bilinear restriction estimates for surfaces with curvatures of different signs recently the sharp l 2 bilinear adjoint restriction estimates for the cone and the paraboloid were established by wolff and tao respectively their results rely on the fact that for the cone and the paraboloid the nonzero principal curvatures have the same sign we generalize those bilinear restriction estimates to surfaces with curvatures of different signs

stretching based diagnostics and reduction of chemical kinetic models with diffusion a new method for diagnostics and reduction of dynamical systems and chemical kinetic models is proposed the method makes use of the local structure of the normal stretching rates by projecting the dynamics onto the local directions of maximal stretching the approach is computationally very simple as it implies the spectral analysis of a symmetric matrix notwithstanding its simplicity stretching based analysis derives from a geometric basis grounded on the pointwise applications of concepts of normal hyperbolicity theory as a byproduct a simple reduction method is derived equivalent to a local embedding algorithm which is based on the local projection of the dynamics onto the most unstable and or slow modes compared to the time scale dictated by the local tangential dynamics this method provides excellent results in the analysis and reduction of dynamical systems displaying relaxation towards an equilibrium point limit cycles and chaotic attractors several numerical examples deriving from typical models of reaction diffusion kinetics exhibiting complex dynamics are thoroughly addressed the application to typical combustion models is also analyzed

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

restriction estimates for some surfaces with vanishing curvatures abstract we obtain bilinear restriction estimates for surfaces with vanishing curvatures as application we also prove new linear restriction estimates for some class of conic surfaces

nekhoroshev theorem for small amplitude solutions in nonlinear schrodinger equations we prove a nekhoroshev type result 26 27 for the nonlinear schrodinger equation with vanishing or periodic boundary conditions on 0 pi here m in mathbb r is a parameter and varphi mathbb r to mathbb r is a function analytic in a neighborhood of the origin and such that varphi 0 0 varphi 0 not 0 more precisely we consider the cauchy problem for 0 1 with initial data which extend to analytic entire functions of finite order and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum the proof is obtained by a method which applies to hamiltonian perturbations of linear pde s with the following properties i the linear dynamics is periodic ii there exists a finite order birkhoff normal form which is integrable and quasi convex as a function of the action variables eq 0 1 satisfies i and ii when restricted to a level surface of left vert u l 2 right vert which is an integral of motion the main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here

quasi equilibrium grid algorithm geometric construction for model reduction the method of invariant grid mig is an iterative procedure for model reduction in chemical kinetics which is based on the notion of slow invariant manifold sim a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 a n gorban i v karlin a y zinovyev invariant grids for reaction kinetics physica a 333 2004 106 154 a n gorban i v karlin invariant manifolds for physical and chemical kinetics lecture notes physics 660 springer berlin heidelberg 2005 doi 10 1007 b98103 important role in that method is played by the initial grid which once refined gives a description of the invariant manifold the invariant grid a convenient way to get a first approximation of the sim is given by the spectral quasi equilibrium manifold sqem a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 in the present paper a flexible numerical method to construct the discrete analog of a quasi equilibrium manifold in any dimension is presented that object is named quasi equilibrium grid qeg while the procedure quasi equilibrium grid algorithm qega extensions of the qem notion are also suggested the qega is a numerical tool which can be used to find a grid based approximation for the locus of minima of a convex function under some linear constraints the method is validated by construction of one and two dimensional grids for a model of hydrogen oxidation reaction

invariant grids for reaction kinetics abstract in this paper we construct low dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold mim mim is based on a formulation of the condition of invariance as an equation and its solution by newton iterations a grid based version of mim is developed the method of invariant grids we describe the newton method and the relaxation method for the invariant grids construction the problem of the grid correction is fully decomposed into the problems of the grid s nodes correction the edges between the nodes appear only in the calculation of the tangent spaces this fact determines high computational efficiency of the method of invariant grids the method is illustrated by two examples the simplest catalytic reaction michaelis menten mechanism and the hydrogen oxidation the algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed generalizations to open systems are suggested the set of methods covered makes it possible to effectively reduce description in chemical kinetics

discrete breathers in honeycomb fermi pasta ulam lattices we consider the two dimensional fermi pasta ulam lattice with hexagonal honeycomb symmetry which is a hamiltonian system describing the evolution of a scalar valued quantity subject to nearest neighbour interactions using multiple scale analysis we reduce the governing lattice equations to a nonlinear schrodinger nls equation coupled to a second equation for an accompanying slow mode two cases in which the latter equation can be solved and so the system decoupled are considered in more detail firstly in the case of a symmetric potential we derive the form of moving breathers we find an ellipticity criterion for the wavenumbers of the carrier wave together with asymptotic estimates for the breather energy the minimum energy threshold depends on the wavenumber of the breather we find that this threshold is locally maximised by stationary breathers secondly for an asymmetric potential we find stationary breathers which even with a quadratic nonlinearity generate no second harmonic component in the breather plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure finally we compare the properties of stationary breathers in the square triangular and honeycomb lattices

