large 2y values are encountered in natural formations the related nonlinear effects could be significant in the velocity statistics and in the overall dispersion process the nonlinearity consequences are here investigated in two dimensional isotropic porous media by the monte carlo technique coupled with a finite element analysis the analysis includes 2y values from 0 05 to 4 enhancing the relevance of the nonlinear effects in the dispersion tensor solution to dissipate the doubts related to the numerical approach the accuracy of the solutions was defined by checking the influence of the factors which can affect the solution and by giving an estimation of the related errors the numerical results confirm the validity of the first order and second order analyses when 2y 0 the second order solution captures the nonlinear effects in a small range of log conductivity variance close to zero the nonlinear terms neglected in the first order formulation for higher 2y values give 1 late travel time longitudinal dispersion values greater than the linear solution 2 notably non gaussian distribution of the lagrangian velocity and particle displacements and 3 travel times to approach the asymptotic fickian regime longer than those obtained using the linear solution

soliton internal mode bifurcations pure power law the bifurcation of internal solitary wave modes from the essential spectrum has been one of the most exciting recent developments in the study of soliton dynamics to date it was believed that the bifurcation of such modes due to discretization has a strict power law dependence on the lattice discreteness parameter in this work we prove that this dependence actually possesses relevant exponentially small terms which distinguish between different solutions for the discrete models the theoretical result is established by using a discrete version of the evans function the predictions presented herein compare very favorably with the numerical study of the linear eigenvalue problem and offer explanations of computational effects not possible on the basis of previous theoretical studies

a finite volume eulerian lagrangian localized adjoint method for solution of the contaminant transport equations in two dimensional multiphase flow systems an eulerian lagrangian localized adjoint method ellam numerical solution is developed for the multiphase contaminant transport equations in two dimensions the ellam uses finite volume test functions in the space time domain defined by the characteristics of the hyperbolic part of the governing equation the use of the characteristics results in an approximation that allows large time steps while still maintaining accurate solutions this greatly reduces the computational effort required to find solutions to the advection dispersion equation combination of the finite volume test functions with the conservative form of the governing equation results in a local conservation of mass property for problems with a constant saturation such as that of contaminant transport in the saturated zone this property is highly advantageous as the resulting numerical approximation conserves mass both globally and locally for problems with a variable saturation the requirement of local conservation of mass is too stringent particularly around first type boundaries where oscillations can occur because of errors inherent in the numerical determination of fluid velocities and in the backtracking routine a combined conservative nonconservative ellam is developed with an ellam formulation based on the nonconservative form of the governing equation being applied to subdomains intersecting first type boundaries and a conservative ellam being used for all other subdomains the combined conservative nonconservative ellam is compared to a galerkin finite element scheme and is found to have greatly superior performance requiring far fewer time steps to obtain a solution of equivalent accuracy

stability of multiple pulses in discrete systems the stability of multiple pulse solutions to the discrete nonlinear schrodinger equation is considered a bound state of widely separated single pulses is rigorously shown to be unstable unless the phase shift delta phi between adjacent pulses satisfies delta phi pi this instability is accounted for by positive real eigenvalues in the linearized system the analysis leading to the instability result does not however determine the linear stability of those multiple pulses for which delta phi pi between adjacent pulses a direct variational approach for a two pulse predicts that it is linearly stable if delta phi pi and if the separation between the individual pulses satisfies a certain condition the variational approach can easily be generalized to study the stability of n pulses for any n or 3 the analysis is supplemented with a detailed numerical stability analysis

