time varying radial basis functions we introduce radial basis functions rbfs whose time varying coefficients determine not only the amplitude and position of each rbf but also their shape the intended use of these time varying rbfs tv rbfs is in the local in time representation of low dimensional approximations of functions that arise in solving spatiotemporal evolution problems in particular for time varying spatially localized solutions with a temporal translation component such as traveling waves modulated pulses or soliton like solutions of evolutionary differential equations this paper is restricted to the one dimensional spatial case we also present an algorithm that places the time varying rbfs tv rbfs over spatiotemporal data that may come from experiments from finely discretized pde simulations or even from multiscale particle based simulations it first approximates the function at a single time instant a temporal snapshot as a sum of rbfs using a novel weighted minimization that causes each rbf to primarily approximate one of the localized parts of the function it then extends that approximation to tv rbfs over a sequence of snapshots of the function at different times we conclude by discussing the potential uses of these tv rbfs

bifurcations of relative periodic orbits in nls gp with a triple well potential abstract the nonlinear schrodinger gross pitaevskii nls gp equation is considered in the presence of three equally spaced potentials the problem is reduced to a finite dimensional hamiltonian system by a galerkin truncation families of oscillatory orbits are sought in the neighborhoods of the system s nine branches of standing wave solutions normal forms are computed in the neighborhood of these branches various hamiltonian hopf and saddle node bifurcations showing how the oscillatory orbits change as a parameter is increased numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and nls gp near the hamiltonian hopf bifurcations and some subtle disagreements near the saddle node bifurcations due to exponentially small terms in the asymptotics

renormalization group reduction of pulse dynamics in thermally loaded optical parametric oscillators we derive a perturbed parametrically forced nonlinear schrodinger equation to model pulse evolution in an optical parametric oscillator with absorption induced heating we apply both a rigorous renormalization group rg technique and the formal wilsonian renormalization group to obtain a low dimensional system of equations which captures the mutual interaction of pulses as well as their response to the thermally induced potential we compare the methodologies of the two approaches and find that the two reduced systems agree to leading order and compare well to simulations of the full equation predicting the formation of stably bound pulse pairs

propagation of reactions in inhomogeneous media consider reaction diffusion equation ut u f x u with x d and general inhomogeneous ignition reaction 0 vanishing at u 0 1 typical solutions 0 u 1 transition from 0 to 1 as time progresses and we study them in the region where this transition occurs under fairly general qualitative hypotheses on f we show that in dimensions d 3 the hausdorff distance of the superlevel sets u e and u 1 e remains uniformly bounded in time for each e 0 1 thus u remains uniformly in time close to the characteristic function of u in the sense of hausdorff distance of superlevel sets we also show that each u e expands with average speed over any long enough time interval between the two spreading speeds corresponding to any x independent lower and upper bounds on f on the other hand these results turn out to be false in dimensions d 4 at least without further quantitative hypotheses on f the proof for d 3 is based on showing that as the solution propagates small values of u cannot escape far ahead of values close to 1 the proof for d 4 is via construction of a counterexample for which this fails such results were before known for d 1 but are new for general non periodic media in dimensions d 2 some are also new for homogeneous and periodic media they extend in a somewhat weaker sense to monostable bistable and mixed reaction types as well as to transitions between general equilibria u u of the pde and to solutions not necessarily satisfying u u u 2016 wiley periodicals inc

uniform compact attractors for the coupled suspension bridge equations in this paper we investigate the uniformly absorbing set for the coupled suspension bridge equations with a new class of external forces and then prove the existence of the uniform attractors for the family of process corresponding to the equations

asymptotic properties of excited states in the thomas fermi limit abstract excited states are stationary localized solutions of the gross pitaevskii equation with a harmonic potential and a repulsive nonlinear term that have zeros on a real axis the existence and the asymptotic properties of excited states are considered in the semi classical thomas fermi limit using the method of lyapunov schmidt reductions and the known properties of the ground state in the thomas fermi limit we show that the excited states can be approximated by a product of dark solitons localized waves of the defocusing nonlinear schrodinger equation with nonzero boundary conditions and the ground state the dark solitons are centered at the equilibrium points where a balance between the actions of the harmonic potential and the tail to tail interaction potential is achieved

