on the geometry of wave solutions of a delayed reaction diffusion equation abstract the aim of this paper is to study the existence and the geometry of positive bounded wave solutions to a non local delayed reaction diffusion equation of the monostable type

dynamics of a nonlocal multi type sis epidemic model with seasonality abstract in this paper we propose and study a multi type sis nonlocal epidemic model with seasonality we first establish the existence of the spreading speed c and the non existence of periodic traveling waves with wave speed c c then we prove the existence and asymptotic behavior of the periodic traveling fronts with speed c c finally the existence of the critical periodic traveling wave fronts is established by using a limiting argument to overcome the difficulty of lack of compactness of solution maps of the nonlocal system with respect to compact open topology we show that the solution sequence is pre compact in l loc r 2 r m

existence of traveling wave solutions for a reaction diffusion equation with distributed delays abstract in this paper we consider a reaction diffusion equation with distributed delays special attention is paid to the existence of traveling wave solutions for special delay kernels using the linear chain trick and the geometric singular perturbation theory we investigate a natural connection between the existence of traveling wave solutions for the reaction diffusion equation with distributed delays and the existence of traveling wave solutions for the corresponding undelayed reaction diffusion equation we show that if the corresponding undelayed reaction diffusion equation has a traveling wave solution then for any sufficiently small delay the delayed reaction diffusion equation also has a traveling wave solution

global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity abstract this paper is concerned with the global stability of wavefronts with minimal speeds for degenerate nonlocal dispersal equations by the method of super and subsolutions and a squeezing technique the wavefront with minimal speed is proved to be globally asymptotically stable with phase shift

asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models abstract the theory of asymptotic speeds of spread and monotone traveling waves is generalized to a large class of scalar nonlinear integral equations and is applied to some time delayed reaction and diffusion population models

travelling waves of the diffusive nicholson s blowflies equation with strong generic delay kernel and non local effect abstract in this paper we consider travelling wave solutions for the diffusive nicholson s blowflies equation incorporating time delay and diffusion special attention is paid to the modelling of the time delay to incorporate associated non local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times for the strong generic delay kernel we show that travelling wave solutions exist provided that the delay is sufficiently small using the geometric singular perturbation theory

stability of traveling wavefronts for a discrete diffusive competition system with three species abstract this paper is concerned with the nonlinear stability of monostable traveling wavefronts for a discrete three species competition diffusion system applying weighted energy method combining with the comparison principle we first show that the traveling wavefronts with large speed are exponentially stable when the initial perturbation around the traveling wave decays exponentially as x but the initial perturbation can be arbitrarily large in other locations then choosing different weight function we improve the stability result to any traveling wavefronts with speed greater than the critical wave speed we have to emphasize that the discrete dispersal operator in the system increases the difficulty of establishing energy estimates

travelling wave solutions for a nonlocal dispersal hiv infection dynamical model abstract this paper is devoted to developing a nonlocal dispersal hiv infection dynamical model the existence of travelling wave solutions is investigated by employing schauder s fixed point theorem that is we study the existence of travelling wave solutions for r 0 1 and each wave speed c c in addition the boundary asymptotic behaviour of travelling wave solutions at is obtained by constructing suitable lyapunov functions and employing lebesgue dominated convergence theorem by employing a limiting argument we investigate the existence of travelling wave solutions for r 0 1 and c c the main difficulties are that the semiflow generated by the model does not have the order preserving property and the solutions lack of regularity

on the existence of traveling waves for delayed reaction diffusion equations abstract we study the existence of traveling wave solutions for reaction diffusion equations with nonlocal delay where reaction terms are not necessarily monotone the existence of traveling wave solutions for reaction diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the schauder fixed point theorem the smoothness of upper and lower solutions is not required in this paper

nonlinear stability of traveling wavefronts for competitive cooperative lotka volterra systems of three species abstract this paper is concerned with nonlinear stability of traveling wavefronts for competitive cooperative lotka volterra systems of three species it is proved that all traveling wavefronts with speed c c are exponentially stable in l space by the weighted energy method and comparison principle moreover the time decay rates are obtained by weighted energy estimates

entire solutions for a multi type sis nonlocal epidemic model in r or z abstract in this paper we study entire solutions for a multi type sis nonlocal epidemic model in r or z the existence and asymptotic behavior of spatially independent solutions are first established some new entire solutions are then constructed by combining traveling wave fronts with different speeds and a spatially independent solution from the viewpoint of epidemiology the results provide some new spread ways of the epidemic

self avoiding polygons on the square lattice we have developed an improved algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 90 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 529 27 1 biased and the critical exponent 0 500 0005 10 unbiased the critical point is indistinguishable from a root of the polynomial 581x4 7x2 13 0 an asymptotic expansion for the coefficients is given for all n there is strong evidence for the absence of any non analytic correction to scaling exponent