exponential estimates of symplectic slow manifolds in this paper we prove the existence of an almost invariant symplectic slow manifold for analytic hamiltonian slow fast systems with finitely many slow degrees of freedom for which the error field is exponentially small we allow for infinitely many fast degrees of freedom the method we use is motivated by a paper of mackay from 2004 the method does not notice resonances and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic we also present a stability result and obtain a generalization of a result of gelfreich and lerman on an invariant slow manifold to finitely many fast degrees of freedom

periodic orbits near a bifurcating slow manifold this paper studies a class of one degree of freedom hamiltonian systems with a slowly varying phase the slowly varying phase is assumed to unfold a hamiltonian pitchfork bifurcation the main result of the paper is that there exists an order of ln 2 1 many periodic orbits that all stay within an o 1 3 distance from the union of the normally elliptic slow manifolds that occur as a result of the bifurcation as they pass through the bifurcation the time scales are comparable here 1 measures the time scale separation these periodic orbits are typically moderately unstable this is in contrast with the periodic orbits that remain an o 1 distance from the slow manifold the effect of approaching the normally elliptic slow manifold is therefore to reduce the stability region the smallest stable orbits that are persistently obtained remain further away from the slow manifold being distant by an order o 1 3 ln 1 2 ln 1 the proofs of these statements are based on averaging of two blow up systems allowing one to estimate the effect of the singularity combined with results on asymptotics of the second painleve equation

lowest landau level functional and bargmann spaces for bose einstein condensates abstract a fast rotating bose einstein condensate can be described by a complex valued wave function minimizing an energy restricted to the lowest landau level or fock bargmann space using some structures associated with this space we study the distribution of zeroes of the minimizer and prove in particular that the number of zeroes is infinite we relate their location to the combination of two problems a confining problem producing an inverted parabola profile and the abrikosov problem of minimizing an energy on a lattice using theta functions

quartic normal forms for the periodic nonlinear schrodinger equation with dispersion management we investigate birkhoff normal forms for the periodic nonlinear schrodinger equation with dispersion management the normalization we describe is related to averaging arguments considered in the literature and has the advantage of producing fewer resonant couplings between high spatial frequency modes one consequence is that the normal form equations have invariant subspaces of large but finite dimension where we can find several classes of periodic orbits the formal arguments apply to other related dispersive systems and to normal forms of high order we also present a rigorous version of the normal form calculation and show that solutions of the quartic normal form equations remain close to solutions of the full system over a time that is inversely proportional to a small nonlinearity parameter

existence and stability of 3 site breathers in a triangular lattice we find conditions for existence and stability of various types of discrete breather concentrated around three central sites in a triangular lattice of one dimensional hamiltonian oscillators with on site potential and nearest neighbour coupling in particular we confirm that it can support non reversible breather solutions despite the time reversible character of the system they carry a net energy flux and can be called vortex breathers we prove that there are parameter regions for which they are linearly stable for example in a lattice consisting of coupled morse oscillators whereas the related reversible breathers are unstable thus non reversible breathers can be physically relevant

estimates on invariant tori near an elliptic equilibrium point of a hamiltonian system abstract we give a precise statement for the kam theorem in a neighbourhood of an elliptic equilibrium point of a hamiltonian system if the frequencies of the elliptic point are nonresonant up to a certain order k 4 and a nondegeneracy condition is fulfilled we get an estimate for the measure of the complement of the kam tori in a neighbourhood of given radius moreover if the frequencies satisfy a diophantine condition with exponent we show that in a neighbourhood of radius r the measure of the complement is exponentially small in 1 r 1 1 we also give a related result for quasi diophantine frequencies which is more useful for practical purposes the results are obtained by putting the system in birkhoff normal form up to an appropiate order and the key point relies on giving accurate bounds for its terms

the optimal trilinear restriction estimate for a class of hypersurfaces with curvature abstract in 4 nearly optimal l 1 trilinear restriction estimates in r n 1 are established under transversality assumptions only in this paper we show that the curvature improves the range of exponents by establishing l p estimates for any p 2 n 4 3 n 2 in the case of double conic surfaces the exponent 2 n 4 3 n 2 is shown to be the universal threshold for the trilinear estimate

higher order corrections in the approximation of low dimensional manifolds and the construction of simplified problems with the csp method in systems of stiff ordinary differential equations odes both fast and slow time scales are encountered the fast time scales are responsible for the development of low dimensional manifolds on which the solution moves according to the slow time scales in this paper methodologies for constructing highly accurate i expressions describing the manifold and ii simplified non stiff equations governing the slow evolution of the solution on the manifold are developed according to an iterative procedure proposed in the computational singular perturbation csp method it is shown that the increasing accuracy achieved with each iteration is directly related to the time rates of change of the csp vectors spanning the manifold along the solution trajectory here an algorithm is presented which implements these calculations and is validated on the basis of two simple examples