estimation of macrodispersion in 2 d highly heterogeneous porous media using the eulerian lagrangian localized adjoint method 1 the eulerian lagrangian localized adjoint method ellam formulation of younes et al 2006 is applied to solve the advection dispersion equation used to describe conservative solute transport in highly heterogeneous porous media heterogeneity is described by a correlated random field with an exponential covariance macrodispersion coefficients are calculated for a broad range of heterogeneities using monte carlo mc simulations on large size domains ellam circumvents some drawbacks of usual particle tracking and eulerian lagrangian methods when local dispersion diffusion is added ellam is also highly efficient and well adapted for advective dominant transport and for mc simulations for pure advection first order approximation provides good estimates of the duration of the preasymptotic regime and of the longitudinal macrodispersion coefficient for a variance of the log conductivity 2 2 25 higher order theories overestimated this coefficient for higher variances computed transverse macrodispersion is equal to 0 for each studied variance of log conductivity 2 0 25 1 0 2 25 4 0 6 25 9 0 local dispersion diffusion affects the macrodispersion for quite low peclet number 100 compared to previous work for a peclet number of 10 it leads to an increase of the longitudinal and transverse macrodispersion for low variances 2 0 25 for higher heterogeneity 2 9 0 for local dispersion and 2 4 0 for local diffusion the longitudinal macrodispersion decreases due to local transverse mixing

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 2 impervious boundary the spatial variability of the head h in a two dimensional flow in the vicinity of an impervious boundary is investigated this study is a continuation of a previous one rubin and dagan 1988 which dealt with the influence of a given head boundary by representing h and the log transmissivity y as space random function and by solving the flow equation first order closed form analytical expressions for the head variogram and the head log transmissivity y covariance are developed these expressions offer the means to evaluate the influence of the boundary on the joint y h random field the assumption of unbounded domain is examined and it is found to be a good approximation at distances larger than three log transmissivity integral scales from the boundary by using conditional probabilities a simple method to estimate the statistical moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytical solution showing favorable results

strong localized perturbations of eigenvalue problems this paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded o

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 1 constant head boundary the spatial variability of log transmissivity y and head h in aquifers assumes a major role in determining the solutions of the direct flow problem as well as of the inverse problem investigation of the joint y h variability has been cast in the past in numerical and analytic frames analytic solutions for the y h statistical moments in two dimensions assumed the flow to occur in an unbounded domain in this study exact analytical expressions for the y h moments are derived which account for the presence of a given head boundary in a half plane this result is achieved by a first order approximation and for an exponential y covariance the result offers the opportunity to examine the effects of boundaries on the semivariogram of h and on the y h cross covariance the assumption of unbounded domain is examined and shown to be accurate at distances larger than two y integral scales from the boundary by using conditional probabilities a general and simple method to estimate the moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytic solution showing excellent results

summing logarithmic expansions for singularly perturbed eigenvalue problems strong localized perturbations of linear and nonlinear eigenvalue problems in a bounded two dimensional domain d are considered the effects on an eigenvalue lambda 0 of the lapla cian and on the fold point lambda c0 of a nonlinear eigenvalue problem of removing a small subdomain d epsilon of radius epsilon from d and imposing a condition on the boundary of the resulting hole are determined using the method of matched asymptotic expansions it is shown that the expansions of the eigenvalues and fold points for these perturbed problems start with infinite series in powers of 1 log epsilon d kappa here d kappa is a constant that depends on the shape of d epsilon and on the precise form of the boundary condition on the hole in each case it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of the hole this related problem is not stiff and can be solved numerically

asymptotic dispersion in 2d heterogeneous porous media determined by parallel numerical simulations we determine the asymptotic dispersion coefficients in 2d exponentially correlated lognormally distributed permeability fields by using parallel computing fluid flow is computed by solving the flow equation discretized on a regular grid and transport triggered by advection and diffusion is simulated by a particle tracker to obtain a well defined asymptotic regime under ergodic conditions initial plume size much larger than the correlation length of the permeability field the characteristic dimension of the simulated computational domains was of the order of 103 correlation lengths with a resolution of ten cells by correlation length we determine numerically the asymptotic effective longitudinal and transverse dispersion coefficients over 100 simulations for a broad range of heterogeneities s 2 0 9 where s 2 is the lognormal permeability variance for purely advective transport the asymptotic longitudinal dispersion coefficient depends linearly on s 2 for s 2 1 and the asymptotic transverse dispersion coefficient is zero addition of homogeneous isotropic diffusion induces an increase of transverse dispersion and a decrease of longitudinal dispersion