existence and stability of spatially localized patterns abstract spatially localized patterns have been observed in numerous physical contexts and their bifurcation diagrams often exhibit similar snaking behavior symmetric solution branches connected by bifurcating asymmetric solution branches wind back and forth in an appropriate parameter previous papers have addressed existence of such solutions here we address their stability taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions we then show that under appropriate assumptions temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane in a companion paper we analyze the behavior of eigenvalues at 0 and inside the essential spectrum our results show that localized snaking solutions are stable if and only if the underlying fronts and backs are stable unlike localized non oscillatory solutions no interaction eigenvalues are present we use the planar swift hohenberg system to illustrate our results

patterns of propagating pulses the complex dynamics that arise in certain nonlinear partial differential equations in time and in one space dimension are studied in the general case considered the equation admits a solitary wave in the form of a pulse tailing off exponentially fore and aft with possibly oscillatory character complicated solutions are described by a superposition of many such solitary structures in interaction the description is asymptotic in terms of a parameter that becomes exponentially small as the ratio of typical pulse separation to pulse width becomes large the outcome is a set of dynamical equations for the motion of the individual pulses with nearest neighbor interactions this system of ordinary differential equations odes admits a wide range of patterns both regular and chaotic the stability theory of such patterns is sketched and the continuum limit of the lattice dynamical equations of the pulses is given

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

the accumulation of eigenvalues in a stability problem abstract when two waves propagating in a one dimensional medium are locked together as a composite wave a natural question arises as to whether the new wave is stable an interesting and novel instability mechanism is exposed here in which a cascade of eigenvalues accumulates at a distinguished point in the unstable half plane the underlying assumption is that the transition between the two waves occurs at an unstable homogeneous steady state of the partial differential equations this causes the individual waves to have an unstable continuous spectrum but the instability of the full wave cannot be predicted from the configuration of these spectra alone

cascades of reversible homoclinic orbits to a saddle focus equilibrium abstract in a reversible system we consider a homoclinic orbit being bi asymptotic to a saddle focus equilibrium as was proved by devaney 1977 there exists a one parameter family of periodic orbits accumulating onto this homoclinic orbit in the present paper we show that for any n 2 there exist infinitely many n homoclinic orbits in a neighborhood of the primary homoclinic orbit each of them is accompanied by one or more families of periodic orbits moreover we indicate how these families of periodic orbits correspond to branches of subharmonic periodic orbits

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

long time behavior of a suspension bridge equations with past history in this paper we study a suspension bridge equation with memory effects for the suspension bridge equation without memory there are many classical results existing results mainly devoted to existence and uniqueness of a weak solution energy decay of solution and existence of global attractors however the existence of global attractors for the suspension bridge equation with memory was no yet considered the object of the present paper is to provide some results on the well posedness and long time behavior to the suspension bridge equation when the unique damping mechanism is given by the memory term

hodge decomposition with degenerate weights and the gross pitaevskii energy abstract we solve a hodge decomposition problem with respect to a weight function that vanishes on the boundary this problem is present in several works on the gross pitaevskii energy and we use our solution to show that a convergence result that links these works is still valid for more general data than previously considered

existence of traveling wave trains in nerve axon equations we establish the existence of two discrete classes of traveling wave solutions to nerve axon equations under conditions on the linearization of the system about the rest state each solution in one class consists of a finite train of pulses while each solution in the other class consists of an infinite train that arises from the rest state in the leading direction the solutions are distinguished by the spacing of the pulses in the train and indexed by n tuples and sequences of integers respectively in the two classes this result is obtained from study of the resulting parametrized dynamical system which has a a single homoclinic orbit for one parameter value these assumptions were previously used to show the existence of the double pulse solutions and are known to hold in for some nerve models

