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

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we consider the transition from this generic situation to a degenerate co dimension 2 case in which the gl approach is no longer valid instead of studying a general underlying model problem we consider a two dimensional system of coupled reaction diffusion equations in one spatial dimension we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

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

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

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

infinitely many large amplitude homoclinic orbits for a class of autonomous hamiltonian systems abstract for a class of fourth order autonomous hamiltonian systems we give geometrical conditions that imply the existence of infinitely many homoclinic orbits to a saddle focus equilibrium our results are global in the sense that they do not rely on any transversality assumption they are applied to two example systems one arises in elasticity and in water wave theory the other describes the motion of a particle in a certain rotating potential

geometric optics approach to reaction diffusion equations the speed of propagation of traveling waves of reaction diffusion equations is calculated by an asymptotic method similar to geometrical optics this also works for reaction diffusion waves in random media

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

mobius mellin and mathematical physics abstract we examine some results and techniques of analytic number theory which have application or potential application in mathematical physics we consider inversion formulae for lattice sums various transformations of infinite series and products functional equations and scaling relations with selected applications in electrostatics and statistical mechanics in the analysis the mellin transform and the riemann zeta function play a key role

closed characteristics on non compact hypersurfaces in r 2n viterbo demonstrated that any 2n 1 dimensional compact hypersur face m r 2n of contact type has at least one closed characteristic this result proved the weinstein conjecture for the standard symplectic space r 2n various extensions of this theorem have been obtained since all for compact hypersurfaces in this paper we consider non compact hypersurfaces m r 2n coming from mechanical hamiltonians and prove an analogue of viterbo s result the main result provides a strong connection between the top half homology groups hi m i n 2n 1 and the existence of closed characteristics in the non compact case including the compact case

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

homoclinic orbits to a saddle center in a fourth order differential equation abstract we study homoclinic orbits to a saddle center of a fourth order ordinary differential equation which is invariant under the transformation x x involving an eigenvalue parameter q and an odd piece wise cubic type nonlinearity it is found that for a sequence of eigenvalues which tends to infinity homoclinic orbits exist whose complexity increases as the eigenvalue becomes larger these orbits are found to be embedded in branches of homoclinic orbits to periodic orbits as x

oscillating solutions of a nonlinear fourth order ordinary differential equation we study the existence of periodic solutions for a nonlinear fourth order ordinary differential equation under suitable conditions we prove the existence of at least one solution of the problem applying coincidence degree theory and the method of upper and lower solutions

the quantum relativistic harmonic oscillator spectrum of zeros of its wave functions the distribution of zeros of the wave functions of the quantum relativistic harmonic oscillator is studied an analytic approximate expression for this distribution is derived within the framework of the wkb approximation using potential theory techniques it is proved that this wkb approach gives the right asymptotic distribution which is explicitly calculated

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

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

metastable patterns for the cahn hilliard equation part ii layer dynamics and slow invariant manifold abstract in this paper we study the dynamics of the 1 dimensional cahn hilliard equation u t 2 u xx w u xx in a finite interval in a neighborhood of an equilibrium with n 1 transition layers where is a small parameter and w is a double well energy density function with equal minima the lower bound of the layer motion speed is given explictly and the layer motion directions are described precisely if a solution of the cahn hilliard equation starts outside a neighborhood of the equilibrium of size o ln 1 it is proved that there is an n dimensional unstable invariant manifold which is a smooth graph over the approximate manifold constructed in j differential equations 111 1994 421 457 with its global lipschitz constant exponentially small and this unstable invariant manifold attracts solutions exponentially fast uniformly in

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 dynamics for the cahn hilliard equation in higher space dimensions the motion of bubbles it is known that the van der waals cahn hilliard w c h dynamics can be approximated by a quasi static stefan problem with surface tension it turns out that the stefan problem has a manifold of equilibria equal in dimension to that of the domain any sphere of fixed radius with interface contained in the domain is an equilibrium indistinguishable from the point of view of the perimeter functional we resolve this degeneracy by showing that at the w c h level this manifold is replaced by a quasi invariant stable manifold on which the typical solution preserves its bubble like shape until it reaches the boundary moreover we show that the bubble moves superslowly we also obtain an equation that determines those special spheres that correspond to equilibria at the w c h level our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions

metastability and stability of patterns in a convolution model for phase transitions where e is a small positive parameter and f is bistable having only three zeros 1 a 1 and is balanced i e r 1 1 f u du 1 4 0 see fig 1 for a typical initial datum j which crosses a several times at the beginning the solution evolves as if the diffusion term euxx were not there and in the time scale o ej n ej the solution gets close to 1 or 1 in intervals separated by thin layers of thickness at the order of o ej n ej 17 28 at this moment a spatial pattern is generated it persists on a time scale o ece but is eventually destroyed typically u x t 1 u x t 1 uniformly for x on 1 20 1 as t 1 it is possible that u x t converges to a or a nonconstant steady state as t 1 but this is an unlikely event because all these steady states are unstable 16 27 research supported in part by the national science foundation and bor leqsf