self consistent four point closure for transport in steady random flows ensemble averaging of advection dispersion equations describing transport of a passive scalar in incompressible random velocity fields requires a closure approximation commonly used approaches such as the direct interaction approximation and large eddy simulations as well as equivalent renormalization schemes employ so called two point or one loop closures these approaches have proven to be adequate for transport in zero mean unbiased time dependent random velocity fields with increasing accuracy for decreasing temporal coherence in the opposite limit of steady velocity fields with finite bias however these schemes fail to predict effective transport properties both quantitatively and qualitatively leading to an obvious inconsistency for transverse dispersion in two spatial dimensions for this case two point closures predict that macroscopic transverse dispersion increases as the square root of the disorder variance while it has been proven rigorously that there is no disorder induced contribution to macroscopic transverse dispersion for purely advective transport furthermore two point closures significantly underestimate the disorder induced contribution to longitudinal dispersion we derive a four point closure for stochastically averaged transport equations that goes beyond classical one loop schemes and demonstrate that it is exact for transverse dispersion and correctly predicts an increase of the longitudinal disorder induced dispersion coefficient with the square of the variance of the strong disorder the predicted values of asymptotic longitudinal dispersion coefficients are consistent with those obtained via monte carlo random walk simulations

solving the advection diffusion equation with the eulerian lagrangian localized adjoint method on unstructured meshes and non uniform time stepping eulerian lagrangian localized adjoint method ellam is used to solve the advection diffusion equation ade which is a very common mathematical model in physics in this work ellam is extended to triangular meshes standard integration schemes which perform well for rectangular grids are improved to reduce oscillations with unstructured triangulations numerical experiments for grid peclet numbers ranking from 1 to 100 show the efficiency of the developed scheme a new algorithm is also developed in order to avoid excessive numerical diffusion when using many time steps with the ellam the basic idea of this approach is to keep the same characteristics for all time steps and to interpolate only the concentration variations due to the dispersion process at the end of each time step although ellam requires a lot of integration points for unstructured meshes it remains a competitive method when using a single or many time steps compared to explicit discontinuous galerkin finite element method

an edge based finite volume scheme for saturated unsaturated groundwater flow abstract the application of a computationally efficient finite volume method for modelling groundwater flow in saturated unsaturated soils is presented the flow equation is solved by applying a vertex centred edge based finite volume scheme and the discretisation of the flow domain is achieved using structured and unstructured grids an acceleration algorithm is employed which utilises an explicit non linear loop for the primary variable update the scheme is employed to solve a two dimensional groundwater flow problem and the results obtained are compared against published solutions it is shown that the edge based finite volume scheme captures the saturation front more efficiently than the published finite element solution comput geosci 27 2 2001 179 moreover an improvement in the solution is shown by the application of a compact stencil scheme to calculate second derivative terms as compared to the standard edge based finite volume scheme

a mixed transfer matrix method for simulating normal conductor perfect insulator perfect conductor random networks we develop a mixed transfer matrix approach for computing the macroscopic conductivity of a three constituent normal conductor perfect insulator perfect conductor random network this is applied to two dimensional and three dimensional samples at a percolation threshold such networks are simulated in order to test whether a diluted percolating network of normal conducting bonds remains in the same universality class of critical behavior when a finite fraction of those bonds have been replaced by perfectly conducting bonds also tested by such simulations is whether a percolating mixture of normal and perfectly conducting bonds remains in the same universality class of critical behavior when a finite fraction of the normal bonds are replaced by perfectly insulating bonds these questions are crucial for some recently published exact results which connect the macroscopic electrical and elastic responses of percolating networks