numerical bifurcation and stability analysis of solitary pulses in an excitable reaction diffusion medium abstract we present a systematic computer assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space the reaction diffusion model used is of the activator inhibitor type only the activator diffuses in this medium the control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold this study focuses on travelling pulses their domain of existence and the bifurcations that render them unstable these pulses are approximated as a homoclinic orbits in a travelling wave ode frame and b as solutions of the full partial differential equation pde with periodic boundary conditions in large domains a variety of bifurcations in the travelling wave ode frame are observed including heteroclinic loops so called t points a r champneys and y a kuznetsov numerical detection and continuation of codimension 2 homoclinic bifurcations int j bif chaos 4 1994 785 h kokobu homoclinic and heteroclinic bifurcations of vectorfields japan j appl math 5 1988 455 instabilities in the full pde frame include both hopf bifurcations to modulated travelling waves involving the discrete pulse spectrum as well as transitions involving the continuous spectrum such as the so called backfiring transition m bar m hildebrand m eiswirth m falcke h engel and m neufeld chemical turbulence and standing waves in a surface reaction model the influence of global coupling and wave instabilities chaos 4 1994 499 the stability of modulated pulses is computed through numerical floquet analysis and a cascade of period doubling bifurcations is observed as well as certain global bifurcations these results corroborated by observations from direct numerical integration provide a skeleton around which many features of the overall complex spatiotemporal dynamics of the pde are organized

existence of strong solutions and global attractors for the coupled suspension bridge equations abstract in this paper we show the existence of the strong solutions for the coupled suspension bridge equations furthermore existence of the strong global attractors is investigated using a new semigroup scheme since the solutions of the coupled equation have no higher regularity and the semigroup associated with the solutions is not continuous in the strong hilbert space the results are new and appear to be optimal

on nonlinear oscillations in a suspension bridge system in this paper we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations by applying the leray schauder degree theory it is proved that the suspension bridge system has at least two solutions one is a near equilibrium oscillation and the other is a large amplitude oscillation

generalized fronts in reaction diffusion equations with mono stable nonlinearity abstract we consider transition fronts generalized traveling fronts of mono stable reaction diffusion equations with spatially inhomogeneous nonlinearity by constructing a cutoff function and using an approximate method we establish the existence of transition fronts of the equation furthermore we give the uniform non degeneracy estimates of the solutions such as a lower bound on the time derivative on some level sets as well as an upper bound on the spatial derivative

front like entire solutions for equations with convection abstract we construct families of front like entire solutions for problems with convection both for bistable and monostable reaction diffusion convection equations and via vanishing viscosity arguments for bistable and monostable balance laws the unified approach employed is inspired by ideas of chen and guo and based on a robust method using front dependent sub and supersolutions unlike convectionless problems the equations studied here lack symmetry between increasing and decreasing travelling waves which affects the choice of sub and supersolutions used our entire solutions include both those that behave like two fronts coming together and annihilating as time increases and for bistable equations those that behave like two fronts merging to propagate like a single front we also characterise the long time behaviour of each family of entire solutions which in the case of solutions constructed from a monostable front merging with a bistable front answers a question that was open even for reaction diffusion equations without convection

existence and stability of n pulses on optical fibers with phase sensitive amplifiers abstract the propagation of pulses in optical communication systems in which attenuation is compensated by phase sensitive amplifiers is investigated a central issue is whether optical fibers are capable of carrying several pieces of information at the same time in this paper multiple pulses are shown to exist for a fourth order nonlinear diffusion model due to kutz and co workers 1994 moreover criteria are derived for determining which of these pulses are stable the pulses arise in a reversible orbit flip a homoclinic bifurcation investigated here for the first time numerical simulations are used to study multiple pulses far away from the actual bifurcation point they confirm that properties of the multiple pulses including their stability are surprisingly well predicted by the analysis carried out near the bifurcation

asymptotic behavior of critical points for a gross pitaevskii energy abstract in this paper we study the asymptotic behavior of critical points of a gross pitaevskii energy which is proposed as a model for rotationally forced bose einstein condensate we prove that the limiting singularity set is one dimensional rectifiable we also establish the convergence result for critical points away from limiting singularities

mathematical analysis of dynamic models of suspension bridges in this paper we present some mathematical analysis of dynamic pde models of suspension bridges as proposed by lazer and mckenna our results are illustrated by numerical simulation with physical interpretation

