on stable nonconstant stationary solutions and mesoscopic patterns for fitzhugh nagumo equations in higher dimensions abstract fitzhugh nagumo equation has been studied extensively in the field of mathematical biology it has the mechanism of lateral inhibition which seems to play a big role in the pattern formation of plankton distribution we consider fitzhugh nagumo equation in high dimension and show the existence of stable nonconstant stationary solutions which have fine structures on a mesoscopic scale we construct spatially periodic stationary solutions moreover we compute the singular limit energy which suggests that the transition from planar structure to droplet pattern can occur when parameters change

periodic solutions and their connecting orbits of hamiltonian systems abstract this article deals with second order periodic hamiltonian systems we apply variational methods to obtain non constant periodic solutions for the time reversible hamiltonian systems there exist connecting orbits joining pairs of periodic solutions our methods can also be used to treat heteroclinic orbits connecting an equilibrium to a periodic solution

some mathematical aspects of the micro phase separation in diblock copolymers abstract a free energy functional of nonlocal type is considered that was originally introduced to describe the micro phase separation of diblock copolymer a mathematical framework is given to the issues of the scaling law for stationary states and the governing equation of morphology the scaling law is equivalent to finding a nice rescaling in order to have a well defined limiting interfacial problem which is free from the interfacial thickness and the total chain length of copolymer and the associated stationary problem becomes a morphology equation that governs the configuration of final patterns although the steady patterns become finer and finer in our scaling limit due to the mesoscopic nature a possible rigorous approach is presented to stability and morphological selection problems for them

a global condition for quasi random behavior in a class of conservative systems devaney has shown that an autonomous hamiltonian system in dimension 4 with an orbit homoclinic to a saddle focus equilibrium admits a chaotic behavior as soon as the homoclinic orbit is the transverse intersection of the stable and unstable manifolds in this paper we deal with two classes of saddle focus systems lagrangian systems defined on a two manifold in the presence of a gyroscopic force and fourth order systems arising in water wave theory we first establish by a standard variational method the existence of a homoclinic orbit then under a weak nondegeneracy condition we show that it gives rise to an infinite family of multibump homoclinic solutions and that the dynamics are chaotic our condition is much easier to check than transversality for example it is automatically satisfied for gyroscopic systems on a two torus for topological reasons 1996 john wiley sons inc

the toda system and clustering interfaces in the allen cahn equation we consider the allen cahn equation varepsilon 2 delta u 1 u 2 u 0 in a bounded smooth domain in mathbb r 2 under zero neumann boundary conditions where varepsilon 0 is a small parameter let 0 be a segment contained in connecting orthogonally the boundary under certain nondegeneracy and nonminimality assumptions for 0 satisfied for instance by the short axis in an ellipse we construct for any given n 1 a solution exhibiting n transition layers whose mutual distances are o varepsilon log varepsilon and which collapse onto 0 as varepsilon to 0 asymptotic location of these interfaces is governed by a toda type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary

minimal lamellar structures in a periodic fitzhugh nagumo system abstract a singular limit of a fitzhugh nagumo system leads to a nonlocal geometric variational problem with periodic boundary conditions we study the stationary lamellar set and give a criterion to select out the one with the lowest energy such an optimal structure is called a minimal lamella while the empty set or the full torus is a global minimizer for appropriate parameter regimes the minimal lamellae beat both in other circumstances the concept of minimal lamella points out that a preferred 1d mesh size is universal

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

the solution of the neumann problem in a strictly pseudoconvex siegel domain in this paper we solve the o neumann problem on 0 q forms q 0 in the strictly pseudoconvex siegel domain the metric we use is invariant under the action of the heisenberg group on the domain we give a formula for a kernel of the neumann operator the fundamental solution of the problem in 1 6 and theorem 4 2 and we prove uniqueness of the neumann operator as far as we know this is the first example of an explicit solution of the neumann problem greiner and stein gs constructed a parametrix for the problem on 0 1 forms in bounded strictly pseudoconvex domains phong p has announced an explicit construction of a parametric on 03 forms in the siegel domain in c 1 n 1 harvey and polking hp have since constructed a neumann kernel for the neumann problem on p q forms on the ball we obtain the kernel of the neumann operator by integrating the fundamental solution of the corresponding heat equation s1 2 with respect to time to carry out the integration we require estimates on the fundamental solution for large time we did not need such estimates in 1 and 2 also to prove that the resulting kernel is locally integrable we require more precise estimates for small time in sect 1 after some preliminaries we state our main theorem we then summarize the results of s1 and 2 we solve the corresponding elliptic problem on c 1 in sect 2 in sect 3 we study the operator which corrects the boundary values of the solution we obtained in sect 2 we solve the o neumann problem in sect 4 most of the work for this paper was done while i was a member of the institute for advanced study i wish to thank the institute for its hospitality

