on bounded positive stationary solutions for a nonlocal fisher kpp equation abstract we study the existence of stationary solutions for a nonlocal version of the fisher kolmogorov petrovskii piscounov fkpp equation the main motivation is a recent study by berestycki et al 2009 where the nonlocal fkpp equation has been studied and it was shown for the spatial domain r and sufficiently small nonlocality that there are only two bounded non negative stationary solutions here we provide a similar result for r d using a completely different approach in particular an abstract perturbation argument is used in suitable weighted sobolev spaces one aim of the alternative strategy is that it can eventually be generalized to obtain persistence results for hyperbolic invariant sets for other nonlocal evolution equations on unbounded domains with small nonlocality i e to improve our understanding in applications when a small nonlocal influence alters the dynamics and when it does not

waves and bumps in neuronal networks with axo dendritic synaptic interactions we consider a firing rate model of a neuronal network continuum that incorporates axo dendritic synaptic processing and the finite conduction velocities of action potentials the model equation is an integral one defined on a spatially extended domain apart from a spatial integral mixing the network connectivity function with space dependent delays arising from non instantaneous axonal communication the integral model also includes a temporal integration over some appropriately identified distributed delay kernel these distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs they are interpreted as green s functions of some linear dierential operator exploiting this green s function description we discuss formal reductions of this non local system to equivalent partial dierential equation pde models we distinguish between those spatial connectivity functions that give rise to local pde models and those that give rise to pde models with delayed non local terms for cases in which local pdes are derived we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions we also calculate exact solutions parameterized by axonal conduction velocity for the heaviside firing rate function the sigmoidal firing rate function in the limit of infinte gain the inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses which we study analytically and numerically in terms of a global homoclinic orbit finally we consider the impact of dendritic interactions on waves and on static spatially localized solutions exact analysis for

localized states in the conserved swift hohenberg equation with cubic nonlinearity the conserved swift hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state the resulting phase field crystal model describes a variety of spatially localized structures in addition to different spatially extended periodic structures the location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases and may lie outside of the binodal for these states the results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential and are expected to carry over to related models with a conserved order parameter 2013 american physical society

discrete solitons in nonlinear schrodinger lattices with a power law nonlinearity abstract we study the discrete nonlinear schrodinger lattice model with the onsite nonlinearity of the general form u 2 u we systematically verify the conditions for the existence and stability of discrete solitons in the one dimensional version of the model predicted by means of the variational approximation va and demonstrate the following monostability of fundamental solitons fss in the case of the weak nonlinearity 2 1 3 68 bistability in a finite range of values of the soliton s power for 3 68 2 1 5 and the presence of a threshold minimum norm of the fs for 2 1 5 we also perform systematic numerical simulations to study higher order solitons in the same general model i e bound states of the fss while all in phase bound states are unstable stability regions are identified for antisymmetric double solitons and their triple counterparts these numerical findings are supplemented by an analytical treatment of the stability problem which allows quantitively accurate predictions for the stability features of such multipulses when these waveforms are found to be unstable we show by means of direct simulations that they self trap into a persistent lattice breather or relax into a stable fs or sometimes decay completely

embedded solitons in lagrangian and semi lagrangian systems we develop the technique of the variational approximation va for solitons in two directions first one may have a physical model which does not admit the usual lagrangian representation as some terms were discarded for various reasons for instance the second harmonic generation shg model considered here which includes the kerr nonlinearity lacks the usual lagrangian representation if one ignores the kerr nonlinearity of the second harmonic as compared to that of the fundamental however we show that with a natural modification one may still apply the va to those seemingly flawed systems as efficiently as it applies to their fully lagrangian counterparts we call such models that do not admit the usual lagrangian representation semi lagrangiansystems second we show that upon adding an infinitesimal tail that does not vanish at infinity to a usual soliton ansatz one can obtain an analytical criterion which within the framework of va gives a condition for finding embedded solitons ess i e isolated truly localized solutions existing inside the continuous spectrum of the radiation modes the criterion takes a form of orthogonality of the radiation mode in the infinite tail to the soliton core to test the criterion we have applied it to both the semi lagrangian truncated version of the shg model and to the same model in its full form in the former case the criterion combined with va for the soliton proper yields an exact solution for the es in the latter case the criterion selects the es with a relative error 1 2003 elsevier b v all rights reserved