pullback attractors for a non autonomous incompressible non newtonian fluid this paper studies the pullback asymptotic behavior of solutions for a non autonomous incompressible non newtonian fluid in two dimensional 2d bounded domains we first prove the existence of pullback attractors av in space v has h2 regularity see notation in section 2 and ah in space h has l2 regularity for the cocycle corresponding to the solutions of the fluid then we verify the regularity of the pullback attractors by showing av ah which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data

thomas fermi approximation for coexisting two component bose einstein condensates and nonexistence of vortices for small rotation we study minimizers of a gross pitaevskii energy describing a two component bose einstein condensate confined in a radially symmetric harmonic trap and set into rotation we consider the case of coexistence of the components in the thomas fermi regime where a small parameter varepsilon conveys a singular perturbation the minimizer of the energy without rotation is determined as the positive solution of a system of coupled pdes for which we show uniqueness the limiting problem for varepsilon 0 has degenerate and irregular behavior at specific radii where the gradient blows up by means of a perturbation argument we obtain precise estimates for the convergence of the minimizer to this limiting profile as varepsilon tends to 0 for low rotation based on these estimates we can show that the ground states remain real valued and do not have vortices even in the region of small density

search extension method for multiple solutions of a nonlinear problem abstract based on the eigensystem j j of multiple solutions of nonlinear problem u u 3 0 min u 0 on are approximated a new method called the search extension method sem composed of three steps is proposed numerical examples in a square and the unit circle are presented to show the efficiency of our method

non existence of vortices in the small density region of a condensate abstract in this paper we answer a question raised by lev pitaevskii and prove that the ground state of the gross pitaevskii energy describing a bose einstein condensate in a rotationally symmetric trap at low rotation does not have vortices in the low density region therefore the first ground state with vortices has its vortices in the bulk in fact we prove something stronger which is that the ground state for the model at low and moderate rotations is equal to the ground state in a condensate with no rotation this is obtained by proving that for small rotational velocities the ground state is multiple of the ground state with zero rotation we rely on sharp bounds of the decay of the wave function combined with weighted jacobian estimates

generalized traveling waves in disordered media existence uniqueness and stability we prove existence uniqueness and stability of transition fronts generalized traveling waves for reaction diffusion equations in cylindrical domains with general inhomogeneous ignition reactions we also show uniform convergence of solutions with exponentially decaying initial data to time translations of the front in the case of stationary ergodic reactions the fronts are proved to propagate with a deterministic positive speed our results extend to reaction advection diffusion equations with periodic advection and diffusion

on nonlinear boundary value problems ghosts parasites and discretizations ghosts are a new class of approximate solutions to nonlinear boundary value problems bvps they are bounded away from true solutions but solve an initial value problem ivp for the differential equations exactly and approximate boundary and internal matching conditions with exponentially decaying error as a control parameter approaches infinity ghosts are characteristic of singularly perturbed differential equations in which solutions are localized and the ivp depends sensitively on initial data they correspond to shifted localized solutions and they can be observed in imperfect physical experiments and be approximated by convergent discretization schemes the latter property suggests that the concept of the numerically irrelevant solution introduced for discretized second order bvps should be re examined

pullback d attractors for non autonomous suspension bridge equations abstract in this paper we prove the existence of a pullback d attractor in v h for the non autonomous suspension bridge equations

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

turning points and traveling waves in fitzhugh nagumo type equations abstract consider the following fitzhugh nagumo type equation u t u x x f u w w t g u w where f u w u u a w 1 u for some smooth function a w and g u w u w by allowing a w to cross zero and one the corresponding traveling wave equation possesses special turning points which result in very rich dynamics in this work we examine the existence of fronts backs and pulses solutions in particular the co existence of different fronts will be discussed

