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

spreading speeds and traveling waves for periodic evolution systems abstract the theory of spreading speeds and traveling waves for monotone autonomous semiflows is extended to periodic semiflows in the monostable case then these abstract results are applied to a periodic system modeling man environment man epidemics a periodic time delayed and diffusive equation and a periodic reaction diffusion equation on a cylinder

the essential spectrum of neumann laplacians on some bounded singular domains in the present paper we consider neumann laplacians on singular domains of the type rooms and passages or combs and we show that in typical situations the essential spectrum can be determined from the geometric data moreover given an arbitrary closed subset s of the non negative reals we construct domains s such that the essential spectrum of the neumann laplacian on is just this set s

an averaging theorem for hamiltonian dynamical systems in the thermodynamic limit it is shown how to perform some steps of perturbation theory if one assumes a measure theoretic point of view i e if one renounces to control the evolution of the single trajectories and the attention is restricted to controlling the evolution of the measure of some meaningful subsets of phase space for a system of coupled rotators estimates uniform in n for finite specific energy can be obtained in quite a direct way this is achieved by making reference not to the sup norm but rather following koopman and von neumann to the much weaker l2 norm

generalized transition waves and their properties in this paper we generalize the usual notions of waves fronts and propagation speeds in a very general setting these new notions which cover all usual situations involve uniform limits with respect to the geodesic distance to a family of hypersurfaces that are parametrized by time we prove the existence of new such waves for some time dependent reaction diffusion equations as well as general intrinsic properties some monotonicity properties and some uniqueness results for almost planar fronts the classification results which are obtained under some appropriate assumptions show the robustness of our general definitions 2012 wiley periodicals inc

travelling fronts and entire solutions of the fisher kpp equation in ℝn this paper is devoted to time global solutions of the fisher kpp equation in n where f is a c 2 concave function on 0 1 such that f 0 f 1 0 and f 0 on 0 1 it is well known that this equation admits a finite dimensional manifold of planar travelling fronts solutions by considering the mixing of any density of travelling fronts we prove the existence of an infinite dimensional manifold of solutions in particular there are infinite dimensional manifolds of nonplanar travelling fronts and radial solutions furthermore up to an additional assumption a given solution u can be represented in terms of such a mixing of travelling fronts

time periodic traveling wave solutions for periodic advection reaction diffusion systems abstract we study the existence uniqueness and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection reaction diffusion systems under certain conditions we prove that there exists a maximal wave speed c such that for each wave speed c c there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system it is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co moving frame coordinate we also show that the traveling wave solutions with wave speed c c are asymptotically stable in certain sense in addition we establish the nonexistence of time periodic traveling waves with speed c c

nonplanar traveling fronts in reaction diffusion equations with combustion and degenerate fisher kpp nonlinearities abstract this paper is concerned with nonplanar traveling fronts in reaction diffusion equations with combustion nonlinearity and degenerate fisher kpp nonlinearity our study contains two parts in the first part we establish the existence of traveling fronts of pyramidal shape in r 3 and in the second part we establish the existence and stability of v shaped traveling fronts in r 2

hp theory on a smooth domain in rn and elliptic boundary value problems in this paper we answer the following two questions question 1 let be an appropriate domain inrn what are the possible natural notions of hp that generalize the usual hardy spaces hp rn we shall see that several versions are possible we concentrate on the two most relevant here which may be considered to be the largest version hpr and the smallest version hpz for the second question consider the dirichlet and neumann problems for the former consists in finding the u g which solves u in and u 0 the latter consists in finding the u g which solves u in and u n 0 question 2 in the context of the relevant hp can one obtain the boundedness of 2g xi xj and g xi xj

nekhoroshev type stability results for hamiltonian systems with an additional transversal component abstract we prove exponential stability theorems of nekhoroshev type for motion in the neighbourhood of an elliptic fixed point in hamiltonian systems having an additional transverse component of arbitrary dimension under certain conditions on this transverse component we consider both the cases i of a strongly constrained motion and ii of a weak perturbation

bistable transition fronts in rn abstract this paper is chiefly concerned with qualitative properties of some reaction diffusion fronts the recently defined notions of transition fronts generalize the standard notions of traveling fronts in this paper we show the existence and the uniqueness of the global mean speed of bistable transition fronts in r n this speed is proved to be independent of the shape of the level sets of the fronts the planar fronts are also characterized in the more general class of almost planar fronts with any number of transition layers these qualitative properties show the robustness of the notions of transition fronts but we also prove the existence of new types of transition fronts in r n that are not standard traveling fronts thus showing that the notions of transition fronts are broad enough to include other relevant propagating solutions