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

numerical simulations of preasymptotic transport in heterogeneous porous media departures from the gaussian limit 1 the objective of this work is to determine whether conventional solute macrodispersion theories adequately predict the behavior of individual subsurface plumes over time and length scales relevant to practical risk assessment and remediation activities this issue is studied with a set of high resolution numerical simulations of conservative tracers moving through saturated two dimensional heterogeneous conductivity fields the simulation experiments are designed to mimic long term field studies spatially correlated statistically stationary lognormal conductivity realizations are generated with a fourier transform procedure steady state velocity solutions are calculated for these fields using an accurate darcian solver velocity contour plots reveal the presence of disconnected networks of preferential pathways over a range of correlation lengths reverse flow cells are rare the velocity probability density functions have exponential tails and strong longitudinal asymmetries solute concentrations are derived from the simulated velocity fields with an accurate taut spline transport code that minimizes numerical dispersion the resulting plumes are tracked for travel distances of over one hundred spatial correlation lengths corresponding to scales of practical interest first moments and macrodispersivities which measure the location and extent of the plume are reasonably well approximated by conventional fickian theories but continue to vary after long travel distances this temporal variability highlights the slow convergence of the plume moments to gaussian limits calculations of the relative dilution index which is a measure of the plume mixing state indicate strongly non gaussian behavior other measures of plume structure suggested by anomalous dispersion theories also reveal the persistence of non gaussian behavior after long travel distances these measures appear to be more sensitive to non gaussian behavior than the spatial moments taken together the simulation results suggest that conservative solute plumes moving through statistically stationary random media may not converge to gaussian limits even after traveling hundreds of log conductivity correlation scales

flow and travel time statistics in highly heterogeneous porous media 1 in this paper we present flow and travel time ensemble statistics based on a new simulation methodology the adaptive fup monte carlo method afmcm as a benchmark case we considered two dimensional steady flow in a rectangular domain characterized by multi gaussian heterogeneity structure with an isotropic exponential correlation and lnk variance y2 up to 8 advective transport is investigated using the travel time framework where lagrangian variables e g velocity transverse displacement or travel time depend on space rather than on time we find that eulerian and lagrangian velocity distributions diverge for increasing lnk variance due to enhanced channeling transverse displacement is a nonnormal for all y2 and control planes close to the injection area but after xiy 20 was found to be nearly normal even for high y2 travel time distribution deviates from the fickian model for large lnk variance and exhibits increasing skewness and a power law tail for large lnk variance the slope of which decreases for increasing distance from the source no anomalous features are found second moment of advective transport is analyzed with respect to the covariance of two lagrangian velocity variables slowness and slope which are directly related to the travel time and transverse displacement variance which are subsequently related to the longitudinal and transverse dispersion we provide simple estimators for the eulerian velocity variance travel time variance slowness and longitudinal dispersivity as a practical contribution of this analysis both two parameter models considered the advection dispersion equation and the lognormal model provide relatively poor representations of the initial part of the travel time probability density function in highly heterogeneous porous media we identify the need for further theoretical and experimental scrutiny of early arrival times and the need for computing higher order moments for a more accurate characterization of the travel time probability density function a brief discussion is presented on the challenges and extensions for which afmcm is suggested as a suitable approach

stability of solitary waves in finite ablowitz ladik lattices we discuss the effects of different types of boundary conditions b c on the linear stability of a solitary wave in a finite length dynamical lattice described by the ablowitz ladik al model types of b c considered are fixed dirichlet no flux neumann and free as well as periodic b c the behaviour of eigenvalues around a stationary nonlinear wave consistent with several types of b c is studied analytically and numerically the translational eigenvalues are found to move away from the origin it is shown that for b c of the fixed type these eigenvalues bifurcate into a neutrally stable oscillatory kind while the no flux and free b c lead to an exponentially weak instability the rotational eigenmodes those related to the gauge invariance of the lattice strictly remain at the origin as gauge invariance is not broken by the linear homogeneous b c these numerical results are compared to the analytical and semi analytical predictions obtained within the approximation of the images generated by the solitary wave beyond the lattice edges a potential effect of continuous spectrum band edge bifurcations on stability is also evaluated with the conclusion that that these bifurcations cannot destabilize the soliton for periodic b c the translational invariance is preserved and so is the lax pair structure sustaining the integrable nature of the problem the al model was chosen so that the effects of boundary conditions are monitored in the absence of non integrable discreteness finally the possibility of the generation of multipulse configurations due to the interplay of finiteness effects with the exponential tail tail interaction of the pulses is examined