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

the γ limit of traveling waves in the fitzhugh nagumo system abstract patterns and waves are basic and important phenomena that govern the dynamics of physical and biological systems a common theme in investigating such systems is to identify the intrinsic factors responsible for such self organization the convergence is a well known technique applicable to variational formulations of concentration phenomena of stable patterns recently a geometric variational functional associated with the limit of standing waves of the fitzhugh nagumo system has been built this article studies the limit of traveling waves to the best of our knowledge this is the first attempt to expand the scope of applicability of convergence to cover non stationary problems

connecting orbits for some classes of almost periodic lagrangian systems abstract the existence is proved by means of variational arguments of infinitely many heteroclinic solutions connecting possibly degenerate equilibria for a class of almost periodic lagrangian system an analogous multiplicity result is then established for homoclinic solutions of systems with an almost periodic singular potential

an application of the modular function in nonlocal variational problems using the modular function and its invariance under the action of a modular group and an heuristic reduction of a mathematical model we present a mathematical account of a hexagonal pattern selection observed in di block copolymer melts

multiple layered solutions of the nonlocal bistable equation abstract the nonlocal bistable equation is a model proposed recently to study materials whose constitutive relations among the variables that describe their states are nonlocal it resembles the local bistable equation the allen cahn equation in some way but contains a much richer set of solutions in this paper we consider two types of solutions the first are the periodic solutions on a finite interval these solutions are observed in materials like elastic crystals undergoing martensitic phase transitions and diblock copolymers at low temperatures they are constructed by a variational method known as the limit technique the second are solutions on the entire real line with transition layers which are found by the formal matched asymptotics argument we construct them to compare with the single layer heteroclinic and traveling wave solutions of the local bistable equation the existence of multiple layered solutions depends on a unique nonlocal feature the presence of two properly balanced competing effects of the constitutive relation the oscillation inhibiting effect and the oscillation forcing effect which coexist at two different length scales

strongly nonlinear vortices in the stokes layer on an oscillating cylinder the stokes layer on a torsionally oscillating circular cylinder is susceptible to axially periodic vortices which are driven by a centrifugal mechanism which has some similarities to that which is responsible for gortler vortices in boundary layers on concave surfaces for small wavelength vortices an analytic description of linear weakly nonlinear and fully nonlinear perturbations is possible and as the growth rates of these modes are asymptotically greater than the timescale of the underlying stokes layer a quasi steady stability analysis is attempted in preference to a floquet type approach it is shown that highly nonlinear vortices can be expected to form on the surface of the cylinder it is noted that these vortices alter the basic mean flow at leading order and occupy a zone which has depth comparable to the stokes thickness depending on the taylor number of the flow three distinct types of vortex behaviour are found in the first of these corresponding to taylor numbers just above that at which the stokes layer becomes unstable over a part of its cycle the vortices are relatively weak and remain attached to the cylinder throughout their lifetime at larger taylor numbers the vortices initially grow whilst remaining on the cylinder but at some stage they break away and concentrate in a region within the stokes layer however their strength reduces with time and they soon decay away in the third regime identified before this vortex activity region dies the residual flow next to the cylinder becomes unstable once more in this case a second vortex zone begins to grow so that for a certain time interval two active zones exist the governing equations for the strongly nonlinear modes take the form of a free boundary value problem subject to ordinary differential constraints numerical solutions are presented and although periodic solutions are by no means guaranteed by use of the quasi steady approach the computations show that remarkably periodic vortex configurations tend to result

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

integral representations on hermitian manifolds the bar partial neumann solution of the cauchy riemann equations its significance for complex analysis lies in the fact that if n g dom d solves o nf and df 0 then u d nf is the unique solution of du which is orthogonal to kerd j j kohn has established existence and regularity properties of the solution operator iv giving the solution of minimal norm the so called 9 neumann operator in case d is strictly pseudoconvex 5 and in more general cases as well 6 the proofs are based on a priori estimates in l sobolev spaces and therefore they do not give any explicit information about the kernels of n or d n in recent years there has been much interest in finding more explicit and concrete representations of the abstractly defined operators n and d n see 2 3 9 10 12 in 7 we began to study d n by using the calculus of cauchy fantappie kernels in c n in analogy to the work of kerzman and stein 4 and ligocka 8 for the szego respectively bergman kernel in contrast to the scalar case the incompatibility of the euclidean metric with the complex geometry of the boundary of d turned out to be a major obstruction in the general case in the present paper we overcome this obstruction by generalizing the results in 7 to arbitrary hermitian manifolds this enables us to then introduce a special levi metric similar to the one in 2 and to establish the required symmetry properties of the kernels our main result gives a new and completely explicit integral representation of the principal part of d n on

the neumann solution to the inhomogeneous cauchy riemann equation in the ball in ⁿ let denote the formal adjoint of the cauchy riemann operator a on c 0 and let n denote the kohn neummn operator on the unit ball in cn the operator o n provides a natural fundamental solution for af g on the ball it is our purpose to present the kernel p for this operator o x explicitly the coeffi cients are exhibited as rational functions 1 introduction the 8 neumann problem was proposed by spencer the motiva tion for the problem comes from a desire to solve the inhomogeneous cauchy riemann equations this problem was successfully carried through by kohn see k and fk however none of the kernels shown to exist by kohn are explictly known even in the simplest cases it is our purpose to present one of these kernels explicitly the coefficients of the kernel are rational functions let d c cn denote the unit ball let denote the formal adjoint of a with respect to the euclidean metric on d and let n denote the kohn neumann operator see 2 for the defining properties of n the operator p o n is a modified fundamental solution for a on d i e if k is the bergman projection and if f is a smooth form on d then