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

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

multi hump pulses in systems with reflection and phase invariance abstract we investigate the existence and stability of standing and travelling multi hump waves in partial differential equations with reflection and phase symmetries we focus on 2 and 3 pulse solutions that arise near bi foci and apply our results to the complex cubic quintic ginzburg landau equation

eigenvalues of a nonlinear ground state in the thomas fermi approximation abstract we study a nonlinear ground state of the gross pitaevskii equation with a parabolic potential in the hydrodynamics limit often referred to as the thomas fermi approximation existence of the energy minimizer has been known in literature for some time but it was only recently when the thomas fermi approximation was rigorously justified the spectrum of linearization of the gross pitaevskii equation at the ground state consists of an unbounded sequence of positive eigenvalues we analyze convergence of eigenvalues in the hydrodynamics limit convergence in norm of the resolvent operator is proved and the convergence rate is estimated we also study asymptotic and numerical approximations of eigenfunctions and eigenvalues using airy functions

a mountain pass method for the numerical solution of semilinear wave equations it is well known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals in the situation that we studied there is usually an obvious solution obtained as a solution of linear problem we formulate a dual variational problem in such a way that the obvious solution is a local minimum we then find additional non obvious solutions via a numerical mountain pass algorithm based on the theorems of ambrosetti rabinowitz and ekeland numerical results are presented

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

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

two component bose einstein condensates abstract recently coupled systems of nonlinear schrodinger equations have been used extensively to describe bose einstein condensates in this paper we study the structure of vortices of the coupled nonlinear equations for two component bose einstein condensates bec in a three dimensional space we show that vortices is 1 rectifiable set and give its mean curvature in particular we show that large interspecies scattering length causes vortices for two component bec

periodic solutions of a nonlinear suspension bridge equation with damping and nonconstant load abstract the paper is concerned with the existence of periodic solutions for the lazer mckenna suspension bridge equation with damping and nonconstant load by using the lyapunov schmidt reduction methods the author discuss the relationship between the sign changing solutions and the source terms the result answers partly the open problem in lazer and mckenna siam rev 32 1990 537 578

random attractors for the coupled suspension bridge equations with white noises the paper is devoted to the investigation of the existence of a compact random attractor for the random dynamical system generated by the coupled suspension bridge equations with white noises

random attractors for the extensible suspension bridge equation with white noise this paper is devoted to the dynamical behavior of a stochastic extensible suspension bridge equation with white noise for a deterministic suspension bridge equation there are many classical results such as existence and uniqueness of a solution and long term behavior of solutions to the best of our knowledge the existence of attractors for the suspension bridge equation with white noise was not yet considered we intend to investigate these problems we first obtain the dissipativeness of a solution in higher energy spaces h 3 h 2 h 0 1 this implies that the random dynamical system generated by the equation has a random attractor in h 2 h 0 1 l 2 which is a tempered random set in the space in h 3 h 2 h 0 1

lower bounds for the energy of unit vector fields and applications we prove lower bounds for the dirichlet energy of a unit vector field defined in a perforated domain of r2with nonzero degree on the outer boundary in terms of the total diameter of the holes we use this to derive lower bounds and then compactness results for sequences ue of minimizers or almost minimizers of ginzburg landau functionals with coupling constant 1 e2tending to

traveling waves in the baer and rinzel model of spine studded dendritic tissue abstract the baer and rinzel model of dendritic spines uniformly distributed along a dendritic cable is shown to admit a variety of regular traveling wave solutions including solitary pulses multiple pulses and periodic waves we investigate numerically the speed of these waves and their propagation failure as functions of the system parameters by numerical continuation multiple pulse waves are shown to occur close to the primary pulse except in certain exceptional regions of parameter space which we identify the propagation failure of solitary and multiple pulse waves is shown to be associated with the destruction of a saddle node bifurcation of periodic orbits the system also supports many types of irregular wave trains these include waves which may be regarded as connections to periodics and bursting patterns in which pulses can cluster together in well defined packets the behavior and properties of both these irregular spike trains is explained within a kinematic framework that is based on the times of wave pulses the dispersion curve for periodic waves is important for such a description and is obtained in a straightforward manner using the numerical scheme developed for the study of the speed of a periodic wave stability of periodic waves within the kinematic theory is given in terms of the derivative of the dispersion curve and provides a weak form of stability that may be applied to solutions of the traveling wave equations the kinematic theory correctly predicts the conditions for period doubling bifurcations and the generation of bursting states moreover it also accurately describes the shape and speed of the traveling front that connects waves with two different periods

