the phase plane picture for a class of fourth order conservative differential equations we study the bounded solutions of a class of fourth order equations u u f u 0 0 we show that when is not too large then the paths in the u u plane of two bounded solutions do not cross moreover the conserved quantity associated with the equation puts an ordering on the bounded solutions in the phase plane and a continuation theorem shows that they fill up part of the phase plane we apply these results to the extended fisher kolmogorov efk equation a fourth order model equation for bi stable systems the uniqueness and ordering results imply that as long as the stable equilibrium points are real saddles the bounded solutions of the stationary efk equation correspond exactly to those of the classical second order fisher kolmogorov equation besides we establish the asymptotic stability of the heteroclinic solution of the efk equation

closed characteristics on singular energy levels of second order lagrangian systems abstract we provide a lower bound on the number of closed characteristics on singular energy levels of second order lagrangian systems in the presence of saddle focus equilibria the hypotheses on the lagrangian are mild and the bound is given in terms of the number of saddle foci and a potential function determined by the lagrangian the method of proof is variational combining techniques to minimize near a saddle focus and an analog of the method of broken geodesics

relations between the minors of a generic matrix abstract it is well known that the plucker relations generate the ideal of relations of the maximal minors of a generic m n matrix in this paper we discuss the relations of t minors for t min m n we will exhibit minimal relations in degrees 2 non plucker in general and 3 and give some evidence for our conjecture that we have found the generating system of the ideal of relations the approach is through the representation theory of the general linear group

on the existence of solutions for a class of fourth order differential equations abstract we study the existence of the homoclinic solution of a class of nonlinear fourth order ordinary differential equations which involve particular cases of the extended fisher kolmogorov and swift hohenberg equations we address the case when the source term possesses a nonlinear dependence on the first order derivative of the solution the main result follows from an iterative method using the sequence of solutions obtained through the mountain pass technique in order to illustrate this method some numerical examples are shown

the motion of a vortex on a closed surface of constant negative curvature the purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one the algorithm is based on a relation between the laplace beltrami green function and the heat kernel the algorithm is used to compute the motion of a vortex on the bolza surface this is the first determination of the orbits of a vortex on a closed surface of genus greater than one the numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the bolza surface some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc the following theorem is proved a weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium

on the transition from the ginzburg landau equation to the extended fisher kolmogorov equation abstract the ginzburg landau gl equation generically describes the behaviour of small perturbations of a marginally unstable basic state in systems on unbounded domains in this paper we consider the transition from this generic situation to a degenerate co dimension 2 case in which the gl approach is no longer valid instead of studying a general underlying model problem we consider a two dimensional system of coupled reaction diffusion equations in one spatial dimension we show that near the degeneration the behaviour of small perturbations is governed by the extended fisher kolmogorov efk equation at leading order the relation between the gl equation and the efk equation is quite subtle but can be analysed in detail the main goal of this paper is to study this relation which we do asymptotically the asymptotic analysis is compared to numerical simulations of the full reaction diffusion system as one approaches the co dimension 2 point we observe that the stable stationary periodic patterns predicted by the gl equation evolve towards various different families of stable stationary but not necessarily periodic so called multi bump solutions in the literature these multi bump patterns are shown to exist as solutions of the efk equation but there is no proof of the asymptotic stability of these solutions our results suggest that these multi bump patterns can also be asymptotically stable in large classes of model problems

uniqueness of the stationary wave for the extended fisher kolmogorov equation abstract the extended fisher kolmogorov equation ut uxxxx uxx u u3 0 models a binary system near the lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria 1 for the classical case 0 the heteroclinic is u x tanh x 2 and is unique up to the obvious symmetries we prove the conjecture that the uniqueness persists all the way to 1 8 where the onset of spatial chaos associated with the loss of monotonicity of the stationary wave is known to occur our methods are non perturbative and employ a global cross section to the hamiltonian flow of the stationary fourth order equation on the energy level of 1 we also prove uniform a priori bounds on all bounded stationary solutions valid for any 0

existence and multiplicity results for heteroclinic orbits of second order hamiltonian systems abstract connecting orbits of nonlinear differential equations have long been studied in the dynamical systems literature generally in a setting involving perturbations and using a melnikov function in this article we consider a class of second order hamiltonian systems which possess infinitely many or finite number of equilibria using variational arguments and penalization methods we obtain the existence of multiple heteroclinic orbits joining pairs of equilibria