asymptotic behaviour of a reaction diffusion model with a quiescent stage this paper is devoted to the investigation of the asymptotic behaviour for a reaction diffusion model with a quiescent stage we first establish the existence of the asymptotic speed of spread and show that it coincides with the minimal wave speed for monotone travelling waves then we obtain a threshold result on the global attractivity of either zero or positive steady state in the case where the spatial domain is bounded

delay induced traveling wave fronts in reaction diffusion equations of kpp fisher type this paper explores the impact of delay on the existence of traveling wave fronts in reaction diffusion equations of kpp fisher type for two such equations one being local and the other with nonlocal effect arising from the age structure of the population we show that delay can induce slow traveling wave fronts

travelling wave fronts in reaction diffusion systems with spatio temporal delays this paper deals with the existence of travelling wave fronts in reaction diffusion systems with spatio temporal delays our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system new iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions following this the existence of travelling wave fronts for reaction diffusion systems with spatio temporal delays is established finally we apply the main results to a single species diffusive model with spatio temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels

traveling wavefronts for delayed reaction diffusion systems via a fixed point theorem abstract by using schauder s fixed point theorem we prove some existence results for traveling wavefronts of reaction diffusion systems with quasimonotonicity reactions more precisely we reduce the existence of traveling wavefronts to the existence of an admissible pair of supersolution and subsolution which are easy to construct in practice finally to illustrate our main results we study the existence of traveling wavefronts for a delayed predator prey model with diffusion as well as the reaction diffusion system with the well known belousov zhabotinskii reaction and the obtained results improve the existing ones

travelling waves for delayed reaction diffusion equations with global response we develop a new approach to obtain the existence of travelling wave solutions for reaction diffusion equations with delayed non local response the approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain banach space coupled with an index formula of the associated fredholm operator and some careful estimation of the nonlinear perturbation the general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non local and delayed response for the matured population

epidemic waves of a spatial sir model in combination with random dispersal and non local dispersal abstract in this paper a spatial sir model in combination with random dispersal and non local dispersal is proposed we establish the existence and non existence of the traveling wave solutions connecting the disease free equilibrium and the endemic equilibrium for the model the main difficulties lie in the fact that the semiflow generated here does not admit the order preserving property and the solutions lack of regularity we use a proper iteration scheme to construct a pair of upper and lower solutions and then apply the schauder s fixed point theorem polar coordinates transform to study the threshold dynamics of the model that is we show that if the basic reproduction number of the model r 0 1 there is a positive constant critical number c such that for each c c the model admits a non trivial and positive traveling wave with wave speed c and if r 0 1 and 0

spreading speeds and traveling wavefronts for second order integrodifference equations this paper is concerned with the spreading speeds and traveling wavefronts for second order integrodifference equations by introducing an auxiliary integrodifference system the spreading speed is established for the integrodifference equation it is shown that the spreading speed coincides with the minimal wave speed for monotonic traveling wavefronts furthermore we prove that the traveling wavefronts are stable by applying the squeezing technique finally we analyze the different effects of the delay term appearing in the integrodifference equation from the viewpoint of ecology

asymptotic stability of traveling wave fronts in nonlocal reaction diffusion equations with delay abstract this paper is concerned with the asymptotic stability of traveling wave fronts of a class of nonlocal reaction diffusion equations with delay under monostable assumption we prove that the traveling wave front is exponentially stable by means of the technical weighted energy method when the initial perturbation around the wave is suitable small in a weighted norm the exponential convergent rate is also obtained finally we apply our results to some population models and obtain some new results which recover complement and or improve a number of existing ones

existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay this paper is concerned with the existence uniqueness and globally asymptotic stability of traveling wave fronts in the quasi monotone reaction advection diffusion equations with nonlocal delay under bistable assumption we construct various pairs of super and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front up to translation which is monotonically increasing and globally asymptotically stable with phase shift the influence of advection on the propagation speed is also considered comparing with the previous results our results recovers and or improves a number of existing ones in particular these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay some new results are obtained

on complex singularities of the 2d euler equation at short times abstract we present a study of complex singularities of a two parameter family of solutions for the two dimensional euler equation with periodic boundary conditions and initial conditions 0 z 1 z 2 f p cos p z f q cos q z in the short time asymptotic regime as has been shown numerically in pauls et al w pauls t matsumoto u frisch j bec nature of complex singularities for the 2d euler equation physica d 219 2006 40 59 the type of the singularities depends on the angle between the modes p and q thus the fourier coefficients of the solutions decrease as g k c k e k with the exponent depending on here we show for the two particular cases of going to zero and to that the type of the singularities can be determined very accurately being characterised by 5 2 and 3 respectively in these two cases we are also able to determine the subdominant corrections furthermore we find that the geometry of the singularities in these two cases is completely different the singular manifold being located over different points in the real domain

entire solutions in monostable reaction diffusion equations with delayed nonlinearity abstract entire solutions for monostable reaction diffusion equations with nonlocal delay in one dimensional spatial domain are considered a comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory key techniques include the characterization of the asymptotic behaviors of solutions as t in term of appropriate subsolutions and supersolutions two models of reaction diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results