periodic traveling curved fronts in reaction diffusion equation with bistable time periodic nonlinearity abstract this paper is concerned with the existence and stability of periodic traveling curved fronts for reaction diffusion equations with time periodic bistable nonlinearity in two dimensional space by constructing supersolution and subsolution we prove the existence of periodic traveling wave fronts furthermore we show that the front is globally stable

the uniqueness and asymptotic stability of pyramidal traveling fronts in the allen cahn equations abstract this paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three dimensional whole space for a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity for this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces moreover we characterize the pyramidal traveling front in another way that is we write it as a combination of two dimensional v form waves on the edges this characterization also uniquely determines a pyramidal traveling front

generalized traveling waves for time dependent reaction diffusion systems traveling wave solutions in general time dependent including time periodic reaction diffusion equations and systems of equations have attracted great attention in the last two decades the aim of this paper is to study the propagation phenomenon in a general time heterogeneous environment more specifically we investigate generalized traveling wave solutions for a two component time dependent non cooperative reaction diffusion system which has applications in epidemiology and ecology sufficient conditions on the existence and nonexistence of generalized traveling wave solutions are established in the susceptible infectious epidemic model setting generalized traveling waves describe the spatio temporal invasion of a disease into a totally susceptible population in the context of predator prey systems the generalized traveling waves describe the spatial invasion of predators introduced into a new environment where the prey population is at its carrying capacity

on the problem of energy equipartition for large systems of the fermi pasta ulam type analytical and numerical estimates abstract we report on some analytical and numerical results on the exchanges of energy in systems of the fermi pasta ulam type in the light of nekhoroshev s theorem with particular attention to the dependence of the estimates on the number n of degrees of freedom for the ordinary fpu problem we look for a control of the single normal mode energies and we find both the analytical and numerical estimates to agree in predicting that the energy exchanges of the single modes cannot be controlled in the thermodynamic limit we consider then a modified fpu model with alternating light and heavy particles which appears as composed of two subsystems of low acoustic frequency and of high optical frequency respectively we try to control the exchange of the total energy of the high frequency modes up to times increasing exponentially with the frequency in this case the numerical estimates are stronger than the available analytical ones and give indications for nonequipartition with constants essentially independent of the number n of degrees of freedom

an averaging technique for highly oscillatory hamiltonian problems in this paper we are concerned with the numerical solution of highly oscillatory hamiltonian systems with a stiff linear part we construct an averaged system whose solution remains close to the exact one over bounded time intervals possesses the same adiabatic and hamiltonian invariants as the original system and is nonstiff we then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in e cances et al numer math 100 2005 pp 211 232 eventually we demonstrate the efficiency of our approach on two test problems with one or several frequencies

classical freezing of plane rotations a proof of the boltzmann jeans conjecture using simple known methods and results of classical perturbation theory especially those due to nekhoroshev and neishtadt we study the energy exchanges between the rotational and the translational degrees of freedom in a particular model representing the planar motion of a rigid body in a bounded analytic potential we prove that if the angular velocity co is initially large then the energy exchanges are small o co 1 for times growing exponentially with co ill exp co we also deduce that in a scattering process from a smooth potential barrier the overall change in the rotational energy of the incoming body is exponentially small in co g exp co the results are interpreted in the light of an old conjecture by boltzmann and jeans on the existence of very large time scales for equilibrium in statistical systems containing high frequency degrees of freedom purely classical freezing of the high frequency degrees of freedom the rotating object is in this interpretation a classical molecule which moves in an external field or collides with the wall of a container two different limits of large co are considered namely the limit of large rotational energy and as is interesting for the molecular interpretation the limit of point mass at finite rotational energy

smoothed langevin dynamics of highly oscillatory systems abstract we consider hamiltonian systems containing a strong constraining potential such systems were first investigated by rubin and ungar who derived the corresponding reduced equations of motion for the slow degree of freedom the purpose of this paper is two fold i we transform the highly oscillatory degree of freedom to action angle variables and use recent results on the truncation error of the corresponding normal form transformations to derive an estimate on the lifespan of the underlying adiabatic invariant ii we investigate thermally embedded hamiltonian systems i e langevin dynamics containing a strong constraining potential using again a transformation to action angle variables and applying the principle of averaging we derive reduced equations of motion for the slowly varying degrees of freedom this includes in particular the derivation of a correcting force term that stands for the coupling of the slow and fast degrees of motion in case of strong thermal coupling our reduced equations of motion suggest that the correcting force term can be approximated by the gradient of the fixman potential as used in statistical mechanics we also discuss smoothing in the context of constant temperature molecular dynamics and provide a numerical example