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

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

metastable patterns for the cahn hilliard equation part i abstract in this paper we study the dynamics of the one dimensional cahn hilliard equation in a neighborhood of an equilibrium having n 1 transition layers an approximation for an n dimensional invariant manifold and slow channel are defined and it is proved that ff a solution of the cahn hilliard equation starts outside but close to the slow channel around the approximate manifold then it will approach the channel at an exponentially large speed after it enters the slow channel it will follow the approximate manifold with speed o e c and stay in the channel for exponentially large time

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

liouville type and uniqueness results for a class of elliptic equations abstract liouville type results are obtained for fourth order elliptic equations of the form 2 u q x g u p x f u 0 and q x g u u p x f u 0 by means of two subharmonic functionals defined appropriately on their respective solutions and a green inequality

relativistic orthogonal polynomials are jacobi polynomials we identify the recently studied relativistic orthogonal polynomials in terms of jacobi polynomials so all their properties follow from the corresponding properties of jacobi polynomials through a change of variable

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

fractional derivative estimates in gevrey spaces global regularity and decay for solutions to semilinear equations in rn abstract we propose a unified functional analytic approach to study the uniform analytic gevrey regularity and the decay of solutions to semilinear elliptic equations on r n first we develop a fractional calculus for nonlinear maps in banach spaces of l p based gevrey functions 1 h p s r n regularity with s scr 0 depending on the nonlinearity next we investigate the type of decay polynomial or exponential of the derivatives of solutions to semilinear elliptic equations provided they decay a priori slowly as o x x for some small 0 the restrictions involved in our results are optimal in particular given a hyperplane l we construct 2d 2 strongly singular solutions locally in h p s r n for s

symmetry and uniqueness of the solitary wave solution for the ostrovsky equation we consider herein the ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water or the capillary waves in a plasma with the effects of rotation using the modified sliding method we prove that the solitary wave moving to the left to the ostrovsky equation is symmetric about the origin and unique up to translations we also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed weak rotation and dispersive parameter

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

the potential constraints of the complex 1 2 dimensional soliton systems and related hamiltonian equations potential complete constraints symmetric and nonsymmetric of the complex kp and mkp systems are derived by using the general theory for a nonlinear equation to be a hamiltonian system all the complex hamiltonian equations corresponding to this kind of constraint are obtained which contain the famous nonlinear schrodinger equations and several new hamiltonian equations in addition the classical kaup newell system and the heisenberg system are also led by a new reduction approach from the mkp system

slow motion for the 1d swift hohenberg equation abstract the goal of this paper is to study the behavior of certain solutions to the swift hohenberg equation on a one dimensional torus t combining results from convergence and ode theory it is shown that solutions corresponding to initial data that is l 1 close to a jump function v remain close to v for large time this can be achieved by regarding the equation as the l 2 gradient flow of a second order energy functional and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of v

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

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

mobius transform pairs a continuous version of a theorem from classical number theory due to mobius has recently been derived and used to solve some inverse problems in physics that are otherwise difficult to solve explicitly using standard results a table of mobius transform pairs is presented that may be useful in identifying other applications of the transform to physical problems

the relativistic hermite polynomial is a gegenbauer polynomial it is shown that the polynomials introduced recently by aldaya bisquert and navarro salas phys lett a 156 381 1991 in connection with a relativistic generalization of the quantum harmonic oscillator can be expressed in terms of gegenbauer polynomials this fact is useful in the investigation of the properties of the corresponding wave function some examples are given in particular related to the asymptotic behavior and to the distribution of zeros of the polynomials for large quantum numbers

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

a new method to introduce the mobius function and the mobius transform the mobius function and mobius transform are introduced and derived in a manner that parallels a typical inverse problem of physics rather than in the usual number theoretic manner

preliminary group classification of a class of fourth order evolution equations we perform preliminary group classification of a class of fourth order evolution equations in one spatial variable following the approach developed by basarab horwath et al acta appl math 69 43 2001 we construct all inequivalent partial differential equations belonging to the class in question which admit semisimple lie groups in addition we describe all fourth order evolution equations from the class under consideration which are invariant under solvable lie groups of dimension n 4 we have constructed all galilei invariant equations belonging to the class of evolution differential equations under study the list of so obtained invariant equations contains both the well known fourth order evolution equations and a variety of new ones possessing rich symmetry and as such may be used to model nonlinear processes in physics chemistry and biology

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

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

the gradient theory of phase transitions for systems with two potential wells on generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel on considere la famille des perturbations e e u w u dx e 2 u 2 dx de la fonctionnelle non convexe e 0 u w u dx ou w r n r supporte deux phases et n 1