a borel transform method for locating singularities of taylor and fourier series given a taylor series with a finite radius of convergence its borel transform defines an entire function a theorem of polya relates the large distance behavior of the borel transform in different directions to singularities of the original function with the help of the new asymptotic interpolation method of van der hoeven we show that from the knowledge of a large number of taylor coefficients we can identify precisely the location of such singularities as well as their type when they are isolated there is no risk of getting artefacts with this method which also gives us access to some of the singularities beyond the convergence disk the method can also be applied to fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the burgers equation large precision on scaling exponents up to twenty accurate digits can be achieved

positive heteroclinics and traveling waves for scalar population models with a single delay the existence of positive heteroclinic solutions is proven for a class of scalar population models with one discrete delay traveling wave solutions for scalar delayed reaction diusion equations are also obtained as perturbations of heteroclinic solutions of the associated equation without diusion as an illustration the results are applied to the nicholson s blowflies equation with diusion n t t x d 2n x2 t x n t x pn t x e an t x in the case of p e for which the nonlinearity is non monotone

the cauchy lagrangian method for numerical analysis of euler flow a novel semi lagrangian method is introduced to solve numerically the euler equation for ideal incompressible flow in arbitrary space dimension it exploits the time analyticity of fluid particle trajectories and requires in principle only limited spatial smoothness of the initial data efficient generation of high order time taylor coefficients is made possible by a recurrence relation that follows from the cauchy invariants formulation of the euler equation zheligovsky and frisch 2014 44 truncated time taylor series of very high order allow the use of time steps vastly exceeding the courant friedrichs lewy limit without compromising the accuracy of the solution tests performed on the two dimensional euler equation indicate that the cauchy lagrangian method is more and occasionally much more efficient and less prone to instability than eulerian runge kutta methods and less prone to rapid growth of rounding errors than the high order eulerian time taylor algorithm we also develop tools of analysis adapted to the cauchy lagrangian method such as the monitoring of the radius of convergence of the time taylor series certain other fluid equations can be handled similarly

asymptotic pattern for a partial neutral functional differential equation abstract in the present article the asymptotic speed of spread and traveling wavefronts of a partial neutral functional differential equation pnfde are investigated a variable transform yields to a partial retarded functional differential equation prfde with infinite number of constant delays the relation between the above pnfde and prfde is studied and an equivalence in a certain meaning of these two equations is verified using this equivalence we in fact obtain the existence and uniqueness of solution for the initial value problem of the pnfde we then consider the truncated system of the above prfde which is a partial retarded functional differential equation with finite number of constant delays we establish the existence of the traveling wave solution and the spreading speed for this truncated system a limit argument and the method of finite delay approximation are used to obtain the existence of the traveling wave solutions and the spreading speed for the prfde with infinite number of constant delays this leads to the existence of the traveling wave solution and the spreading speed for the considered pnfde by the above equivalence at last we confirm that the spreading speed is coincidental with the minimal wave speed for all equations an example is given to illustrate the application of our results

travelling wave and global attractivity in a competition diffusion system with nonlocal delays in this paper we are concerned with a two competition model described by a reaction diffusion system with nonlocal delays which account for the drift of individuals to their present position from their possible positions at previous times by using the iterative technique recently developed in wang et al 2006 14 the sufficient conditions are established for the existence of travelling wave solutions connecting the semi trivial steady state to the coexistence steady state of the considered system when the domain is bounded we investigate the global attractivity of the coexistence steady state of the system under homogeneous neumann boundary conditions as well the approach used is the upper lower solutions and monotone iteration technique

on analytic properties of periodic solutions for equation mathcal h u_x u u p 0 the paper focuses on 2l periodic solutions ul x of the nonlocal equation p 2 is integer is the hilbert transform we give numerical evidence for the existence of a continuous family of these solutions parametrized by l for p 3 4 5 the features of the solutions ul x are discussed in particular we offer strong numerical arguments that ul x can be continued analytically from the real axis to some strip in the complex plane ul x ul z the singularities which arise under analytical continuation of these solutions into the complex plane are considered it is proved theorem 1 that ul z cannot be any power of a meromorphic function the formula which describes the asymptotic behavior of ul z in a neighborhood of the singularity is given it agrees with the numerical results computations also allow us to locate the closest to the real axis singularity of ul z and estimate the width of the analyticity strip

a common generalization of convolved generalized fibonacci and lucas polynomials and its applications the purpose of this paper is to give a common generalization of convolved generalized fibonacci and lucas polynomials and some recurrence relations are established as applications we obtain some computational formulas of mixed multiple sums for h x fibonacci polynomials and lucas polynomials

traveling waves in a delayed sir epidemic model with nonlinear incidence we study the traveling waves of a diffusive sir epidemic model with a general nonlinear incidence we obtain the threshold dynamics for spatial spread of the disease we show that c is the minimum wave speed for the existence of traveling wave solutions we establish the existence and non existence of traveling wave solutions for a diffusive sir model with a general nonlinear incidence the existence proof is shown by introducing an auxiliary system applying schauder s fixed point theorem and then a limiting argument the nonexistence proof is obtained by two sided laplace transform when the speed is less than the critical velocity numerical simulations support the theoretical results we also point out the effects of the delay and the diffusion rate of the infective individuals on the spreading speed