exponential averaging for hamiltonian evolution equations we derive estimates on the magnitude of non adiabatic interaction between a hamiltonian partial differential equation and a high frequency nonlinear oscillator assuming spatial analyticity of the initial conditions we show that the dynamics can be transformed to the uncoupled dynamics of an infinite dimensional hamiltonian system and an anharmonic oscillator up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator the result is derived from an abstract averaging theorem for infinite dimensional analytic evolution equations in gevrey spaces refining upon a similar result by neishtadt for analytic ordinary differential equations the temporal estimate crucially depends on the spatial regularity of the initial condition the result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data an application is provided by a system of nonlinear schrodinger equations coupled to a rapidly forcing single mode representing small scale oscillations we provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations

endpoint bilinear restriction theorems for the cone and some sharp null form estimates recently wolff 25 obtained a nearly sharp l 2 bilinear restriction theorem for bounded subsets of the cone in general dimension we obtain the endpoint of wolff s estimate and generalize to the case when one of the subsets is large as a consequence we are able to deduce some nearly sharp l p null form estimates

the quasi steady state assumption a case study in perturbation the quasi steady state assumption qssa of biochemistry is studied as an approximation that is important in itself and also that exemplifies an approach to ode systems with an initial fast transient simple estimates of the two relevant time scales of the underlying problem are made these estimates lead to a dimensionless formulation containing a different small parameter from that which is customary earlier results on the qssa are surveyed in the context of the new nondimensionalization it is shown that more comprehensive and simpler error estimates can now be made some general methodological hints are drawn from this example

the hidden dynamics of the rossler attractor abstract the fast slow dynamics of the rossler model in the chaotic regime are compared to the dynamics of the reduced slow model governing the flow along the slow invariant manifold on which the trajectory is restrained to evolve during the slow part of the attractor it is shown that the dynamics of the reduced model incorporate the slow time scales of the full model however instead of the fast dissipative time scale of the full model that restrains the trajectory on the slow invariant manifold the reduced model generates a new time scale that i relates to the curvature of the manifold and ii becomes faster as higher order corrections are incorporated in the approximation of the manifold and the reduced model it is shown that this new time scale does not characterize the motion of the solution on the manifold but it can be employed as a signal for the strengthening or weakening of the manifold depending on whether it relates to components of the reduced model that tend to lead its solution towards equilibrium or away from it finally it is demonstrated that the new time scale introduced by the reduced model has a significant role in determining the maximum accuracy that can be delivered by the singular perturbation methodology

high order breather solutions to a discrete nonlinear klein gordon model abstract a general discrete nonlinear klein gordon model is studied with regards to the movability of breathers approximations to moving breather solutions are found by a semi discrete multiple scale perturbation expansion to 3rd order the higher order effects of discreteness have a remarkable connection to nonlinear fiber optics effects such as higher order dispersion self steepening of the pulse edge and self frequency shift are found modifying both the shape velocity and frequency of the solution here only their combined influence is studied discreteness is found to trap breathers with initial amplitudes higher than a critical value which depends on the internal state of the breather this shows that no strict definition of the peierls nabarro potential can be given for breathers furthermore we find that the envelope velocity of the breather cannot exceed the maximum group velocity which converges to zero as the degree of discreteness is increased thus no moving breathers exist in a highly discrete system

explicit time scale splitting algorithm for stiff problems auto ignition of gaseous mixtures behind a steady shock a new explicit algorithm based on the computational singular perturbation csp method is presented this algorithm is specifically designed to solve stiff problems and its performance increases with stiffness the key concept in its structure is the splitting of the fast from the slow time scales in the problem realized by embedding csp concepts into an explicit scheme in simple terms the algorithm marches in time with only the terms producing the slow time scales while the contribution of the terms producing the fast time scales is taken into account at the end of each integration step as a correction the new algorithm is designed for the integration of stiff systems of pdes by means of explicit schemes for simplicity in the presentation and discussion of the different features of the new algorithm a simple test case is considered involving the auto ignition of a methane air mixture behind a normal shock wave which is described by a system of odes the performance of the new algorithm accuracy and computational efficiency is then compared with the well known lsode package its merits when used for the solution of systems of pdes are discussed although when dealing with a stiff system of odes the new algorithm is shown to provide equal accuracy with that delivered by lsode at the cost of higher execution time the results indicate that its performance could be superior when facing a stiff system of pdes

asymptotic solution of stiff pdes with the csp method the reaction diffusion equation the computational singular perturbation csp method is employed for the solution of stiff pdes and for the acquisition of the most important physical understanding the usefulness of the method is demonstrated by analyzing a transient reaction diffusion problem it is shown that in the regions where the solution exhibits smooth spatial slopes a simple nonstiff system of equations can be used instead of the full governing equations from the simplified system which is numerically provided by csp and whose structure varies with space and time important physical information comes to light the relation of this method to the class of asymptotic expansion methods is explored it is shown that the csp results are identical to the ones obtained by the asymptotic methods the identifications of the nondimensional parameters and the tedious manipulations needed by the asymptotic methods are performed by programmable numerical or analytic computations specified by csp preliminary numerical results are presented validating the theoretical aspects of the proposed algorithm and providing a measure of its usefulness and its accuracy