sharp global nonlinear stability for temperature dependent viscosity convection with regard to the dependence of viscosity on temperature an unconditional non linear energy stability analysis for thermal convection according to navier stokes theory has not yet been developed we here analyse three models of fluid behaviour proposed by o a ladyzhenskaya we show that by using these theories we can develop an unconditional analysis directly two of the models lead directly to an unconditional development employing l2 theory while the third necessitates the introduction of a generalized energy also involving an l3 term the nonlinear stability boundaries in all three cases are sharp when compared with the instability thresholds of linear theory

a rigorous reduction of the l2 stability of the solutions to a nonlinear binary reaction diffusion system of pde s to the stability of the solutions to a linear binary system of ode s abstract a basic peculiar lyapunov functional v is introduced for the dynamical systems generated by a pair of nonlinear reaction diffusion pde s with nonconstant coefficients the sign of v and of its derivative along the solutions is linked through an immediate simple relation to the eigenvalues by using v and the l 2 norm the non linear l 2 stability instability is rigorously reduced to the stability instability of the solutions to a linear binary system of ode s

heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible hamiltonian hopft bifurcation this paper considers an unfolding of a degenerate reversible 1 1 resonance or hamiltonian hopf bifurcation for four dimensional systems of time reversible ordinary differential equations odes this bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become imaginary pairs the degeneracy occurs via the vanishing of a normal form coefficient q2 0 that determines whether the bifurcation is super or subcritical of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits a partial unfolding of such solutions already occurs in the work of dias and iooss eur j mech b fluids 15 1996 367 393 given a sign of the coefficient of a higher order term q4 0 and 1 q2 0 the normal form then shows a small amplitude bifurcating branch of homoclinic solutions terminating at a heteroclinic connection to a simple periodic orbit a genericity argument shows that this connection is not structurally stable and should break into a pair of heteroclinic tangencies this is confirmed by numerics which shows that branches of the simplest homoclinic orbits undergo an infinite snaking sequence of limit points accumulating on the parameter values of the two tangencies at each limit point the homoclinic orbit generates another bump close to the periodic orbit as q2 is further decreased from zero the snake widens until at a critical value a branch is formed of heteroclinic connections from the origin to a nontrivial equilibrium a kink for q2 less than this value the kinks heteroclinic tangencies and snake like curves no longer occur

collective coordinates and length scale competition in spatially inhomogeneous soliton bearing equations perturbed one dimensional integrable i e soliton bearing equations arise in many applied contexts when trying to improve the usually highly idealized description of problems of interest in terms of the purely integrable equations in particular when the assumption of perfect homogeneity is dropped to account for unavoidable impurities or defects perturbations depending on the spatial coordinate must be added to the original equation in this review we use the one dimensional sine gordon sg equation perturbed by a spatially periodic term as a generic paradigm to discuss the main perturbative techniques available for the study of this class of problems to place the work in context we summarize the approaches developed to date and focus on the collective coordinate approach as one of the most useful tools we introduce several versions of this perturbative method and relate them to more involved procedures we analyze in detail the application to the sg equation but the procedure is very general to illustrate this other examples of the application of collective coordinates are briefly revisited in our case study this approach helps us identify perturbative and nonperturbative regimes yielding a very simple picture of the former beyond perturbative calculations the same example of the inhomogeneous sg equation allows us to introduce a phenomenon termed length scale competition which we show to be a rather general mechanism for the appearance of complex spatiotemporal behavior in perturbed integrable systems as other instances discussed in the review show such nonperturbative results are obtained by means of numerical simulations of the full perturbed problem numerical linear stability analysis is also used to clarify the origins of the instability originated by this competition to complement our description of the techniques employed in these studies computational details of our numerical simulations are also included finally the paper closes with a discussion of the above ideas and a speculative outlook on general questions concerning the interplay of nonlinearity with disorder