asymptotic patterns of a structured population diffusing in a two dimensional strip abstract in this paper we derive a population model for the growth of a single species on a two dimensional strip with neumann and robin boundary conditions we show that the dynamics of the mature population is governed by a reaction diffusion equation with delayed global interaction using the theory of asymptotic speed of spread and monotone traveling waves for monotone semiflows we obtain the spreading speed c the non existence of traveling waves with wave speed 0 c c and the existence of monotone traveling waves connecting the two equilibria for c c

asymptotic behavior and traveling wave solutions for parabolic functional differential equations on presente une generalisation de la theorie des equations kpp et de diffusion bistables non lineaires on montre que des solutions onde de propagation existent pour des equations fonctionnelles paraboliques non lineaires

traveling wavefronts of a prey predator diffusion system with stage structure and harvesting from a biological point of view we consider a prey predator type free diffusion fishery model with stage structure and harvesting first we study the stability of the nonnegative constant equilibria in particular the effect of harvesting on the stability of equilibria is discussed and supported with numerical simulation then employing the upper and lower solution method we show that when the wave speed is large enough there exists a traveling wavefront connecting the zero solution to the positive equilibrium of the system numerical simulation is also carried out to illustrate the main result

perron theorem in the monotone iteration method for traveling waves in delayed reaction diffusion equations abstract in this paper we revisit the existence of traveling waves for delayed reaction diffusion equations by the monotone iteration method we show that perron theorem on existence of bounded solution provides a rigorous and constructive framework to find traveling wave solutions of reaction diffusion systems with time delay the method is tried out on two classical examples with delay the predator prey and belousov zhabotinskii models

traveling waves for a time delayed newtonian filtration equation abstract this paper deals with traveling wave solutions for a degenerate parabolic equation with time delay we first establish the necessary and sufficient conditions to the existence of monotone nonincreasing and nondecreasing traveling wave solutions which correspond to a unique wave speed c respectively it will be shown that the traveling wave may be a semi finite or infinite traveling wave under different exponents furthermore we give an accurate estimation on the convergent rate for the semi finite or infinite traveling waves

a phase plane discussion of convergence to travelling fronts for nonlinear diffusion the paper is concerned with the asymptotic behavior as t of solutions u x t of the equation u t u xx f left u right 0 x in left infty infty right in the case f 0 f 1 0 with f u non positive for u 0 sufficiently close to zero and f u non negative for u 1 sufficiently close to 1 this guarantees the uniqueness but not the existence of a travelling front solution u u x ct u 0 u and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front if it exists and to a stacked combination of travelling fronts if it does not the approach is to use the monotonicity to take u and t as independent variables and p ux as the dependent variable and to apply ideas of sub and super solutions to the diffusion equation for p

accelerating propagation in a recursive system arising from seasonal population models with nonlocal dispersal abstract the system of interest is defined by iterations of the poincare map of a time periodic population model with symmetric nonlocal dispersal and periodic time delay under the kpp type setting we show that the system admits the spreading speed c 0 that coincides with the minimal speed of traveling waves where c if the nonlocal dispersal kernel j is exponentially bounded and c if j is exponentially unbounded further the latter case gives rises to an accelerating propagation that the level set of solutions with compactly supported initial values locate in between two exponential functions in time

nature of complex singularities for the 2d euler equation abstract a detailed study of complex space singularities of the two dimensional incompressible euler equation is performed in the short time asymptotic regime when such singularities are very far from the real domain this allows an exact recursive determination of arbitrarily many spatial fourier coefficients using high precision arithmetic we find that the fourier coefficients of the stream function are given over more than two decades of wavenumbers by f k c k e k where k k cos sin the prefactor exponent typically between 5 2 and 8 3 is determined with an accuracy better than 0 01 it depends on the initial condition but not on the vorticity diverges as s where 7 2 and s is the distance to the complex singular manifold this new type of non universal singularity is permitted by the strong reduction of nonlinearity depletion which is associated to incompressibility spectral calculations show that the scaling reported above persists well beyond the time of validity of the short time asymptotics a simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent 5 2

stability of strong travelling waves for a non local time delayed reaction diffusion equation the paper is concerned with a non local time delayed reaction diffusion equation we prove the time asymptotic stability of a travelling wavefront without a smallness assumption on its wavelength i e the so called strong wavefront by means of the technical weighted energy method when the initial perturbation around the wave is small the exponential convergent rate is also given selection of a suitable weight plays a key role in the proof

traveling waves solutions for partial neutral differential equations abstract using monotonicity techniques and the schauder fixed point theorem we study the existence of traveling waves for neutral partial differential equations in the last section we illustrate our results on the neutral versions of the kpp fisher and the kobayashi equations