numerical computation of travelling breathers in klein gordon chains abstract we numerically study the existence of travelling breathers in klein gordon chains which consist of one dimensional networks of nonlinear oscillators in an anharmonic on site potential linearly coupled to their nearest neighbors travelling breathers are spatially localized solutions having the property of being exactly translated by p sites along the chain after a fixed propagation time t these solutions generalize the concept of solitary waves for which p 1 in the case of even on site potentials the existence of small amplitude travelling breathers superposed on a small oscillatory tail has been proved recently g james y sire travelling breathers with exponentially small tails in a chain of nonlinear oscillators commun math phys 2005 in press available online at http www springerlink com the tail being exponentially small with respect to the central oscillation size in this paper we compute these solutions numerically and continue them into the large amplitude regime for different types of even potentials we find that klein gordon chains can support highly localized travelling breather solutions superposed on an oscillatory tail we provide examples where the tail can be made very small and is difficult to detect at the scale of central oscillations in addition we numerically observe the existence of these solutions in the case of non even potentials

integral bounds for radar ambiguity functions and wigner distributions an upper bound is proved for the lp norm of woodward s ambiguity function in radar signal analysis and of the wigner distribution in quantum mechanics when p 2 a lower bound is proved for 1 p 2 in addition a lower bound is proved for the entropy these bounds set limits to the sharpness of the peaking of the ambiguity function or wigner distribution the bounds are best possible and equality is achieved in the lp bounds if and only if the functions f and g that enter the definition are both gaussians

operator splitting for chemical reaction systems with fast chemistry abstract splitting methods are frequently used for large scale chemical reaction systems the main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem however a splitting error is introduced the most popular splitting schemes are the first order lie trotter splitting and the second order strang splitting however in case of stiff differential equations the strang splitting suffers from order reduction and both schemes have order one for stiff differential equations therefore the step size restrictions due to the low order can result in a prohibitive small step size hence splitting schemes with order larger than one are necessary for stiff differential equations the richardson extrapolation of the lie trotter splitting is a second order scheme in this paper we examine differential equations with a fast chemical source term and a slow transport term thus all stiffness of the system is related to the chemical source term we show that the richardson extrapolation does not suffer from order reduction in this case furthermore we perform a stability analysis of the extrapolated splitting scheme for a linear test problem thereby the operators of the test problem do not commute

approximate momentum conservation for spatial semidiscretizations of semilinear wave equations we prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions analytic nonlinearity and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic the method of proof is that of backward error analysis whereby we construct a modified equation which is itself lagrangian and translation invariant and therefore also conserves momentum this modified equation interpolates the semidiscrete system for all time and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system these properties directly imply approximate momentum conservation for the semidiscrete system we also consider discretizations that are not variational as well as discretizations on non uniform grids through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

slow integral manifolds for lagrangian fluid dynamics in unsteady geophysical flows abstract the authors consider lagrangian motion of fluid particles in unsteady gravity currents in geophysical flows the vertical motion of fluid particles especially the induced vertical mixing in these currents is partially responsible for the ocean thermohaline circulation and thus plays a role in the global climate dynamics first a reduced dynamic system for slow variables is derived for a nonautonomous multiscale system the reduced system still nonautonomous is the original system restricted to a centre like nonautonomous invariant manifold so called slow manifold which holds slow motions of the system an algorithm is also presented to obtain an approximation of the nonautonomous slow manifold a novelty here is that the reduction principle applies to nonautonomous multiscale systems which satisfy conditions that are true only locally in space as in many physical cases this makes the reduction principle applicable to real physical systems then this invariant manifold reduction principle is applied to an approximate conceptual lagrangian model of gravity currents and a reduced nonautonomous system for slow vertical motion is obtained this reduced system may be useful as a conceptual tractable tool for understanding some features of vertical mixing in unsteady gravity currents

physical understanding of complex multiscale biochemical models via algorithmic simplification glycolysis in saccharomyces cerevisiae abstract large scale models of cellular reaction networks are usually highly complex and characterized by a wide spectrum of time scales making a direct interpretation and understanding of the relevant mechanisms almost impossible we address this issue by demonstrating the benefits provided by model reduction techniques we employ the computational singular perturbation csp algorithm to analyze the glycolytic pathway of intact yeast cells in the oscillatory regime as a primary object of research for many decades glycolytic oscillations represent a paradigmatic candidate for studying biochemical function and mechanisms using a previously published full scale model of glycolysis we show that due to fast dissipative time scales the solution is asymptotically attracted on a low dimensional manifold without any further input from the investigator csp clarifies several long standing questions in the analysis of glycolytic oscillations such as the origin of the oscillations in the upper part of glycolysis the importance of energy and redox status as well as the fact that neither the oscillations nor cell cell synchronization can be understood in terms of glycolysis as a simple linear chain of sequentially coupled reactions