dynamic instabilities in scalar neural field equations with space dependent delays in this paper we consider a class of scalar integral equations with a form of space dependent delay these non local models arise naturally when modelling neural tissue with active axons and passive dendrites such systems are known to support a dynamic oscillatory turing instability of the homogeneous steady state in this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability the appropriate amplitude equations are found to be the coupled mean field ginzburg landau equations describing a turing hopf bifurcation with modulation group velocity of o 1 importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio temporal kernels defining the neural field equation of interest indeed our results cover not only models with axonal or dendritic delays but those which are described by a more general distribution of delayed spatio temporal interactions we illustrate the predictive power of this form of analysis with comparison against direct numerical simulations paying particular attention to the competition between standing and travelling waves and the onset of benjamin feir instabilities

the effect of density dependent dispersal on the stability of populations abstract the nonlinear stability of two populations diffusing in a common bounded domain with nonlinear density dependent dispersal of a general type is studied under robin boundary conditions the stability and instability of the positive equilibrium are studied by the linear eigenvalues method the nonlinear stability is studied with the lyapunov direct method in a particular case global stability has been obtained

a nonlinear diffusion system with convection abstract this paper deals with a coupled nonlinear diffusion system with convection a characteristic matrix equation covering all the six nonlinear exponents of the system is introduced to get a simple description for the blow up criteria of the system in addition the influence of the four coefficients of the diffusion and the convection as well as the geometry of on global boundedness or not of the solutions is shown clearly especially for the situation with the critical sign of the characteristic parameters

neron models lie algebras and reduction of curves of genus one let k be a discrete valuation field let ok denote the ring of integers of k and let k be the residue field of ok of characteristic p 0 let s specok let x k be a smooth geometrically connected projective curve of genus 1 over k denote by ek the jacobian of x k let x s and e s be the minimal regular models of x k and ek respectively in this article we investigate the possible relationships between the special fibers xk and ek in doing so we are led to study the geometry of the picard functor pic x s when x s is not necessarily cohomologically flat as an application of this study we are able to prove in full generality a theorem of gordon on the equivalence between the artin tate and birch swinnertondyer conjectures recall that when k is algebraically closed the special fibers of elliptic curves are classified according to their kodaira type which is denoted by a symbol t in i n n z 0 ii ii iii iii iv iv given a type t and a positive integer m we denote by mt the new type obtained from t by multiplying all the multiplicities of t by m when k is algebraically closed the relationships between the type of a curve of genus 1 and the type of its jacobian can be summarized as follows

the swift hohenberg equation with a nonlocal nonlinearity it is well known that aspects of the formation of localised states in a one dimensional swift hohenberg equation can be described by ginzburg landau type envelope equations this paper extends these multiple scales analyses to cases where an additional nonlinear integral term in the form of a convolution is present the presence of a kernel function introduces a new lengthscale into the problem and this results in additional complexity in both the derivation of envelope equations and in the bifurcation structure when the kernel is short range weakly nonlinear analysis results in envelope equations of standard type but whose coefficients are modified in complicated ways by the nonlinear nonlocal term nevertheless these computations can be formulated quite generally in terms of properties of the fourier transform of the kernel function when the lengthscale associated with the kernel is longer our method leads naturally to the derivation of two different novel envelope equations that describe aspects of the dynamics in these new regimes the first of these contains additional bifurcations and unexpected loops in the bifurcation diagram the second of these captures the stretched out nature of the homoclinic snaking curves that arises due to the nonlocal term

a nonlinear fourth order diffusion problem convergence to the steady state and non negativity of solutions the paper deals with nonlinear diffusion both time dependent and time independent the spatial terms in the partial differential equation p d e contain a second order nonlinear part where the non negative diffusivity depends on the dependent variable and a fourth order linear part in the context of non null time independent boundary conditions convergence of the unsteady to the steady state is established the second part of the paper discusses criteria on data ensuring non negativity of the solutions this is done for the steady state irrespective of the spatial dimension and it is done for the unsteady state for the one dimensional rectilinear case only using a result from the first part of the paper

variational approximation and the use of collective coordinates we consider propagating spatially localized waves in a class of equations that contain variational and nonvariational terms the dynamics of the waves is analyzed through a collective coordinate approach motivated by the variational approximation we show that there is a natural choice of projection onto collective variables for reducing the governing nonlinear partial differential equation pde to coupled ordinary differential equations odes this projection produces odes whose solutions are exactly the stationary states of the effective lagrangian that would be considered in applying the variational approximation method we illustrate our approach by applying it to a modified fisher equation for a traveling front containing a non constant coefficient nonlinear term we present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the pde much more closely than previously proposed projections