travelling wavefronts in the diffusive single species model with allee effect and distributed delay in this paper we consider the diffusive single species model with allee effect and distributed delay time special attention is paid to the existence of travelling wavefront solutions first we shall show that such fronts exist when the convolution kernel assumes the strong generic delay kernel and the delay is sufficiently small then in the non local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times we shall show that such fronts still exist for the weak generic delay kernel and small delay the approach used in this paper is the geometric singular perturbation theory

on travelling wavefronts of nicholson s blowflies equation with diffusion this paper is devoted to the study of nicholson s blowflies equation with diffusion a kind of time delayed reaction diffusion for any travelling wavefront with speed c c c is the minimum wave speed we prove that the wavefront is time asymptotically stable when the delay time is sufficiently small and the initial perturbation around the wavefront decays to zero exponentially in space as x but it can be large in other locations the result develops and improves the previous wave stability obtained by mei et al in 2004 the new approach developed in this paper is the comparison principle combined with the technical weighted energy method numerical simulations are also carried out to confirm our theoretical results

traveling wave solutions in delayed reaction diffusion systems with mixed monotonicity this paper deals with the existence of traveling wave solutions in delayed reaction diffusion systems with mixed monotonicity based on two different mixed quasi monotonicity reaction terms we propose new conditions on the reaction terms and new definitions of upper and lower solutions by using schauder s fixed point theorem and a new cross iteration scheme we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions the general results obtained have been applied to type k monotone and type k competitive diffusive lotka volterra systems

wave front solutions of a diffusive delay model for populations of daphnia magna abstract in this paper we consider a diffusional food limited population model incorporating a discrete time delay for the case when the delay is small we show that monotone travelling front solutions exist connecting the two uniform steady states of the model the effect of a larger delay is studied numerically suggesting that in this case travelling fronts still exist but lose their monotonicity

some identities for the generalized fibonacci numbers and the generalized lucas numbers in this paper we study some properties of the generalized fibonacci numbers and the generalized lucas numbers these numbers are equal to the total numbers of k independent sets in special graphs we give some identities for the generalized fibonacci numbers and the generalized lucas numbers which can be useful also in problems of counting of k independent sets in graphs

some identities of the generalized fibonacci and lucas sequences abstract the purpose of this paper is to study generalized fibonacci and lucas sequences we first introduce generalized lucas sequences section 2 contains a list of elementary relationships about generalized fibonacci and lucas sequences in section 3 we give a generalization of the binet s formulas of generalized fibonacci lucas sequences and its applications section 4 is devote to derive many identities and congruence relations for generalized fibonacci lucas sequences by using operator method

entire solutions in bistable reaction diffusion equations with nonlocal delayed nonlinearity this paper is concerned with entire solutions for bistable reaction diffusion equations with nonlocal delay in one dimensional spatial domain here the entire solutions are defined in the whole space and for all time t r assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the x axis and annihilating at a finite time furthermore we show that such an entire solution is unique up to space time translations and is liapunov stable a key idea is to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions in order to illustrate our main results two models of reaction diffusion equations with nonlocal delay arising from mathematical biology are considered

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

analytic structure of solutions of the one dimensional burgers equation with modified dissipation we use the one dimensional burgers equation to illustrate the effect of replacing the standard laplacian dissipation term by a more general function of the laplacian of which hyperviscosity is the best known example in equations of hydrodynamics we analyze the asymptotic structure of solutions in the fourier space at very high wave numbers by introducing an approach applicable to a wide class of hydrodynamical equations whose solutions are calculated in the limit of vanishing reynolds numbers from algebraic recursion relations involving iterated integrations we give a detailed analysis of their analytic structure for two different types of dissipation a hyperviscous and an exponentially growing dissipation term our results obtained in the limit of vanishing reynolds numbers are validated by high precision numerical simulations at non zero reynolds numbers we then study the bottleneck problem an intermediate asymptotics phenomenon which in the case of the burgers equation arises when ones uses dissipation terms such as hyperviscosity growing faster at high wave numbers than the standard laplacian dissipation term a linearized solution of the well known boundary layer limit of the burgers equation involving two numerically determined parameters gives a good description of the bottleneck region

travelling waves for a non monotone bistable equation with delay existence and oscillations we consider a bistable 0 theta 1 being the three constant steady states delayed reaction diffusion equation which serves as a model in population dynamics the problem does not admit any comparison principle this prevents the use of classical technics and as a consequence it is far from obvious to understand the behaviour of a possible travelling wave in infty combining refined it a priori estimates and a leray schauder topological degree argument we construct a travelling wave connecting 0 in infty to lq lq something which is strictly above the unstable equilibrium theta in infty furthemore we present situations additional bound on the nonlinearity or small delay where the wave converges to 1 in infty whereas the wave is shown to oscillate around 1 in infty when typically the delay is large

traveling wave solutions for integro difference systems abstract this paper is concerned with the traveling wave solutions for integro difference systems of higher order by using schauder fixed point theorem the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions then the asymptotic behavior of traveling wave solutions is studied by the idea of contracting rectangles to illustrate our results the traveling wave solutions of three systems are considered which completes some known results