discrete breathers localization and transfer of energy in discrete hamiltonian nonlinear systems we present a partial review of results concerning the theory of discrete breathers in nonlinear hamiltonian and discrete systems these special time periodic solutions gained much interest during the last decade because they may be involved in numerous and various phenomena in physics and biophysics where they could produce nontrivial effects of energy focusing and transfer we first review the principles which govern their existence and which are used in the existence proofs available up to now next we discuss their linear stability and the interaction of discrete breathers with small amplitude waves showing also how they could grow or decay we also briefly discuss the existence of intraband dbs in systems with a linear discrete spectrum which are not spatially periodic random or otherwise and where linear waves cannot propagate we also show that nonlinearity may restore the existence of solutions that propagate energy we review some results concerning energy transportation by dbs and especially the main features associated with their mobility we briefly discuss new perspectives opened up by the theory for applications that in particular look especially interesting for biophysics

moving discrete breathers abstract we give definitions for different types of moving spatially localized objects in discrete nonlinear lattices we derive general analytical relations connecting frequency velocity and localization length of moving discrete breathers and kinks in nonlinear one dimensional lattices then we propose a new numerical algorithm to find these solutions

discrete breathers classical and quantum abstract discrete breathers are time periodic spatially localised motions in networks of oscillators their occurrence in three principal contexts is reviewed i autonomous hamiltonian or reversible systems ii autonomous forced damped systems and iii time periodically forced systems the main proposed area of application however is to molecular crystals where quantum effects must be taken into account thus an idea for a theory of quantum discrete breathers is also presented though some details remain to be worked out

travelling breathers with exponentially small tails in a chain of nonlinear oscillators we study the existence of travelling breathers in klein gordon chains which consist of one dimensional networks of nonlinear oscillators in an anharmonic on site potential linearly coupled to their nearest neighbors travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity approximate solutions of this type have been constructed in the form of modulated plane waves whose envelopes satisfy the nonlinear schrodinger equation m remoissenet phys rev b33 n 4 2386 1986 j giannoulis and a mielke nonlinearity17 p 551 565 2004 in the case of travelling waves where the phase velocity of the plane wave equals the group velocity of the wave packet the existence of nearby exact solutions has been proved by iooss and kirchgassner who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail g iooss k kirchgassner commun math phys 211 439 464 2000 however a rigorous existence result has been lacking in the more general case when phase and group velocities are different this situation is examined in the present paper in a case when the breather period and the inverse of its velocity are commensurate we show that the center manifold reduction method introduced by iooss and kirchgassner is still applicable when the problem is formulated in an appropriate way this allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations whose principal part admits homoclinic solutions to quasi periodic orbits under general conditions on the potential for an even potential using the additional symmetry of the system we obtain homoclinic orbits to small periodic ones for the full reduced system for the oscillator chain these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail their principal part excluding the tail coincides at leading order with the nonlinear schrodinger approximation

normally elliptic singular perturbations and persistence of homoclinic orbits abstract we consider a dynamical system possibly infinite dimensional or non autonomous with fast and slow time scales which is oscillatory with high frequencies in the fast directions we first derive and justify the limit system of the slow variables assuming a steady state persists we construct the stable unstable center stable center unstable and center manifolds of the steady state of a size of order o 1 and give their leading order approximations finally using these tools we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems

the proof of the l 2 decoupling conjecture we prove the l 2 decoupling conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone this has a wide range of important consequences one of them is the validity of the discrete restriction conjecture which implies the full range of expected l p strichartz estimates for both the rational and up to n losses the irrational torus another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere various applications to additive combinatorics incidence geometry and number theory are also discussed our argument relies on the interplay between linear and multilinear restriction theory

invariant manifolds in singular perturbation problems for ordinary differential equations based on fenichel s geometric idea invariant manifold theory is applied to singular perturbation problems this approach clarifies the nature of outer and inner solutions a specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds a method to approximate the connections is also presented

bounds for the chaotic region in the lorenz model abstract in a previous paper the authors made an extensive numerical study of the lorenz model changing all three parameters of the system we conjectured that the region of parameters where the lorenz model is chaotic is bounded for fixed r in this paper we give a theoretical proof of the conjecture by obtaining theoretical bounds for the chaotic region and by using fenichel theory the theoretical bounds are complemented with numerical studies performed using the maximum lyapunov exponent and ofli2 techniques and a comparison of both sets of results is shown finally we provide a complete three dimensional model of the chaotic regime depending on the three parameters