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

shooting methods and topological transversality we show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds the interest of such results is twofold first these orbits persist under perturbations which destroy the structure allowing the shooting method and second topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics we shall focus on heteroclinic solutions in the extended fisher kolmogorov equation

the moving curve ideal and the rees algebra the rees algebra of an ideal in a commutative ring is the quotient of a polynomial ring by its ideal of defining relations for a polynomial ring in two variables this ideal was discovered independently by the geometric modeling community where it is called the moving curve ideal we review some properties of the rees algebra and discuss one result and one conjecture concerning the structure of the moving curve ideal and its relation to adjoint curves some parts of the paper are purely expository

existence and asymptotic behavior of nontrivial solutions to the swift hohenberg equation abstract in this paper we discuss several results regarding existence non existence and asymptotic properties of solutions to u q u f u 0 under various hypotheses on the parameter q and on the potential f t 0 t f s d s generally assumed to be bounded from below we prove a non existence result in the case q 0 and an existence result of periodic solution for 1 almost every suitably small depending on f positive values of q 2 all suitably large depending on f values of q finally we describe some conditions on f which ensure that some or all solutions u q to the equation satisfy u q 0 as q 0

thomson s heptagon a case of bifurcation at infinity abstract vortex modeling has a long history descartes 1644 used it as a model for the solar systems j j thomsom 1883 used it as a model for the atom we consider point vortex systems which can be regarded as discrete solutions of the euler equation their dynamics is described by a hamiltonian system of equations we are interested in polygonal configurations and how their stability depends upon various dynamical variables in the plane a polygon with seven vortices has been shown to be a special boundary case polygons with n 7 vortices are linearly and nonlinearly stable while polygons with n 7 vortices are unstable why should n 7 be special celestial mechanics helped us to simplify a problem that has been studied for over a century and to show that the case of thomson s heptagon is actually a case of bifurcation at infinity this becomes particularly clear when considering the corresponding problem of a ring on a sphere with two polar vortices of variable intensities n and s at the north and south pole respectively

the motion of point vortices on closed surfaces we develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere when the sum of the vortex circulations is non ze

morse theory on spaces of braids and lagrangian dynamics in the first half of the paper we construct a morse type theory on certain spaces of braid diagrams we define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces parabolic flows a type of one dimensional lattice dynamics evolve singular braid diagrams in such a way as to decrease their topological complexity algebraic lengths decrease monotonically this topological invariant is derived from a morse conley homotopy index in the second half of the paper we apply this technology to second order lagrangians via a discrete formulation of the variational problem this culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits

the possible dimensions of the higher secant varieties let x c p be an irreducible closed subvariety of projective space the kth higher secant variety of x denoted x is the closure of the union of all linear spaces spanned by k points of x let a ix dim x dim x l it is known that the sequence ty x is nonincreasing and that there may be at most one 1 in such a sequence we prove that these two conditions are also sufficient conditions for an arbitrary sequence a of nonnegative integers to be the fl x s for an irreducible variety x to this end we exhibit the ideals of the higher secant varities of rational normal scrolls as determinantal ideals this allows us to compute their dimensions showing that there is in fact always a rational normal scroll x such that ak x a for any sequence a satisfying the two conditions above adlandsvik al a2 and flenner and vogel f v much of this work has centered on the behavior of the differences ak x dimx dimx 1 of the dimensions of consecutive higher secant varieties it is well known that this sequence is nonincreasing and that there is at most one 1 in the sequence palatini claimed moreover that once ak x 1 were counterexamples a2 we answer here the question of what sequences actually appear as the ai x s of some variety x some of the guiding examples of varieties with degenerate higher secant varieties come from rational normal scrolls explicit equations for the scrolls themselves have been known for some time cf acgh we give a determi nantal presentation of the ideals of their higher secant varieties generalizing a result for rational normal scrolls in w this presentation enables us to compute the dimensions of all the higher secant varieties of a rational normal scroll and gives examples of varieties x with a prescribed nonincreasing ak x with at most one 1