a note on generalized fibonacci sequences consider the generalized fibonacci sequence qn n 0 having initial conditions q0 0 q1 1 and recurrence relation qn aqn 1 qn 2 when n is even or qn bqn 1 qn 2 when n is odd where a and b are nonzero real numbers these sequences arise in a natural way in the study of continued fractions of quadratic irrationals and combinatorics on words or dynamical system theory some well known sequences are special cases of this generalization the fibonacci sequence is a special case of qn with a b 1 pell s sequence is qn with a b 2 and the k fibonacci sequence is qn with a b k in this article we study numerous new properties of these sequences and investigate a sequence closely related to these sequences which can be regarded as a generalization of lucas sequence of the first kind

stability of traveling waves in a population dynamics model with spatio temporal delay abstract in this paper we are concerned with the stability of all traveling waves for a population dynamics model with spatio temporal delay by the weighted energy method combining comparison principle the global exponential stability of all traveling waves for the model is established even including the slower waves whose wave speed is close to the critical speed

bistable wavefronts in a diffusive and competitive lotka volterra type system with nonlocal delays abstract a diffusive lotka volterra type model with nonlocal delays for two competitive species is considered the existence of a traveling wavefront analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a four dimensional system without delay furthermore in order to prove the asymptotic stability up to translation of bistable wavefronts of the system the existence regularity and comparison theorem of solutions of the corresponding cauchy problem are first established for the systems on r by appealing to the theory of abstract functional differential equations the asymptotic stability up to translation of bistable wavefronts are then proved by spectral methods in particular we also prove that the spreading speed is unique by upper and lower solutions technique from the point of view of ecology our results indicate that the nonlocal delays appeared in the interaction terms are not sensitive to the invasion of species of spatial isolation

traveling waves for time delayed reaction diffusion equations with degenerate diffusion abstract this paper is concerned with time delayed reaction diffusion equations with degenerate diffusion when the term for birth rate is a nonlocal integral with a heat kernel the family of minimum wave speeds corresponding to all the degenerate diffusion coefficients is proved to admit a uniform positive infimum however when the term for birth rate is local there is no positive infimum of all the minimum wave speeds this difference indicates that the nonlocal effect plays a role as laplacian such that a positive lower bound independent of the degenerate diffusion exists for the minimum wave speeds the approach adopted for the proof is the monotone technique with the viscosity vanishing method the degeneracy of diffusion for the equation causes us essential difficulty in the proof a number of numerical simulations are also carried out at the end of the paper which further numerically confirm our theoretical results

spreading speeds and traveling waves for periodic evolution systems abstract the theory of spreading speeds and traveling waves for monotone autonomous semiflows is extended to periodic semiflows in the monostable case then these abstract results are applied to a periodic system modeling man environment man epidemics a periodic time delayed and diffusive equation and a periodic reaction diffusion equation on a cylinder

singularity formation for prandtl s equations abstract we consider prandtl s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method we classify van dommelen and shen s singularity as a cubic root singularity we introduce a class of initial data uniformly bounded in h 1 which have a dipole singularity in the complex plane these data lead to a solution blow up whose time can be made arbitrarily short within the class this is numerical evidence of the ill posedness of the prandtl equations in h 1 the presence of a small viscosity in the streamwise direction changes the behavior of the singularities they stabilize at a distance from the real axis which depends on the amount of viscosity we show that the van dommelen and shen singularity and the singularity predicted by e and engquist in w e b engquist blowup of the solutions to the unsteady prandtl s equations comm pure appl math 50 1997 1287 1293 have different complex structures

traveling wavefronts for time delayed reaction diffusion equation ii nonlocal nonlinearity abstract in this paper we study a class of time delayed reaction diffusion equation with local nonlinearity for the birth rate for all wavefronts with the speed c c where c 0 is the critical wave speed we prove that these wavefronts are asymptotically stable when the initial perturbation around the traveling waves decays exponentially as x but the initial perturbation can be arbitrarily large in other locations this essentially improves the stability results obtained by mei so li and shen m mei j w h so m y li s s p shen asymptotic stability of traveling waves for the nicholson s blowflies equation with diffusion proc roy soc edinburgh sect a 134 2004 579 594 for the speed c 2 d m e p d m with small initial perturbation and by lin and mei c k lin m mei on travelling wavefronts of the nicholson s blowflies equations with diffusion submitted for publication for c c with sufficiently small delay time r 0 the approach adopted in this paper is the technical weighted energy method used in m mei j w h so m y li s s p shen asymptotic stability of traveling waves for the nicholson s blowflies equation with diffusion proc roy soc edinburgh sect a 134 2004 579 594 but inspired by gourley s a gourley linear stability of travelling fronts in an age structured reaction diffusion population model quart j mech appl math 58 2005 257 268 and based on the property of the critical wavefronts the weight function is carefully selected and it plays a key role in proving the stability for any c c and for an arbitrary time delay r 0