some integral identities and inequalities for entire functions and their application to the coherent state transform let be an entire function on cn and for any h 0 and r 0 define fr z r e 2 z 2h let d h denote h n times lebesgue measure on cn frs2 2 d h n sh frsd h from this and a logarithmic sobolev inequality we easily deduce qnq fr q pnp fr p for all 0 p q t8 where the lp norms are taken with respect to the measure d h above we apply these results to the study of the spaces ap consisting of all entire functions satisfying z pe 2 z 2h d h obtaining sharp bounds for some associated operators and proving denseness of analytic polynomials in ap for 1 p we then apply our results to the coherent state transform extending and simplifying some previously known results

time averaging under fast periodic forcing of parabolic partial differential equations exponential estimates the phases of a large class of parabolic partial differential equations with rapid time periodic forcing can be separated up to exponential small errors the originally nonautonomous equation is transformed such that the nonautonomous terms are exponentially small in the period h of the forcing this is a counterpart for partial differential equations of the theorem by a neishtadt 1984 j appl math mech 48 134 139 for ordinary differential equations in our case the exponential rate depends on time t and the estimates have the form h exp c t h 1 3

a regularized model equation for discrete breathers in anharmonic lattices with symmetric nearest neighbor potentials abstract we propose a regularized continuum model equation for describing discrete breathers or intrinsic localized modes in one dimensional anharmonic lattices with symmetric nearest neighbor potentials exact stationary breather solutions with purely hard quartic anharmonicity as well as approximate stationary breather solutions in the general case are found the application of the multiple scales analysis indicates the movability of the small amplitude breather solutions the results of numerical simulations for the model equation fully support the analytical solutions as regards the breather breather collisions the continuum model shares many common features with its discrete counterpart which provides an opportunity to clarify the energy exchange mechanism for collisions between discrete breathers in lattices

long periodic orbits and invariant tori in a singularly perturbed hamiltonian system in this paper we study a singularly perturbed two degree of freedom hamiltonian system with a normally elliptic slow manifold we prove that the slow manifold persists but can have a large number e 1 of exponentially small e c e gaps we demonstrate the existence of kam tori in a neighborhood of the slow manifold in addition we investigate a bifurcation which describes the creation of a gap in the slow manifold and derive its normal form 2002 elsevier science b v all rights reserved

properties of the michaelis menten mechanism in phase space abstract we study the two dimensional reduction of the michaelis menten reaction of enzyme kinetics first we prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines second we determine the concavity of all solutions in the first quadrant third we establish the asymptotic behaviour of all solutions near the origin which generally is not given by a taylor series finally we determine the asymptotic behaviour of the slow manifold at infinity to this end we show that the slow manifold can be constructed as a centre manifold for a fixed point at infinity

a direct construction of a slow manifold for a semilinear wave equation of klein gordon type abstract we study a semilinear wave equation whose linear part corresponds to the linear klein gordon equation in the non relativistic limit augmented with a nonlinearity that is frechet differentiable over the complex numbers we show that this equation possesses an almost invariant manifold in phase space that generalizes the slow manifold which is known to exist for finite dimensional galerkin truncations of the system this manifold is shown to be almost invariant to any algebraic order and can be constructed in the h s 1 h s phase space of the equation uniformly in the order of the approximation in particular we prove that the dynamics on this slow manifold shadows orbits of the full system over a finite interval of time

drift of slow variables in slow fast hamiltonian systems abstract we study the drift of slow variables in a slow fast hamiltonian system with several fast and slow degrees of freedom keeping the slow variables frozen for any periodic trajectory of the fast subsystem we define an action for a family of periodic orbits the action is a scalar function of the slow variables and can be considered as a hamiltonian function which generates some slow dynamics these dynamics depend on the family of periodic orbits assuming that for the frozen slow variables the fast system has a pair of hyperbolic periodic orbits connected by two transversal heteroclinic trajectories we prove that for any path composed of a finite sequence of slow trajectories generated by action hamiltonians there is a trajectory of the full system whose slow component shadows the path

the endpoint case of the bennett carbery tao multilinear kakeya conjecture we prove the endpoint case of the multilinear kakeya conjecture of bennett carbery and tao the proof uses the polynomial method introduced by dvir

the use of algebraic sets in the approximation of inertial manifolds and lumping in chemical kinetic systems abstract the use of inertial manifolds for the derivation of reduced representations of complex chemical models is investigated using a method due to foias and temam physica d 32 1988 163 the method which is based on the approximation of inertial manifolds using algebraic sets is extended for use in pde systems and is shown to provide new and interesting results in reaction diffusion systems both the diffusion term and the initial fast transient behaviour are seen to have an effect on the long time dynamics of the remaining slow variable comparison with other methods for the reduction of complex chemical systems is made including some comments about the application of the quasi steady state approximation qssa in reaction diffusion systems the method is illustrated using as examples some systems which are of interest to applications in chemical and enzyme kinetics