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

slow motion in higher order systems and g convergence in one space dimension the slow motion in higher order systems and convergence in one space dimension was studied some of the theorems and computations that support and proves the stability of the equations are presented

oscillatory solutions of fourth order conservative systems via the conley index in this paper we investigate periodic solutions of second order lagrangian systems which oscillate around equilibrium points of center type the main ingredients are the discretization of second order lagrangian systems that satisfy the twist property and the theory of discrete braid invariants developed by ghrist et al 2003 5 the problem with applying this topological theory directly is that the braid types in our analysis are so called improper this implies that the braid invariants do not entirely depend on the topology the relevant braid classes are non isolating neighborhoods of the flow so that their conley index is not universal in first part of this paper we develop the theory of the braid invariant for improper braid classes and in the second part this theory is applied to second order lagrangian system and in particular to the swift hohenberg equation

a variational construction of chaotic trajectories for a reversible hamiltonian system abstract a geometric criterion for the existence of chaotic trajectories of a reversible hamiltonian system with the configuration space t n with n a compact manifold is given the main result is a variational version of the theorem of d v turayev and l p shilnikov dokl akad nauk sssr 304 1989 811 814 on the symbolic representation of trajectories of hamiltonian systems possessing several homoclinics to a saddle equilibrium

on the equations of the moving curve ideal of a rational algebraic plane curve given a parametrization of a rational plane algebraic curve c some explicit adjoint pencils on c are described in terms of determinants moreover some generators of the rees algebra associated to this parametrization are presented the main ingredient developed in this paper is a detailed study of the elimination ideal of two homogeneous polynomials in two homogeneous variables that form a regular sequence

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

cremona transformations and some related algebras abstract one proves a general characteristic free criterion for a rational map between projective varieties to be birational in terms of ideal theoretic and modulo theoretic conditions this criterion is more inclusive than that of f russo a simis compositio math 126 2001 335 358 and moreover differs from previous criteria in its nature in that the syzygies of the base ideal of the map are not directly involved in its formulation however a great deal of the consequences are phrased by means of those very syzygies avoided in the formulation of the criterion in any case the criterion is stated in effective terms so it yields an efficient computable test of birationality one also introduces a so called linear obstruction principle for base ideals of linear type thus raising a basic question concerning the structure of a certain related bilinear algebra

one dimensional infinite horizon variational problems arising in continuum mechanics in this paper we study a variational problem for real valued functions defined on an infinite semiaxis of the line to wit given x 2 we seek a minimal solution to the problem minimize the functional given by i left w left cdot right right int limits 0 infty f left w left s right ddot w left s right right ds w in ax left upsilon in w loc 2 1 left 0 infty right left upsilon left 0 right dot upsilon left 0 right x right right p

the variational contribution of the periodic orbits provided by the birkhoff lewis theorem with applications to convex hamiltonians and to three body type problems

orthogonal cubic spline collocation method for the extended fisher kolmogorov equation a second order splitting combined with orthogonal cubic spline collocation method is formulated and analysed for the extended fisher kolmogorov equation with the help of lyapunov functional a bound in maximum norm is derived for the semidiscrete solution optimal error estimates are established for the semidiscrete case finally using the monomial basis functions we present the numerical results in which the integration in time is performed using radau 5 software library

heteroclinic solutions for the extended fisher kolmogorov equation abstract we study the extended fisher kolmogorov efk equation and its variants by variational approach we show that for every global minimum of the potential function v there is a pair of heteroclinic solutions one emanating from and the other terminating at we require neither superquadratic growth of v nor the presence of saddle focus equilibria moreover v is allowed to approach its minimal level near infinity

closed characteristics of second order lagrangians we study the existence of closed characteristics on three dimensional energy manifolds of second order lagrangian systems these manifolds are always noncompact connected and not necessarily of contact type using the specific geometry of these manifolds we prove that the number of closed characteristics on a prescribed energy manifold is bounded below by its second betti number which is easily computable from the lagrangian

n body dynamics on closed surfaces the axioms of mechanics a major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of n body and n vortex dynamics on riemann surfaces in this paper we show how the two problems are in fact closely related when considering the role played by the intrinsic geometry of the surface this enables a straightforward deduction of the dynamics of point masses using recently derived results for point vortices on general closed differentiable surfaces m endowed with a metric g we find generally that kepler s laws do not hold what is more even newton s first law the law of inertia fails on closed surfaces with variable curvature e g the ellipsoid