nelkin scaling for the burgers equation and the role of high precision calculations nelkin scaling the scaling of moments of velocity gradients in terms of the reynolds number is an alternative way of obtaining inertial range information it is shown numerically and theoretically for the burgers equation that this procedure works already for reynolds numbers of the order of 100 or even lower when combined with a suitable extended self similarity technique at moderate reynolds numbers for the accurate determination of scaling exponents it is crucial to use higher than double precision similar issues are likely to arise for three dimensional navier stokes simulations

sharp oscillatory traveling waves of structured population dynamics model with degenerate diffusion abstract we consider a population dynamics model with degenerate diffusion and time delay we discover sharp oscillatory waves with sharp edges and non decaying oscillations arising from density dependent dispersal and reproduction with age structure degenerate diffusion and bad effect of time delay prevent the use of existing approaches here we develop a new delayed iteration framework to show the existence of these peculiar waves in particular the estimate of the admissible wave speed is highly nontrivial we employ a new phase transform method coupled with the detailed analysis of phase energy furthermore we give a complete characterization of the dynamical behaviors of various kinds of waves our results indicate that simple invasion rules can generate complex wave patterns and provide some interesting insights into the ecological dynamics

monotone traveling wavefronts of the kpp fisher delayed equation abstract in the early 2000 s gourley 2000 wu et al 2001 ashwin et al 2002 initiated the study of the positive wavefronts in the delayed kolmogorov petrovskii piskunov fisher equation u t t x u t x u t x 1 u t h x u 0 x r m since then this model has become one of the most popular objects in the studies of traveling waves for the monostable delayed reaction diffusion equations in this paper we give a complete solution to the problem of existence and uniqueness of monotone waves in eq we show that each monotone traveling wave can be found via an iteration procedure the proposed approach is based on the use of special monotone integral operators which are different from the usual wu zou operator and appropriate upper and lower solutions associated to them the analysis of the asymptotic expansions of the eventual traveling fronts at infinity is another key ingredient of our approach

spreading speed and wavefronts for parabolic functional differential equations with spatio temporal delays the existence of traveling wavefronts and the spreading speed for a kind of general reaction diffusion system with spatio temporal delays are investigated we obtain the conclusion that there is c 0 being the asymptotic speed of spread and the minimal speed for the system some more delicate analysis techniques have been developed to make the forms of the kernel function g t x and the function f more flexible in our paper a pair of super solution and sub solution have been constructed in order that the existence of traveling wavefronts can be guaranteed there is a biological model as an example in this article which is coincident with this system

traveling wavefronts for a time delayed non newtonian filtration equation abstract this paper is concerned with traveling wavefronts for the evolutionary p laplacian with time delay we first establish some necessary and sufficient conditions for the existence of monotone nondecreasing and nonincreasing traveling wavefronts which correspond to definite wave speeds c 1 and c 2 respectively moreover special attention is also paid to the asymptotic behavior of traveling fronts connecting two uniform steady states

traveling waves for nonlinear cellular neural networks with distributed delays abstract in this paper we will establish the existence and nonexistence of traveling waves for nonlinear cellular neural networks with finite or infinite distributed delays the dynamics of each given cell depends on itself and its nearest m left or l right neighborhood cells where delays exist in self feedback and left or right neighborhood interactions our approach is to use schauder s fixed point theorem coupled with upper and lower solutions of the integral equation in a suitable banach space further we obtain the exponential asymptotic behavior in the negative infinity and the existence of traveling waves for the minimal wave speed by the limiting argument our results improve and cover some previous works

travelling waves for a biological reaction diffusion model with spatio temporal delay in this paper we consider the reaction diffusion equations with spatio temporal delay which models the microbial growth in a flow reactor nonlocal spatial term a weighted average in space arises when the individuals have not necessarily been at the same point in space at previous time by employing linear chain technique geometric singular perturbation and the center manifold theorem we prove that the steady travelling wave does not only persist but also it looks qualitatively the same as it do with no delay at all under the introduction of delays at least for small delay

existence and nonexistence of monotone traveling waves for the delayed fisher equation abstract in this paper a new approach based on a shooting method in a half line coupled with the technique of upper lower solution pair is used to study the existence and nonexistence of monotone waves for one form of the delayed fisher equation that does not have the quasimonotonicity property a necessary and sufficient condition is provided this new method can be extended to investigate many other nonlocal and non monotone delayed reaction diffusion equations

persistence of wavefronts in delayed nonlocal reaction diffusion equations we develop a perturbation argument based on existing results on asymptotic autonomous systems and the fredholm alternative theory that yields the persistence of traveling wavefronts for reaction diffusion equations with nonlocal and delayed nonlinearities when the time lag is relatively small this persistence result holds when the nonlinearity of the corresponding ordinary reaction diffusion system is either monostable or bistable we then illustrate this general result using five different models from population biology epidemiology and bio reactors