effective hamiltonian for travelling discrete breathers hamiltonian chains of oscillators in general probably do not sustain exact travelling discrete breathers however solutions which look like moving discrete breathers for some time are not difficult to observe in numerics in this paper we propose an abstract framework for the description of approximate travelling discrete breathers in hamiltonian chains of oscillators the method is based on the construction of an effective hamiltonian enabling one to describe the dynamics of the translation degree of freedom of moving breathers error estimate on the approximate dynamics is also studied the concept of the peierls nabarro barrier can be made clear in this framework we illustrate the method with two simple examples namely the salerno model which interpolates between the ablowitz ladik lattice and the discrete nonlinear schrodinger system and the fermi pasta ulam chain

restriction estimates using polynomial partitioning ii we improve the estimates in the restriction problem in dimension n geqslant 4 to do so we establish a weak version of a k linear restriction estimate for any k the exponents in this weak k linear estimate are sharp for all k and n

hs class construction of an almost invariant slow subspace for the klein gordon equation in the non relativistic limit we consider the linear klein gordon equation in one spatial dimension with periodic boundary conditions in the non relativistic limit where e 2 mc2 tends to zero it is classical that the equation is well posed for example in the sense of possessing a continuous semiflow into spaces hs 1 hs for wave function and momentum respectively in this paper we iteratively construct a family of bounded operators fslown hs 1 hs whose graphs are o en invariant subspaces under the klein gordon evolution for o 1 times contrary to a naive asymptotic series there is no loss of derivatives in the iterative step i e the sobolev index s can be chosen independent of n this is achieved by solving an operator sylvester equation at each step of the construction

averaging slaving and balance dynamics in a simple atmospheric model we report numerical results from a study of balance dynamics using a simple model of atmospheric motion that is designed to help address the question of why balance dynamics is so stable the non autonomous hamiltonian model has a chaotic slow degree of freedom representing vortical modes coupled to one or two linear fast oscillators representing inertia gravity waves the system is said to be balanced when the fast and slow degrees of freedom are separated we find adiabatic invariants that drift slowly in time this drift is consistent with a random walk behaviour at a speed which qualitatively scales even for modest time scale separations as the upper bound given by neishtadt s and nekhoroshev s theorems moreover a similar type of scaling is observed for solutions obtained using a singular perturbation slaving technique in resonant cases where nekhoroshev s theorem does not apply we present evidence that the smaller lyapunov exponents of the system scale exponentially as well the results suggest that the observed stability of nearly slow motion is a consequence of the approximate adiabatic invariance of the fast motion

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

multiple time scales in nonlinear flight mechanics diagnosis and modeling there are often disparate time scales in the dynamics of flight creating the potential for reduced order modeling to simplify simulation analysis and design there have been notable successes in developing reduced order models however in the case of nonlinear dynamics which one must typically deal with in guidance problems there has not been a systematic reliable means of diagnosing disparate time scales and developing reduced order models focusing on two time scale behavior in nonlinear dynamical systems we recall fenichel s characterization of the geometric structure of the flow and his theorem establishing the existence and properties of coordinates adapted to this structure adapted coordinates are difficult to construct directly without an appropriate singularly perturbed model of the dynamics we discuss the use of lyapunov exponents and vectors to diagnose two time scale behavior and to determine the corresponding tangent space structure for the linearized dynamics the structure of the linearized flow can then be translated into the manifold structure of the nonlinear flow we briefly mention the use of lyapunov vectors to locate a slow manifold and contrast this approach with two existing approaches the minimum time to climb problem provides an example of two time scale behavior and motivates the discussion

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

natural tangent dynamics with recurrent biorthonormalizations a geometric computational approach to dynamical systems exhibiting slow manifolds and periodic chaotic limit sets abstract this article develops a geometric method referred to as natural tangent dynamics with recurrent biorthonormalizations ntdrb that unifies into a single computational technique the analysis of dynamical systems possessing either a slow manifold associated with an equilibrium point or exhibiting nontrivial limit sets limit cycles and chaotic attractors for the first class of systems the main focus relies on the identification of the slow fast timescales while for the latter dynamics within nontrivial attractors the central issue is the temporal dichotomy between unstable stable components as described by the structure of the lyapunov spectrum the method is based on vector and covector dynamics in the tangent and cotangent bundles induced by the phase space flow and on the periodic application of a biorthonormalization procedure which contrasts the natural tendencies of vectors and covectors to align towards the most unstable slow subspaces the ntdrb method provides the most convenient and natural vector and covector bases for i the characterization of the fast timescales controlling the dynamic relaxation towards a slow manifold ii the identification of the stable and unstable components of the dynamics on a limit set iii the construction of a low dimensional reduced model reproducing the asymptotic behavior of the original systems

small generalized breathers with exponentially small tails for klein gordon equations abstract we consider a class of nonlinear klein gordon equation u t t u x x u f u and show that generically there exist small breathers with exponentially small tails