global attractors for a semilinear hyperbolic equation in viscoelasticity a semilinear partial differential equation of hyperbolic type with a convolution term describing simple viscoelastic materials with fading memory is considered regarding the past history memory of the displacement as a new variable the equation is transformed into a dynamical system in a suitable hilbert space the dissipation is extremely weak and it is all contained in the memory term longtime behavior of solutions is analyzed in particular in the autonomous case the existence of a global attractor for solutions is achieved

the critical velocity for vortex existence in a two dimensional rotating bose einstein condensate we investigate a model corresponding to the experiments for a two dimensional rotating bose einstein condensate it consists in minimizing a gross pitaevskii functional defined in r 2 under the unit mass constraint we estimate the critical rotational speed 1 for vortex existence in the bulk of the condensate and we give some fundamental energy estimates for velocities close to 1

nonlinear stability of gardner breathers abstract we show that breather solutions of the gardner equation a natural generalization of the kdv and mkdv equations are h 2 r stable through a variational approach we characterize gardner breathers as minimizers of a new lyapunov functional and we study the associated spectral problem through i the analysis of the spectrum of explicit linear systems spectral stability and i i controlling degenerated directions by using low regularity conservation laws

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

the semistrong limit of multipulse interaction in a thermally driven optical system abstract we consider the semistrong limit of pulse interaction in a thermally driven parametrically forced nonlinear schrodinger tdnls system modeling pulse interaction in an optical cavity the tdnls couples a parabolic equation to a hyperbolic system and in the semistrong scaling we construct pulse solutions which experience both short range tail tail interactions and long range thermal coupling we extend the renormalization group rg methods used to derive semistrong interaction laws in reaction diffusion systems to the hyperbolic parabolic setting of the tdnls system a key step is to capture the singularly perturbed structure of the semigroup through the control of the commutator of the resolvent and a re scaling operator the rg approach reduces the pulse dynamics to a closed system of ordinary differential equations for the pulse locations

multiple periodic solutions for suspension bridge systems abstract this paper studies the existence of multiple periodic oscillations of the suspension bridge system governed by the coupled nonlinear beam and wave equations by using the degree theory it is proved that the system has at least three periodic solutions

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 of global attractors for the coupled system of suspension bridge equations a mathematical model of the suspension bridge describes the vibration of the road bed in the vertical plain and that of the main cable we show the existence of an absorbing set for the solution of the problem furthermore the global attractors of the coupled system of suspension bridge are studied by a new semigroup approach

existence of strong global attractors for hyperbolic equation with linear memory we prove that the existence of global attractors of strong solutions for the hyperbolic equations with linear memory using the semigroup approach

solitary wave interactions in dispersive equations using manton s approach we generalize the approach first proposed by manton nucl phys b 150 397 1979 to compute solitary wave interactions in translationally invariant dispersive equations that support such localized solutions the approach is illustrated using as examples solitons in the korteweg de vries equation standing waves in the nonlinear schr odinger equation and kinks as well as breathers of the sine gordon equation

lyapunov schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies abstract this article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits in addition tangencies in the intersection of un stable manifolds are allowed the main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time and their parameter values a leading order expansion is given in terms of the time spent near vertices and if applicable the location on the non trivial tangent directions the only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit the essential assumptions are hyperbolicity of the vertices and transversality of parameters using the result conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven finally equilibrium to periodic orbit heteroclinic cycles of various types are considered

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

persistence and stability of discrete vortices in nonlinear schrodinger lattices abstract we study discrete vortices in the two dimensional nonlinear schrodinger lattice with small coupling between lattice nodes the discrete vortices in the anti continuum limit of zero coupling represent a finite set of excited nodes on a closed discrete contour with a non zero charge using the lyapunov schmidt reductions we analyze continuation and termination of the discrete vortices for small coupling between lattice nodes an example of a square discrete contour is considered that includes the vortex cell also known as the off site vortex we classify families of symmetric and asymmetric discrete vortices that bifurcate from the anti continuum limit we predict analytically and confirm numerically the number of unstable eigenvalues associated with each family of such discrete vortices