traveling waves for the fitzhugh nagumo system on an infinite channel abstract we are concerned with the traveling wave solutions for the fitzhugh nagumo system on an infinite channel based on a variational formulation in which a non local term depends on a parameter the speed of a traveling wave can be selected out furthermore to show the existence of a traveling wave solution with such a speed we seek a minimizer subject to a constraint in the way of solving the variational problem we apply a truncation technique to the nonlocal term to obtain a minimizer located in a bounded invariant region

large wavenumber vortices in time periodic flows the linear stability properties of short wavelength vortices within time periodic flows are examined using a wkb based asymptotic analysis such vortices typically have an instantaneous growth rate considerably larger than the time scale of the underlying flow and thus have often been studied using quasi steady techniques a major drawback of these quasi steady approaches is that although they are able to determine the instantaneous flow structure they are of no use in describing the evolution of the modes it is shown how rational asymptotic analysis based on wkb ideas overcomes this deficiency and enables the form of the neutral stability curve to be obtained for short wavelengths the technique is applied to three problems first the key features of the analysis are exemplified by careful consideration of a model problem which is motivated by the physical situation of the sinusoidal heating of a horizontal flat plate bounding a semi infinite layer of fluid the governing linearized stability equations take the form of a system which is sixth order in the spatial variable and second order in time however the important features of this system are captured by our model which is only second order in the spatial coordinate and first order in time the second physically important flow examined is the motion induced in a viscous fluid surrounding a long torsionally oscillating cylinder a curved stokes flow detailed asymptotic and numerical studies of both the model and the two physically motivated flows are undertaken and it is shown how the analytical results provide good qualitative agreement with the numerical findings at quite modest wavenumbers the methodology adopted here may be used as a basis for the rational study of the stability properties of other time periodic flows a key result forthcoming from this study is the demonstration that the first few terms in the relevant high wavenumber form of the neutral curve for modes in an oscillatory flow may be derived relatively quickly with minimal computational effort

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

standing pulse solutions to fitzhugh nagumo equations reaction diffusion systems serve as relevant models for studying complex patterns in several fields of nonlinear sciences a localized pattern is a stable non constant stationary solution usually located far away from neighborhoods of bifurcation induced by turing s instability in the study of fitzhugh nagumo equations we look for a standing pulse with a profile staying close to a trivial background state except in one localized spatial region where the change is substantial this amounts to seeking a homoclinic orbit for a corresponding hamiltonian system and we utilize a variational formulation which involves a nonlocal term such a functional is referred to as helmholtz free energy in modeling microphase separation in diblock copolymers while its global minimizer does not exist in our setting of dealing with standing pulse the homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions in contrast with the known results for positive standing pulses found in the literature a new technique is attempted by seeking a standing pulse solution with a sign change

the structure of highly nonlinear vortices in curved channel flow the structure of highly nonlinear small wavelength gortler vortices is analysed for fully developed flow in a curved channel the taylor number is taken to be t 0 epsilon 5 which is asymptotically larger than the neutral stability taylor number for vortices of small wavelength 2 pi epsilon over most of the flow domain the motion is determined by the interaction of the mean flow and terms proportional to exp iz epsilon whereas adjacent to the outer channel wall there is a boundary layer of thickness o epsilon which requires all the harmonics of the disturbance to be considered numerical solutions of the wall layer equations are presented for t 0 up to 14 000 with the main feature of these results being the development of a weak vortex on the boundary layer lengthscale

the fully nonlinear development of gortler vortices in growing boundary layers the fully nonlinear development of small wavelength gortler vortices in a growing boundary layer is investigated by a combination of asymptotic and numerical methods the starting point for the analysis is the weakly nonlinear theory of hall j inst math applies 29 173 1982 who discussed the initial development of large wavenumber small amplitude vortices in a neighbourhood of the location where they first become linearly unstable that development is unusual in the context of nonlinear stability theory in that it is not described by the stuart watson approach in fact the development is governed by a pair of coupled nonlinear partial differential evolution equations for the vortex flow and the mean flow correction here the further development of this interaction is considered for vortices so large that the mean flow correction driven by them is as large as the basic state surprisingly it is found that such a nonlinear interaction can still be described by asymptotic means it is shown that the vortices spread out across the boundary layer and effectively drive the boundary layer in fact the system obtained by the equations for the fundamental component of the vortex generates a differential equation for the basic state thus the mean flow adjusts so as to make these large amplitude vortices locally neutral moreover in the region where the vortices exist the mean flow has a square root profile and the vortex velocity field can be written down in closed form the upper and lower boundaries of the region of vortex activity are determined by a free boundary problem involving the boundary layer equations in general it is found that this region ultimately includes almost all of the original boundary layer and much of the free stream in this situation the mean flow has essentially no relation to the flow that exists in the absence of the vortices