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

non smooth atomic decomposition of 2 microlocal spaces and application to pointwise multipliers abstract we provide non smooth atomic characterizations for 2 microlocal besov and triebel lizorkin spaces with variable integrability b p q w r n and f p q w r n these spaces cover the classical besov and triebel lizorkin spaces as well as spaces of variable smoothness and integrability as an application we state a pointwise multipliers result for these spaces

existence and global stability of traveling curved fronts in the allen cahn equations abstract this paper is concerned with existence and stability of traveling curved fronts for the allen cahn equation in the two dimensional space by using the supersolution and the subsolution we construct a traveling curved front and show that it is the unique traveling wave solution between them our supersolution can be taken arbitrarily large which implies some global asymptotic stability for the traveling curved front

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

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

long time energy conservation of numerical methods for oscillatory differential equations we consider second order differential systems where high frequency oscillations are generated by a linear part we present a frequency expansion of the solution and we discuss two invariants of the system that determine the coefficients of the frequency expansion these invariants are related to the total energy and the oscillatory harmonic energy of the original system for the numerical solution we study a class of symmetric methods that discretize the linear part without error we are interested in the case where the product of the step size with the highest frequency can be large in the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system this allows us to prove the near conservation of the total and the oscillatory energy over very long time intervals

stability of motions near resonances in quasi integrable hamiltonian systems nekhoroshev s theorem on the stability of motions in quasi integrable hamiltonian systems is revisited at variance with the proofs already available in the literature we explicitly consider the case of weakly perturbed harmonic oscillators furthermore we prove the confinement of orbits in resonant regions in the general case of nonisochronous systems by using the elementary idea of energy conservation instead of more complicated mechanisms an application of nekhoroshev s theorem to the study of perturbed motions inside resonances is also provided

realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory part ii the so called problem of the realization of the holonomic constraints of classical mechanics is here revisited in the light of nekhoroshev like classical perturbation theory precisely if constraints are physically represented by very steep potential wells with associated high frequency transversal vibrations then one shows that within suitable assumptions the vibrational energy and the energy associated to the constrained motion are separately almost constant for a very long time scale growing exponentially with the frequency i e with the rigidity of the constraint one aims to realize this result can also be applied to microscopic physics providing a possible entirely classical mechanism for the freezing of the high frequency degrees of freedom in terms of non equilibrium statistical mechanics according to some ideas expressed by boltzmann and jeans at the turn of the century in this part i we introduce the problem and prove a first theorem concerning the realization of a single constraint within a system of any number of degrees of freedom the problem of the realization of many constraints will be considered in a forthcoming part ii

transition fronts for the fisher kpp equation this paper is concerned with transition fronts for reaction diffusion equations of the fisher kpp type basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich in the paper we describe the class of transition fronts and we study their qualitative dynamical properties in particular we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate we also classify the transition fronts in the class of measurable superpositions of standard traveling fronts

traveling waves with multiple and nonconvex fronts for a bistable semilinear parabolic equation we construct new examples of traveling wave solutions to the bistable and balanced semilinear parabolic equation in input amssym bbb r n 1 our first example is that of a traveling wave solution with two non planar fronts that move with the same speed our second example is a traveling wave solution with a nonconvex moving front to our knowledge no existence results of traveling fronts with these type of geometric characteristics have been previously known our approach explores a connection between solutions of the semilinear parabolic pde and eternal solutions to the mean curvature flow in input amssym bbb r n 1

pulsating fronts for bistable on average reaction diffusion equations in a time periodic environment abstract this paper is devoted to reaction diffusion equations with bistable nonlinearities depending periodically on time these equations admit two linearly stable states however the reaction terms may not be bistable at every time these may well be a periodic combination of standard bistable and monostable nonlinearities we are interested in a particular class of solutions namely pulsating fronts we prove the existence of such solutions in the case of small time periods of the nonlinearity and in the case of small perturbations of a nonlinearity for which we know there exist pulsating fronts we also study uniqueness monotonicity and stability of pulsating fronts

transition fronts of time periodic bistable reaction diffusion equations in rn abstract this paper is concerned with the existence and qualitative properties of transition fronts for time periodic bistable reaction diffusion equations in r n we first show that any almost planar transition front is actually planar regardless of the number of transition layers then we prove that all transition fronts admit a global mean speed and it holds c where c is the speed of the planar traveling front finally we establish the existence of a transition front in r n that is not a standard traveling front such a front behaves like three moving time periodic planar fronts as time goes to and like a time periodic v shaped traveling front as time goes to