variational approximations to homoclinic snaking in continuous and discrete systems localized structures appear in a wide variety of systems arising from a pinning mechanism due to the presence of a small scale pattern or an imposed grid when there is a separation of length scales the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods we show how this behavior can be obtained using a variational method for two systems in the case of the quadratic cubic swift hohenberg equation this gives results that are in agreement with recent work using exponential asymptotics in addition the method is applied to a discrete system with cubic quintic nonlinearity giving results that agree well with numerical simulations

weakly subcritical stationary patterns eckhaus instability and homoclinic snaking the transition from subcritical to supercritical stationary periodic patterns is described by the one dimensional cubic quintic ginzburg landau equation at a axx i a1 a 2 ax a2a 2 a b a 2 a a 4 a w here a x t represents the pattern amplitude and the coefficients a1 a2 a ndb are real the conditions for eckhaus instability of periodic solutions are determined and the resulting spatially modulated states are computed some of these evolve into spatially localized structures in the vicinity of a maxwell point while others resemble defect states the results are used to shed light on the behavior of localized structures in systems exhibiting homoclinic snaking during the transition from subcriticality to supercriticality

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude

on the validity of the variational approximation in discrete nonlinear schrodinger equations abstract the variational approximation is a well known tool to approximate localized states in nonlinear systems in the context of a discrete nonlinear schrodinger equation with a small coupling constant we prove error estimates for the variational approximations of site symmetric bond symmetric and twisted discrete solitons this is shown for various trial configurations which become increasingly more accurate as more parameters are taken it is also shown that the variational approximation yields the correct spectral stability result and controls the oscillatory dynamics of stable discrete solitons for long but finite time intervals

internal modes of discrete solitons near the anti continuum limit of the dnls equation discrete solitons of the discrete nonlinear schrodinger dnls equation are compactly supported in the anti continuum limit of the zero coupling between lattice sites eigenvalues of the linearization of the dnls equation at the discrete soliton determine its spectral stability small eigenvalues bifurcating from the zero eigenvalue near the anti continuum limit were characterized earlier for this model here we analyze the resolvent operator and prove that it is bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti continuum limit this result rules out the existence of internal modes neutrally stable eigenvalues of the discrete spectrum near the anti continuum limit

reaction diffusion front crossing a local defect the interaction of a zeldovich frank kamenetsky reaction diffusion front with a localized defect is studied numerically and analytically for the analysis we start from conservation laws and develop simple collective variable ordinary differential equations for the front position and width their solutions are in good agreement with the solutions of the full problem finally using this reduced model we explain the pinning of the front on a large defect and obtain a quantitative criterion

approximating the ideal free distribution via reaction diffusion advection equations abstract we consider reaction diffusion advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness where fitness is defined as a logistic local population growth rate we show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics the modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution the analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability the models are quasilinear parabolic equations with nonlinear boundary conditions

asymptotic and other properties of a nonlinear diffusion model let r 3 be a sufficiently smooth bounded fixed domain ensuring the validity of divergence like theorems and let us consider the initial boundary value problem i b v p u t delta f u quad x t in omega times r 1 u t left x 0 right u 0 left x right quad x in omega 2 u t left x t right u 1 left x right quad x in partial omega times r 3 where f c 2 r and u 0 in c left overline omega right u 1 c are assigned functions

exceptional curves on rational numerically elliptic surfaces a numerically elliptic surface is a complete smooth algebraic surface x over an algebraically closed field k with a proper morphism f x c to a smooth curve c such that the general fiber off is an integral curve of arithmetic genus 1 if the generic fiber is smooth the surface is called elliptic otherwise it is called quasi elliptic the latter only can occur if k has characteristic 2 or 3 in which case the general fiber is a rational curve with an ordinary cusp bm if no fiber off contains an exceptional cwue i e a smooth irreducible curve isomorphic to fp and having self intersection l then x is said to be minimal all elliptic surfaces will hereafter be assumed to be minimal one says that an elliptic surface x is jacobian if the smooth points of the generic fiber x comprise the jacobian curve of x it is equivalent for x to have a rational point or for f to have a section for a rational jacobian numerically elliptic surface the exceptional curves are precisely the sections of the libration which provides a tool by which an enumeration of the exceptional curves on a rational jacobian numerically elliptic surface can be carried out cf mp hl mop whether jacobian or not a rational