fast rotating bose einstein condensates in an asymmetric trap abstract we investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating bose einstein condensate this is done in the framework of the 2d gross pitaevskii equation and requires a symplectic reduction of the quadratic form defining the energy this reduction allows us to simplify the energy on a bargmann space and study the asymptotics of large rotational velocity we characterize two regimes of velocity and anisotropy in the first one where the behaviour is similar to the isotropic case we construct an upper bound a hexagonal abrikosov lattice of vortices with an inverted parabola profile the second regime deals with very large velocities a case in which we prove that the ground state does not display vortices in the bulk with a 1d limiting problem in that case we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a gaussian in the other direction the features of this second regime appear as new phenomena

an efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems the relation between the iterative algorithms based on the computational singular perturbation csp and the invariance equation ie methods is examined the success of the two methods is based on the appearance of fast and slow time scales in the dynamics of stiff systems both methods can identify the low dimensional surface in the phase space slow invariant manifold sim where the state vector is attracted under the action of fast dynamics it is shown that this equivalence of the two methods can be expressed by simple algebraic relations csp can also construct the simplified non stiff system that models the slow dynamics of the state vector on the sim an extended version of ie is presented which can also perform this task this new ie version is shown to be exactly similar to a modified version of csp which results in a very efficient algorithm especially in cases where the sim dimension is small so that significant model simplifications are possible

symmetry and resonance in periodic fpu chains the symmetry and resonance properties of the fermi pasta ulam chain with periodic boundary conditions are exploited to construct a near identity transformation bringing this hamiltonian system into a particularly simple form this birkhoff gustavson normal form retains the symmetries of the original system and we show that in most cases this allows us to view the periodic fpu hamiltonian as a perturbation of a nondegenerate liouville integrable hamiltonian according to the kam theorem this proves the existence of many invariant tori on which motion is quasiperiodic experiments confirm this qualitative behaviour we note that one can not expect this in lower order resonant hamiltonian systems so the periodic fpu chain is an exception and its special features are caused by a combination of special resonances and symmetries

asymptotic analysis of two reduction methods for systems of chemical reactions abstract this paper concerns two methods for reducing large systems of chemical kinetics equations namely the method of intrinsic low dimensional manifolds ildms due to maas and pope combust flame 88 1992 239 and an iterative method due to fraser j chem phys 88 1988 4732 and further developed by roussel and fraser j chem phys 93 1990 1072 both methods exploit the separation of fast and slow reaction time scales to find low dimensional manifolds in the space of species concentrations where the long term dynamics are played out the asymptotic expansions of these manifolds e 0 where e measures the ratio of the reaction time scales are compared with the asymptotic expansion of m e the slow manifold given by geometric singular perturbation theory it is shown that the expansions of the ildm and m e agree up to and including terms of o e the former has an error at o e 2 that is proportional to the local curvature of m 0 the error vanishes if and only if the curvature is zero everywhere the iterative method generates term by term the asymptotic expansion of m e starting from m 0 the ith application of the algorithm yields the correct expansion coefficient at o e i while leaving the lower order coefficients invariant thus after l applications the expansion is accurate up to and including the terms of o e l the analytical results are illustrated on a planar system from enzyme kinetics michaelis menten henri and a model planar system due to davis and skodje

near integrability of periodic fpu chains abstract the fpu chain with periodic boundary conditions is studied by birkhoff gustavson normalisation in the cases of up to 6 particles and for chains with an odd number of particles the normal forms are integrable which permits us to apply kam theory this leads to the presence of many invariant tori on which the motion is quasi periodic thus we explain the recurrence phenomena and the small size of chaos observed in experiments furthermore we find a certain clustering of modes

stabilizing effect of large average initial velocity in forced dissipative pdes invariant with respect to galilean transformations abstract we describe a topological method to study the dynamics of dissipative pdes on a torus with rapidly oscillating forcing terms we show that a dissipative pde which is invariant with respect to the galilean transformations with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms we apply the technique to the viscous burgers equation and the incompressible 2d navier stokes equations with a time dependent forcing we prove that for a large initial average speed the equation admits a bounded eternal solution which attracts all other solutions forward in time for the incompressible 3d navier stokes equations we establish the existence of a locally attracting solution

multibreathers in klein gordon chains with interactions beyond nearest neighbors abstract we study the existence and stability of multibreathers in klein gordon chains with interactions that are not restricted to nearest neighbors we provide a general framework where such long range effects can be taken into consideration for arbitrarily varying as a function of the node distance linear couplings between arbitrary sets of neighbors in the chain by examining special case examples such as three site breathers with next nearest neighbors we find crucial modifications to the nearest neighbor picture of one dimensional oscillators being excited either in or anti phase configurations with nontrivial phase profiles emerge from or collide with the ones with standard 0 or phase difference profiles through supercritical or subcritical bifurcations respectively similar bifurcations emerge when examining four site breathers with either next nearest neighbor or even interactions with the three nearest one dimensional neighbors the latter setting can be thought of as a prototype for the two dimensional building block namely a square of lattice nodes which is also examined