calderon reproducing formulas and new besov spaces associated with operators abstract let l be the generator of an analytic semigroup whose heat kernel satisfies an upper bound of poisson type acting on l 2 x where x is a possibly non doubling space of polynomial upper bound on volume growth the aim of this paper is to introduce a new class of besov spaces associated with the operator l so that when l is the laplace operator or its square root acting on the euclidean space r n the new besov spaces are equivalent to the classical besov spaces depending on the choice of l the new besov spaces are natural settings for generic estimates for certain singular integral operators such as the fractional powers l since our approach does not require the doubling volume property of the underlying space it is applicable to any subset of euclidean spaces without any smoothness requirement on the boundary we will also develop a number of calderon reproducing formulas which play an important role in the theory of function spaces and are of independent interest as an application we study besov spaces associated with schrodinger operators with non negative potentials satisfying reverse holder estimates

hardy spaces associated with magnetic schrodinger operators on strongly lipschitz domains abstract let p 0 1 be a strongly lipschitz domain in r n and a i a i a v a magnetic schrodinger operator on l 2 satisfying the dirichlet boundary condition where a a 1 a n l loc 2 r n and 0 v l loc 1 in this paper the authors introduce the hardy space h a p by the lusin area function associated with a and establish its equivalent characterization via the non tangential maximal function associated with e t a t 0 as applications the authors obtain the boundedness of the riesz transforms l k a 1 2 k 1 n from h a p to l p for p 0 1 and the fractional integral a from h a p to h a q for 0 p q 1 and n 2 1 p 1 q where l k is the closure of x k i a k in l 2

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

hardy spaces and divergence operators on strongly lipschitz domains of rn abstract let be a strongly lipschitz domain of r n consider an elliptic second order divergence operator l including a boundary condition on and define a hardy space by imposing the non tangential maximal function of the extension of a function f via the poisson semigroup for l to be in l1 under suitable assumptions on l we identify this maximal hardy space with h 1 r n if r n with h r 1 under the dirichlet boundary condition and with h z 1 under the neumann boundary condition

non smooth atomic decompositions traces on lipschitz domains and pointwise multipliers in function spaces abstract we provide non smooth atomic decompositions for besov spaces b p q s r n s 0 0 p q defined via differences the results are used to compute the trace of besov spaces on the boundary of bounded lipschitz domains with smoothness s restricted to 0 s 1 and no further restrictions on the parameters p q we conclude with some more applications in terms of pointwise multipliers

slow dynamics via degenerate variational asymptotics we introduce the method of degenerate variational asymptotics for a class of singularly perturbed ordinary differential equations in the limit of strong gyroscopic forces such systems exhibit dyna

bistable traveling waves around an obstacle we consider traveling waves for a nonlinear diffusion equation with a bistable or multistable nonlinearity the goal is to study how a planar traveling front interacts with a compact obstacle that is placed in the middle of the space n as a first step we prove the existence and uniqueness of an entire solution that behaves like a planar wave front approaching from infinity and eventually reaching the obstacle this causes disturbance on the shape of the front but we show that the solution will gradually recover its planar wave profile and continue to propagate in the same direction leaving the obstacle behind whether the recovery is uniform in space is shown to depend on the shape of the obstacle 2008 wiley periodicals inc

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

symmetry of traveling wave solutions to the allen cahn equation in mathbb r 2 in this paper we prove even symmetry of monotone traveling wave solutions to the balanced allen cahn equation in the entire plane related results for the unbalanced allen cahn equation are also discussed

existence and stability of time periodic traveling waves for a periodic bistable lotka volterra competition system abstract this paper is concerned with the time periodic lotka volterra competition diffusion system u t d 1 t u x x u r 1 t a 1 t u b 1 t v v t d 2 t v x x v r 2 t a 2 t u b 2 t v x r t 0 where d i t r i t a i t b i t c 2 r are t periodic functions 0 1 i 1 2 under certain conditions the system admits two stable semi trivial periodic solutions p t 0 and 0 q t and a unique coexistence periodic solution u 0 t v 0 t which is unstable and satisfies 0 u 0 t p t and 0 v 0 t q t for t 0 t in this paper we prove that the system admits a time periodic traveling wave solution u x t v x t u x c t t v x c t t connecting two periodic solutions p t 0 and 0 q t as x where c is the wave speed by using a dynamical method we show that the time periodic traveling wave solution u x c t t v x c t t is asymptotically stable and unique modulo translation for front like initial values

two dimensional curved fronts in a periodic shear flow abstract this paper is devoted to the study of traveling fronts of reaction diffusion equations with periodic advection in the whole plane r 2 we are interested in curved fronts satisfying some conical conditions at infinity we prove that there is a minimal speed c such that curved fronts with speed c exist if and only if c c moreover we show that such curved fronts are decreasing in the direction of propagation that is they are increasing in time we also give some results about the asymptotic behaviors of the speed with respect to the advection diffusion and reaction coefficients