subalgebras of bigraded koszul algebras abstract we show that diagonal subalgebras and generalized veronese subrings of a bigraded koszul algebra are koszul we give upper bounds for the regularity of side diagonal and relative veronese modules and apply the results to symmetric algebras and rees rings

nonlinear stability of a latitudinal ring of point vortices on a nonrotating sphere we study the nonlinear stability of relative equilibria of configurations of identical point vortices on the surface of a sphere in particular we study how the stability changes as a function of the colatitude theta and of the number of vortices n by using the integrals of motion we view the system in asuitable corotating frame where the polygonal vortex configuration is at rest then after a sufficient criterion due to dirichlet the stability ranges are the theta intervals for which the hessian of the hamiltonian evaluated at the equilibrium configuration is positive or negative definite we find that the stability intervals coincide with those for linear stability determined by polvani and dritschel j fluid mech 255 1993 pp 35 64 for n 3 we recover the result previously established by pekarsky and marsden j math phys 39 1998 pp 5894 5907

global branches of multi bump periodic solutions of the swift hohenberg equation in this paper we study the existence of single and multi bump periodic solutions of a class of fourth order ordinary differential equations arising in problems of pattern formation measuring the tendency to form patterns by a parameter q we view the problem as a nonlinear eigenvalue problem with the use of analytical as well as numerical methods branches of periodic solutions are investigated both locally and globally

periodic solutions for extended fisher kolmogorov and swift hohenberg equations by truncature techniques abstract combining truncature techniques with a variational approach we establish an existence result for nontrivial periodic solutions for a class of fourth order ordinary differential equations involving extended fisher kolmogorov and swift hohenberg equations

statistics of point vortex turbulence in non neutral flows and in flows with translational and rotational symmetries a theory esler and ashbee in j fluid mech 779 275 308 2015 describing the statistics of n freely evolving point vortices in a bounded two dimensional domain is extended first the case of a non neutral vortex gas is addressed and it is shown that the density of states function can be identified with the probability density function of an infinite sum of independent non central chi squared random variables the details of which depend only on the shape of the domain equations for the equilibrium energy spectrum and other statistical quantities follow the validity of which are verified against direct numerical simulations of the equations of motion second domains with additional conserved quantities associated with a symmetry e g circle periodic channel are investigated and it is shown that the treatment of the non neutral case can be modified to account for the additional constraint

grobner bases and invariant theory abstract in this paper we study the relationship between buchberger s grobner basis method and the straightening algorithm in the bracket algebra these methods will be introduced in a self contained overview on the relevant areas from computational algebraic geometry and invariant theory we prove that a certain class of van der waerden syzygies forms a grobner basis for the syzygy ideal in the bracket ring we also give a description of a reduced grobner basis in terms of standard and non standard tableaux some possible applications of straightening for symbolic computations in projective geometry are indicated

hyperdeterminantal relations among symmetric principal minors abstract the principal minors of a symmetric n n matrix form a vector of length 2 n we characterize these vectors in terms of algebraic equations derived from the 2 2 2 hyperdeterminant

homoclinic solutions for swift hohenberg and suspension bridge type equations we establish the existence of homoclinic solutions for a class of fourth order equations which includes the swift hohenberg model and the suspension bridge equation in the first case the nonlinearity has three zeros corresponding to a double well potential while in the second case the nonlinearity is asymptotically constant on one side the swift hohenberg model is a higher order extension of the classical fisher kolmogorov model its more complicated dynamics give rise to further possibilities of pattern formation the suspension bridge equation as studied by chen and mckenna j differential equations 136 1997 325 355 we give a positive answer to an open question raised by the authors c 2002 elsevier science usa

second order lagrangian twist systems simple closed characteristics we consider a special class of lagrangians that play a fundamental role in the theory of second order lagrangian systems twist systems this subclass of lagrangian systems is defined via a convenient monotonicity property that such systems share this monotonicity property twist property allows a finite dimensional reduction of the variational principle for finding closed characteristics in fixed energy levels this reduction has some similarities with the method of broken geodesics for the geodesic variational problem on riemannian manifolds on the other hand the monotonicity property can be related to the existence of local twist maps in the associated hamiltonian flow the finite dimensional reduction gives rise to a second order monotone recurrence relation we study these recurrence relations to find simple closed characteristics for the lagrangian system more complicated closed characteristics will be dealt with in future work furthermore we give conditions on the lagrangian that guarantee the twist property