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

spreading speed and traveling waves for a multi type sis epidemic model abstract the theory of asymptotic speeds of spread and monotone traveling waves for monotone semiflows is applied to a multi type sis epidemic model to obtain the spreading speed c and the nonexistence of traveling waves with wave speed c c then the method of upper and lower solutions is used to establish the existence of monotone traveling waves connecting the disease free and endemic equilibria for c c this shows that the spreading speed coincides with the minimum wave speed for monotone traveling waves we also give an affirmative answer to an open problem presented by rass and radcliffe l rass j radcliffe spatial deterministic epidemics math surveys monogr 102 amer math soc providence ri 2003

complex singularities and the lorenz attractor the lorenz attractor is one of the best known examples of applied mathematics however much of what is known about it is a result of numerical calculations and not of mathemat ical analysis as a step toward mathematical analysis we allow the time variable in the three dimensional lorenz system to be complex hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane knowledge of singularities being fundamental to any investigation in the complex plane we build upon earlier work and give a complete and consistent formal development of complex singular ities of the lorenz system using the psi series the psi series contain two undetermined constants in addition the location of the singularity is undetermined as a consequence of the autonomous nature of the lorenz system we prove that the psi series converge using a technique that is simpler and more powerful than that of hille thus implying a two parameter family of singular solutions of the lorenz system we pose three ques tions answers to which may bring us closer to understanding the connection of complex singularities to lorenz dynamics

on convolved generalized fibonacci and lucas polynomials we define the convolved h x fibonacci polynomials as an extension of the classical convolved fibonacci numbers then we give some combinatorial formulas involving the h x fibonacci and h x lucas polynomials moreover we obtain the convolved h x fibonacci polynomials from a family of hessenberg matrices

traveling wave solutions in a delayed diffusive competition system abstract in this paper we first investigate the existence of traveling wave fronts in a delayed diffusive competition system by constructing a pair of upper and lower solutions then we consider the asymptotic behavior of traveling wave solutions at the minus plus infinity by means of the bilateral laplace transform finally the monotonicity and uniqueness up to the translation of traveling wave solutions are also obtained by the strong comparison principle and the sliding method

pyramidal traveling fronts in a nonlocal delayed diffusion equation abstract recently there have been great progresses on the study of nonplanar traveling wave solutions of reaction diffusion equations in this paper we study the pyramidal traveling fronts of nonlocal delayed diffusion equation in r n with n 2 by using the comparison principle and establishing super and subsolution since the effect of nonlocal delay we show that nonlocal delayed diffusion equation in n dimensional space has a pyramidal traveling front u t x v x x n s t toward x n axis for each s c 0 in particular two dimensional traveling curved front and three dimensional pyramidal fronts for nonlocal delayed diffusion equation in r 2 and r 3 are also established respectively moreover we also extend our results to generally nonlocal delayed reaction diffusion equation

critical traveling waves in a diffusive disease model abstract in this paper the existence of a non trivial positive and bounded critical traveling wave solution of a diffusive disease model whose reaction system has infinitely many equilibria is obtained for the first time this gives an affirmative answer to an open problem left in wang et al 2012 19 our result shows that the critical traveling wave in this model is a mix of front s component and pulse i component type

traveling wave solutions in n dimensional delayed reaction diffusion systems with mixed monotonicity this paper deals with the existence of traveling wave solutions for n components delayed reaction diffusion systems with mixed monotonicity based on a certain kind of mixed quasimonotonicity reaction terms of higher dimension we propose new conditions on the reaction terms and new definitions of upper and lower solutions by using schauder s fixed point theorem and a cross iteration scheme we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions the general results obtained will be applied to type k monotone diffusive lotka volterra systems of higher dimension

a constructive approach to regularity of lagrangian trajectories for incompressible euler flow in a bounded domain the 3d incompressible euler equations are an important research topic in the mathematical study of fluid dynamics not only is the global regularity for smooth initial data an open issue but the behaviour may also depend on the presence or absence of boundaries for a good understanding it is crucial to carry out besides mathematical studies high accuracy and well resolved numerical exploration such studies can be very demanding in computational resources but recently it has been shown that very substantial gains can be achieved first by using cauchy s lagrangian formulation of the euler equations and second by taking advantage of analyticity results of the lagrangian trajectories for flows whose initial vorticity is holder continuous the latter has been known for about 20 years serfati in j math pures appl 74 95 104 1995 but the combination of the two which makes use of recursion relations among time taylor coefficients to obtain constructively the time taylor series of the lagrangian map has been achieved only recently frisch and zheligovsky in commun math phys 326 499 505 2014 podvigina et al in j comput phys 306 320 342 2016 and references therein here we extend this methodology to incompressible euler flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity non constructive regularity results for these cases have already been obtained by glass et al ann sci ec norm sup 45 1 51 2012 using the invariance of the boundary under the lagrangian flow we establish novel recursion relations that include contributions from the boundary this leads to a constructive proof of time analyticity of the lagrangian trajectories with analytic boundaries which can then be used subsequently for the design of a very high order cauchy lagrangian method