variational approximations to homoclinic snaking we investigate the snaking of localized patterns seen in numerous physical applications using a variational approximation this method naturally introduces the exponentially small terms responsible for the snaking structure which are not accessible via standard multiple scales asymptotic techniques we obtain the symmetric snaking solutions and the asymmetric ladder states and also predict the stability of the localized states the resulting approximate formulas for the width of the snaking region show good agreement with numerical results

localized states in the generalized swift hohenberg equation the swift hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states the presence of these states is related to a phenomenon called homoclinic snaking numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states the evolution of the localized states once they lose stability is illustrated using direct simulations in time i introduction ever since the observation that the subcritical complex ginzburg landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states the presence of these states has important consequences for other systems described by partial differential equations on the line since the ginzburg landau equation describes the behavior of such systems near bifurcation from the trivial state of the system specifically the complex ginzburg landau equation describes the evolution of a long wavelength oscillatory instability as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry in contrast near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real ginzburg landau equation and this equation possesses only unstable spatially localized states it is of interest therefore to examine what happens to these unstable states at larger amplitude where the real ginzburg landau equation no longer provides an adequate description of the system in this paper we show that the localized states can become stable at such amplitudes and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions we are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1 1 resonance and use this theory to construct a large number of such states the stability properties of these states are also determined and the evolution of nonstationary localized states is studied by numerical integration in time it is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation the theory was originally developed in the context of water waves where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ode the resulting localized states are called solitary waves and in some cases turn out to be solitons kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area specifically the ode is viewed as a dynamical system in space and localized states are sought as homoclinic orbits connecting the trivial state to itself whether such orbits are possible depends in part on the stability properties of the trivial state eigenvalues with positive real part indicate that a nontrivial state can grow from x while eigenvalues with negative real part indicate that such a state may return under appropriate conditions back to the trivial state as x the spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system in many cases and in particular in the case considered here the ode is reversible as a result the bifurcations that are encountered as a parameter is increased are nongeneric in the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1 1 resonance the unfolding of this resonance has been worked out in detail by iooss and peroueme 3 and can be used to understand the appearance of a variety of homoclinic orbits in this system and hence of localized states with different spatial structure

stability of discrete solitons in nonlinear schrodinger lattices we consider the discrete solitons bifurcating from the anti continuum limit of the discrete nonlinear schrodinger nls lattice the discrete soliton in the anti continuum limit represents an arbitrary finite superposition of in phase or anti phase excited nodes separated by an arbitrary sequence of empty nodes by using stability analysis we prove that the discrete solitons are all unstable near the anti continuum limit except for the solitons which consist of alternating anti phase excited nodes we classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons

surfaces of globally f regular and f split type we prove that normal projective surfaces of dense globally f split type respectively globally f regular type are of calabi yau type respectively fano type

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

modulated traveling fronts for a nonlocal fisher kpp equation a dynamical systems approach abstract we consider a nonlocal generalization of the fisher kpp equation in one spatial dimension as a parameter is varied the system undergoes a turing bifurcation we study the dynamics near this turing bifurcation our results are two fold first we prove the existence of a two parameter family of bifurcating stationary periodic solutions and derive a rigorous asymptotic approximation of these solutions we also study the spectral stability of the bifurcating stationary periodic solutions with respect to almost co periodic perturbations second we restrict to a specific class of exponential kernels for which the nonlocal problem is transformed into a higher order partial differential equation in this context we prove the existence of modulated traveling fronts near the turing bifurcation that describe the invasion of the turing unstable homogeneous state by the periodic pattern established in the first part both results rely on a center manifold reduction to a finite dimensional ordinary differential equation