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

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

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

invariant manifolds for metastable patterns in ut e2uxx f u we consider the above equation on the interval 0 x 1 subject to neumann boundary conditions with f u f u where f is a double well energy density function with equal minima our previous work 3 proved the existence and persistence of very slowly evolving patterns metastable states in solutions with two phase initial data here we characterise these metastable states in terms of the global unstable manifolds of equilibria as conjectured by fusco and hale 6

limit theorems for large deviations and reaction diffusion equations the equation u t u xx u 1 u is the simplest reaction diffusion equation introduction of a small parameter allows construction of geometric optics approximations for the solutions of such equations these solutions are approximated by step functions with the values 0 and 1 the region where the solution is close to 1 propagates according to the huygens principle for the corresponding velocity field v x e which is calculated via the equation new effects may emerge such as stops and jumps of the wave front the feynman kac formula implies that the solutions of certain cauchy problems obey some integral equations in the space of trajectories of the corresponding markov processes examination of this equation requires the study of laplace type asymptotics for functional integrals these asymptotics are defined by large deviations for the corresponding family of processes and are expressed through action functionals

minimal heteroclinics for a class of fourth order o d e systems abstract we prove the existence of minimal heteroclinic orbits for a class of fourth order o d e systems with variational structure in our general set up the set of equilibria of these systems is a union of manifolds and the heteroclinic orbits connect two disjoint components of this set

slow motion for gradient systems with equal depth multiple well potentials abstract for scalar reaction diffusion in one space dimension it has been known for a long time that fronts move with an exponentially small speed for potentials with several distinct minimizers the purpose of this paper is to provide a similar result in the case of systems our method relies on a careful study of the evolution of localized energy this approach also has the advantage of relaxing the preparedness assumptions on the initial datum

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

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

positivity and the attractor dimension in a fourth order reaction diffusion equation in this paper we investigate the semilinear partial differential equation ut fuxxxx uxx u 1 u2 with a view particularly to obtaining some insight into how one might establish positivity preservation results for equations containing fourth order spatial derivatives the maximum principle cannot be applied to such equations however progress can be made by employing some very recent best possible interpolation inequalities due to the third named author in which the interpolation constants are both explicitly known and sharp these are used to estimate the lx distance between u and 1 during the evolution a positivity preservation result can be obtained under certain restrictions on the initial datum we also establish an explicit two sided estimate for the fractal dimension of the attractor which is sharp in terms of the physical parameters

on the slow dynamics for the cahn hilliard equation in one space dimension we study the limiting behaviour of the solution of the cahn hilliard equation using energy type methods we assume that the initial data has a transition layer structure i e u epsilon approx pm 1 except near finitely many transition points we show that in the limit as epsilon rightarrow 0 the solution maintains its transition layer structure and the transition layers move slower than any power of epsilon

on three phase boundary motion and the singular limit of a vector valued ginzburg landau equation we present a formal asymptotic analysis which suggests a model for three phase boundary motion as a singular limit of a vector valued ginzburg landau equation we prove short time existence and uniqueness of solutions for this model that is for a system of three phase boundaries undergoing curvature motion with assigned angle conditions at the meeting point such models pertain to grain boundary motion in alloys the method we use based on linearization about the initial conditions applies to a wide class of parabolic systems we illustrate this further by its application to an eutectic solidification problem

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

a study of the orthogonal polynomials associated with the quantum harmonic oscillator on constant curvature spaces recently carinena et al ann phys 322 434 2007 introduced a new family of orthogonal polynomials that appear in the wave functions of the quantum harmonic oscillator in two dimensional constant curvature spaces they are a generalization of the hermite polynomials and will be called curved hermite polynomials in the following we show that these polynomials are naturally related to the relativistic hermite polynomials introduced by aldaya et al phys lett a 156 381 1991 and thus are jacobi polynomials moreover we exhibit a natural bijection between the solutions of the quantum harmonic oscillator on negative curvature spaces and on positive curvature spaces at last we show a maximum entropy property for the ground states of these oscillators

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

population extinction on a random fitness seascape we explore the role of stochasticity and noise in the statistical outcomes of commonly studied population dynamics models within a space independent mean field perspective specifically we consider a distributed population with logistic growth at each location subject to seascape noise wherein the population s fitness randomly varies with location and time despite its simplicity the model actually incorporates variants of directed percolation and directed polymers in random media within a mean field perspective probability distributions of the population can be computed self consistently and the extinction transition is shown to exhibit novel critical behavior with exponents dependent on the ratio of the strengths of migration and noise amplitudes the results are compared and contrasted with the more conventional choice of demographic noise due to stochastic changes in reproduction

nonconvex variational problems with general singular perturbations we study the effect of a general singular perturbation on a non convex variational problem with infinitely many solutions using a scaling argument and the theory of t convergence of nonlinear functionals we show that if the solutions of the perturbed problem converge in l1 as the perturba tion parameter goes to zero then the limit function satisfies a classical minimal surface problem

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