global existence of homoclinic and periodic orbits for a class of autonomous hamiltonian systems abstract when nonlinear reversible fourth order hamiltonian systems have a saddle focus at a hyperbolic equilibrium of zero energy geometric conditions on the hamiltonian are given to ensure the existence of a symmetric homoclinic orbit which is the limit of certain specific zero energy periodic orbits uniformly on compact time intervals when the equilibrium is a centre the existence of a large amplitude zero energy periodic orbit which is not given by lyapunov s centre theorem is proved

homoclinic snaking near a heteroclinic cycle in reversible systems snaking curves of homoclinic orbits have been found numerically in a number of ode models from water wave theory and structural mechanics along such a curve infinitely many fold bifurcation of homoclinic orbits occur thereby the corresponding solutions spread out and develop more and more bumps oscillations about their own centre a common feature of the examples is that the systems under consideration are reversible in this paper it is shown that such a homoclinic snaking can be caused by a heteroclinic cycle between two equilibria one of which is a bi focus using lin s method a snaking of 1 homoclinic orbits is proved to occur in an unfolding of such a cycle further dynamical consequences are discussed as an application a system of boussinesq equations is considered where numerically a homoclinic snaking curve is detected and it is shown that the homoclinic orbits accumulate along a heteroclinic cycle between a real saddle and a bi focus equilibrium

vortex dynamics in complex domains on a spherical surface we consider the motion of both point vortices and uniform vortex patches in arbitrary possibly multiply connected regions bounded by impenetrable walls on the surface of a sphere by exploiting knowledge of the functional form of the relevant green s function in a pre image circular domain that is conformally equivalent to a stereographic projection of the fluid domain on the spherical surface we first generalize kirchhoff routh theory for point vortex motion in the plane to point vortex motion on a spherical shell next we study vortex patch motion and show that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected domain on a spherical shell bounded by impenetrable walls we describe a novel numerical scheme whereby this motion can be computed some illustrative calculations are shown

spatial chaotic structure of attractors of reaction diffusion systems the dynamics described by a system of reaction diffusion equations with a nonlinear potential exhibits complicated spatial patterns these patterns emerge from preservation of homotopy classes of solutions with bounded energies chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree this group corresponds to level sets of the potential the estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem

vortex motion on surfaces with constant curvature vortex motion on two dimensional riemannian surfaces with constant curvature is formulated by way of the stereographic projection the relation and difference between the vortex motion on a sphere s 2 and on a hyperbolic plane h 2 can be clearly analysed the hamiltonian formalism is presented for the motion of point vortices on s 2 and h 2 the set of first integrals for each hamiltonian shows a corresponding algebraic property in terms of the poisson bracket defined respectively for s 2 and h 2 as an example of analytic solutions the motion of a vortex pair dipole is considered it is shown that a dipole draws a geodesic curve as its trajectory on s 2 and h 2

stable patterns for fourth order parabolic equations we consider fourth order parabolic equations of gradient type for the sake of simplicity the analysis is carried out for the specific equation u

rational normal scrolls and the defining equations of rees algebras consider a height two ideal i which is minimally generated by m homogeneous forms of degree d in the polynomial ring r k x y suppose that one column in the homogeneous presenting matrix of i has entries of degree n and all of the other entries of are linear we identify an explicit generating set for the ideal a which defines the rees ring r r it so r s a for the polynomial ring s r t1 tm we resolve r as an s module and is as an r module for all powers s the proof uses a rational normal scroll ring a s h with aa isomorphic to the nth symbolic power of a height one prime ideal k of a the ideal k n is generated by monomials whenever possible we study a k n in place of a aa because the generators of k n are much less complicated then the generators of aa we obtain a filtration of k n in which the factors are polynomial rings hypersurface rings or modules resolved by eagon northcott complexes the generators of i give rise to an algebraic curve c in projective m 1 space the defining equations of the fiber ring r x y r yield a solution of the implicitization problem for c introduction let i be a height two ideal in the polynomial ring r k x y with i minimally generated by m forms of degree d we consider the following questions what is the defining ideal a of the rees ring r r it r it it what is the hilbert function of r the rees ring r is equal to r t1 tm a what is the minimal resolution of r what are the defining equations of the fiber 1supported in part by the national security agency 2supported in part by the national science foundation and the national security agency 3supported in part by the national science foundation 2000 mathematics subject classification primary 14q05 secondary 13c20 13d02 14e05 14m12 68w30