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

exact relations between elastic and electrical response of d dimensional percolating networks with angle bending forces arguments are presented to demonstrate that exact equality relations exist between the critical exponents which characterize the macroscopic conductivity e and the macroscopic elastic stiffness moduli c e of percolating systems of any dimensionality using the notation e p t c e p t for the critical behavior of a randomly diluted system slightly above the percolation threshold p c p p p c 0 and e p s c e p s for the critical behavior of a random mixture of normal and perfectly conducting or normal and perfectly rigid constituents slightly below that threshold p p p c 0 we show that t t 2 and s s where is the percolation correlation length critical exponent p p 0

a mean field statistical theory for the nonlinear schrodinger equation abstract a statistical model of self organization in a generic class of one dimensional nonlinear schrodinger nls equations on a bounded interval is developed the main prediction of this model is that the statistically preferred state for such equations consists of a deterministic coherent structure coupled with fine scale random fluctuations or radiation the model is derived from equilibrium statistical mechanics by using a mean field approximation of the conserved hamiltonian and particle number l2 norm squared for finite dimensional spectral truncations of the nls dynamics the continuum limits of these approximated statistical equilibrium ensembles on finite dimensional phase spaces are analyzed holding the energy and particle number at fixed finite values the analysis shows that the coherent structure minimizes total energy for a given value of particle number and hence is a solution to the nls ground state equation and that the remaining energy resides in gaussian fluctuations equipartitioned over wave numbers some results of direct numerical integration of the nls equation are included to validate empirically these properties of the most probable states for the statistical model moreover a theoretical justification of the mean field approximation is given in which the approximate ensembles are shown to concentrate on the associated microcanonical ensemble in the continuum limit

flow and transport in highly heterogeneous formations 2 semianalytical results for isotropic media 1 flow and transport of nonreactive solutes in heterogeneous porous media is studied by adopting a multi indicator model of permeability structure the porous formation is modeled as a collection of blocks of uniform permeability k implanted at random in a matrix of constant conductivity k0 the multi indicator model leads to simple semianalytical solutions based on the self consistent argument which are valid for a high degree of heterogeneity the methodology is applied to isotropic formations to derive a few statistical moments of the velocity field and of the solute particles trajectory as functions of time and of the log conductivity variance y2 along the common models of aquifer permeability distribution the distribution of y ln k is assumed to be normal all the semianalytical results degenerate in the well known first order results when y2 1 in particular it is shown that the first order longitudinal dispersivity l seems to hold for values of the log conductivity variance much larger than expected up to y2 4 this results from a compensation of errors associated with the first order approximation in contrast for y2 1 the asymptotic l grows exponentially with y2 the effect of molecular diffusion is considered in a simple manner by introducing a cutoff c for the conductivity contrast the time to reach the asymptotic l grows with the log conductivity variance it is thus observed that transport in highly heterogeneous formations can be characterized by a very prolonged non fickian stage with dispersivity l growing continuously with time further analysis of third order moment of trajectory indicates that for growing values of y2 the time needed for the plume to become gaussian can be quite large

surface solitons in three dimensions we study localized modes on the surface of a three dimensional dynamical lattice the stability of these structures on the surface is investigated and compared to that in the bulk of the lattice typically the surface makes the stability region larger an extreme example of that being the three site horseshoe shaped structure which is always unstable in the bulk while at the surface it is stable near the anticontinuum limit we also examine effects of the surface on lattice vortices for the vortex placed parallel to the surface the increased stability region feature is also observed while the vortex cannot exist in a state normal to the surface more sophisticated localized dynamical structures such as five site horseshoes and pyramids are also considered

kink dynamics in the highly discrete sine gordon system abstract we study kink dynamics in a very discrete sine gordon system where the kink width is of the order of the lattice spacing numerical simulations exhibit new properties of kinks in this case they lose the memory of their initial velocity and propagate preferentially at well defined velocities which correspond to quasi steady states while a kink moving at other velocities suffers relatively high rates of radiation of small amplitude oscillations when a small external driving force is applied to the system the same velocities appear as plateus in the strongly nonlinear mobility of the kink the energy radiated by the kink is calculated for a simple model that preserves the discrete character of the system and the preferential velocities for the kink are obtained to good accuracy similar results may be expected to be valid for other discrete systems manifesting topological solitons the numerical simulations reveal also new stable multiple kink excitations which can propagate almost freely in extremely discrete systems where ordinary simple kinks are pinned to the lattice by discreteness the stability of the multiple kinks is discussed