multistable solitons in the cubic quintic discrete nonlinear schrodinger equation we analyze the existence and stability of localized solutions in the one dimensional discrete nonlinear schrodinger dnls equation with a combination of competing self focusing cubic and defocusing quintic onsite nonlinearities we produce a stability diagram for different families of soliton solutions that suggests the co existence of infinitely many branches of stable localized solutions bifurcations that occur with an increase in the coupling constant are studied in a numerical form a variational approximation is developed for accurate prediction of the most fundamental and next order solitons together with their bifurcations salient properties of the model which distinguish it from the well known cubic dnls equation are the existence of two different types of symmetric solitons and stable asymmetric soliton solutions that are found in narrow regions of the parameter space the asymmetric solutions appear from and disappear back into the symmetric ones via loops of forward and backward pitchfork bifurcations

on torsors under elliptic curves and serre s pro algebraic structures let mathcal o k be a complete discrete valuation ring with algebraically closed residue field of positive characteristic and field of fractions k let x k be a torsor under an elliptic curve a k over k x the proper minimal regular model of x k over s hbox spec mathcal o k and j the identity component of the neron model of mathrm pic x k k 0 we study the canonical morphism q mathrm pic 0 x s rightarrow j which extends the natural isomorphism on generic fibres we show that q is pro algebraic in nature with a construction that recalls serre s work on local class field theory furthermore we interpret our results in relation to shafarevich s duality theory for torsors under abelian varieties

snakes and ladders in an inhomogeneous neural field model continuous neural field models with inhomogeneous synaptic connectivities are known to support traveling fronts as well as stable bumps of localized activity we analyze stationary localized structures in a neural field model with periodic modulation of the synaptic connectivity kernel and find that they are arranged in a snakes and ladders bifurcation structure in the case of heaviside firing rates we construct analytically symmetric and asymmetric states and hence derive closed form expressions for the corresponding bifurcation diagrams we show that the ideas proposed by beck and co workers to analyze snaking solutions to the swift hohenberg equation remain valid for the neural field model even though the corresponding spatial dynamical formulation is non autonomous we investigate how the modulation amplitude affects the bifurcation structure and compare numerical calculations for steep sigmoidal firing rates with analytic predictions valid in the heaviside limit

the evolution to a steady state for a porous medium model abstract an initial boundary value problem is considered for a nonlinear diffusion equation the diffusivity being a function of the dependent variable dirichlet boundary conditions independent of time are considered and positive solutions are assumed this paper is mainly concerned with the rate of convergence in time of the unsteady to the steady state this is done by obtaining an upper estimate for a positive definite integral measure of the perturbation i e unsteady steady state using differential inequality techniques a previous result is recalled where the diffusivity k n n being a positive constant appropriate to mass transport or filtration in a porous medium the present paper treats an alternative model sharing some of the characteristics of the previous one k e 1 being non negative the paper concludes by considering a backwards in time initial boundary value problem for the perturbation amenable to the same techniques and establishes that the solution ceases to exist beyond a critical computable time

variational approximations in discrete nonlinear schrodinger equations with next nearest neighbor couplings solitons of a discrete nonlinear schrodinger equation which includes the next nearest neighbor interactions are stud ied by means of a variational approximation and numerical computations a large family of multi humped solutions including those with a nontrivial phase structure which are a feature particular to the next nearest neighbor interac tion model are accurately predicted by the variational approximation bifurcations linking solutions with the trivial and nontrivial phase structures are also captured remarkably well including a prediction of critical parameter values