sums of squares and varieties of minimal degree the study of nonnegativity and its relation with sums of squares is a basic challenge in real algebraic geometry the classification of varieties of minimal degree is one of the milestones of classical complex algebraic geometry the goal of this paper is to establish the deep connection between these apparently separate topics to achieve this let x p be an embedded real projective variety with homogeneous coordinate ring r the variety x has minimal degree if it is nondegenerate not contained in a hyperplane and deg x 1 codim x a homogeneous element f r of even degree is nonnegative if its evaluation at each real point of x is at least zero our main theorem is a broad generalization of hilbert s 1888 classification of nonnegative forms and provides a tight connection between real and complex algebraic geometry

homotopy classes for stable connections between hamiltonian saddle focus equilibria for a class of hamiltonian systems in 4 the set of homoclinic and heteroclinic orbits which connect saddle focus equilibria is studied using a variational approach the oscillatory properties of a saddle focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the configuration plane punctured at the saddle foci this variational approach does not require any assumptions on the intersections of stable and unstable manifolds such as transversality moreover these connections are shown to be local minimizers of an associated action functional this result has applications to spatial pattern formation in a class of fourth order bistable evolution equations

existence and numerical approximations of periodic solutions of semilinear fourth order differential equations abstract a multiplicity result of existence of periodic solutions with prescribed wavelength for a class of fourth order nonautonomous differential equations related either to the extended fisher kolmogorov or to the swift hohenberg equation is proved variational approach is used some numerical solutions are calculated via the finite element method

point vortex dynamics a classical mathematics playground the idealization of a two dimensional ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics several examples are given including the integrability of the three vortex problem the interplay of relative equilibria of identical vortices and the roots of certain polynomials addition formulas for the cotangent and the weierstras function projective geometry and other topics the hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community

a non crossing standard monomial theory abstract the second author has introduced non crossing tableaux objects whose non nesting analogues are semi standard young tableaux we relate non crossing tableaux to gelfand tsetlin patterns and develop the non crossing analogue of standard monomial theory leclerc and zelevinsky s weakly separated sets are special cases of non crossing tableaux and we suggest that non crossing tableaux may help illuminate the theory of weakly separated sets

homotopy classes for stable periodic and chaotic patterns in fourth order hamiltonian systems we investigate periodic and chaotic solutions of hamiltonian systems in 4 which arise in the study of stationary solutions of a class of bistable evolution equations under very mild hypotheses variational techniques are used to show that in the presence of two saddle focus equilibria minimizing solutions respect the topology of the configuration plane punctured at these points by considering curves in appropriate covering spaces of this doubly punctured plane we prove that minimizers of every homotopy type exist and characterize their topological properties

a characteristic free criterion of birationality abstract one develops ab initio the theory of rational birational maps over reduced but not necessarily irreducible projective varieties in arbitrary characteristic a virtual numerical invariant of a rational map is introduced called the jacobian dual rank it is proved that a rational map in this general setup is birational if and only if the jacobian dual rank is well defined and attains its maximal possible value even in the classical case where the source variety is irreducible there is some gain for this invariant over the degree of the map because on one hand it relates naturally to constructs in commutative algebra and on the other hand is effectively computable applications are given to results only known so far in characteristic zero one curious byproduct is an alternative approach to deal with the result of dolgachev concerning the degree of a plane polar cremona map

a hermite spectral method for the computation of homoclinic orbits and associated functionals we present a spectral method for the computation of homoclinic orbits in ordinary differential equations the method is based on hermite fourier expansions of the complete homoclinic solution and exhibits exponential convergence in addition our method can be used to approximate nonlinear functionals which depend on the complete homoclinic solution this is demonstrated using examples from phase separation dynamics and metastability

existence and stability of traveling fronts in the extended fisher kolmogorov equation abstract we study traveling wave solutions to a general fourth order differential equation that is a singular perturbation of the fisher kolmogorov equation we apply the geometric method for singularly perturbed systems to show that for every positive wavespeed there exists a traveling wave also we find that there exists a critical wavespeed c which divides these solutions into monotonic c c and oscillatory c c solutions we show that the monotonic fronts are locally stable under perturbations in appropriate weighted sobolev spaces by using various energy functionals