multiple fibers on rational elliptic surfaces our main result theorem 0 1 classifies multiple fibers on rational elliptic surfaces over algebraically closed fields of arbitrary characteristic one result of this is the existence in positive characteristics of tame multiple fibers of additive type for several of the kodaira fiber types for which no examples were previously known 0 introduction we work in this paper over an algebraically closed field k of arbitrary characteristic p an elliptic surface f x b is a fibration of a complete smooth surface x over a complete smooth curve b such that the generic fiber is a smooth curve of genus one it is natural to wonder what kind of special fibers arise we answer this in the case of rational surfaces giving as a result examples of tame multiple fibers of additive type for every p o and for all such types except ib b 4 cf 5 to our knowledge the only example heretofore recognized is of multiplicity 2 and kodaira type iox due to katsura ka l1 although since our results were obtained we have seen a preprint of katsura and ueno ku giving other examples mostly for p 7 2 3 and kodaira type not ib b 0 any special fiber is an effective divisor on x and so can be written as mf menjcj where each curve cj is reduced and irreducible and the coefficients are positive integers with g c d nj 1 the positive integer m is the rnultiplicity of the fiber mf which is said to be a multiple fiber whenever m 1 the form of the divisor f enjcj is very restricted such a divisor is an indecomposable curve of canonical type mu a topological classification of all such curves has been given by kodaira ko cf 5 we refer to the classes which arise in this classification as the kodaira fiber types more generally the kodaira type of the fiber mf is mt where t is the type of f over an algebraically closed field k it is known that any multiple fiber mf of multiplicity m on a surface x is of the form mt where t is a kodaira type and m is divisible by and in characteristic 0 equal to the order of the normal bundle ox f s of e pic f of f in x but the extent to which these conditions are sufficient for there to occur on some elliptic surface a multiple fiber of type mt is not known of particular interest to us here are the tame multiple fibers of additive type these being fibers mf of multiplicity m where pic f is the additive group ga of the ground field k and m equals the order in ga f ox f s of these are possible only if p the characteristic of k is positive since ga only then has ptorsion but their occurrence is poorly understood received by the editors march 21 1987 1980 mathematzcs subject classzficatzon 1985 re szon primary 14j25 14m20 14d99

asymptotic behavior of the nonlinear diffusion equation nt n 1nx x the asymptotic behavior of the equation nt ln n xx is studied on the finite interval 0 x 1 with the boundary conditions n 0 t n 1 t n0 and initial data n x 0 n0 we prove that asymptotically ln n x t n0 a exp 2t n0 21 2 sin x and also provide rigorous upper and lower bounds on the asymptotic amplitude a in terms of integrals of nonlinear functions of the initial data the rigorous bounds are compared to values of a obtained from computer experiments the lower bound l 23 2 exp li 1 q where li is the logarithmic integral is euler s constant and q 2 f n x 0 n0 1 sin x dx is found to be the best known estimate of a

analysis of nonlocal neural fields for both general and gamma distributed connectivities abstract this work studies the stability of equilibria in spatially extended neuronal ensembles we first derive the model equation from statistical properties of the neuron population the obtained integro differential equation includes synaptic and space dependent transmission delay for both general and gamma distributed synaptic connectivities the latter connectivity type reveals infinite finite and vanishing self connectivities the work derives conditions for stationary and nonstationary instabilities for both kernel types in addition a nonlinear analysis for general kernels yields the order parameter equation of the turing instability to compare the results to findings for partial differential equations pdes two typical pde types are derived from the examined model equation namely the general reaction diffusion equation and the swift hohenberg equation hence the discussed integro differential equation generalizes these pdes in the case of the gamma distributed kernels the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges as a novel finding we obtain turing instabilities in fields with local inhibition lateral excitation while wave instabilities occur in fields with local excitation and lateral inhibition numerical simulations support the analytical results

validity of amplitude equations for nonlocal nonlinearities amplitude equations are used to describe the onset of instability in wide classes of partial differential equations pdes one goal of the field is to determine simple universal generic pdes to which many other classes of equations can be reduced at least on a sufficiently long approximating time scale in this work we study the case when the reaction terms are nonlocal in particular we consider quadratic and cubic convolution type nonlinearities as a benchmark problem we use the swift hohenberg equation the resulting amplitude equation is a ginzburg landau pde where the coefficients can be calculated from the kernels our proof relies on separating critical and noncritical modes in fourier space in combination with suitable kernel bounds amplitude equations are used to describe the onset of instability in wide classes of partial differential equations pdes one goal of the field is to determine simple universal generic pdes to which many other classes of equations can be reduced at least on a sufficiently long approximating time scale in this work we study the case when the reaction terms are nonlocal in particular we consider quadratic and cubic convolution type nonlinearities as a benchmark problem we use the swift hohenberg equation the resulting amplitude equation is a ginzburg landau pde where the coefficients can be calculated from the kernels our proof relies on separating critical and noncritical modes in fourier space in combination with suitable kernel bounds