multiple normalized standing wave solutions to the scalar non linear klein gordon equation with two competing powers abstract in this work we prove the existence of standing wave solutions to the scalar non linear klein gordon equation in dimension one and the stability of the ground state the set which contains all the minima of the energy constrained to the manifold of the states sharing a fixed charge for non linearities which are combinations of two competing powers we prove that standing waves in the ground state are orbitally stable we also show the existence of a degenerate minimum and the existence of two positive and radially symmetric minima having the same charge

variational properties and orbital stability of standing waves for nls equation on a star graph we study standing waves for a model of nonlinear schr odinger equation on a graph the graph is obtained joining n halflines at a vertex i e it is a star graph at the vertex an interaction occurs due to a boundary condition of delta type with strength alpha leqslant 0 which includes the free or kirchhoff case alpha 0 there imposed the nonlinearity is of focusing power type the dynamics is given by an equation of the form i frac d dt psi t h psi t psi t 2 mu psi t where h is the s a operator which generates the linear schr odinger dynamics on the graph in the hilbert space l 2 gg we show the existence of several families of standing waves solutions of the form phi t e i omega t psi omega where psi omega is a vector amplitude parametrized by a frequency omega they exist for every sign of the coupling at the vertex attractive alpha 0 and for every omega frac alpha 2 n 2 the number of the families depends on the number n of the edges of the graph we moreover determine the ground state which is the standing wave the amplitude of which minimizes the action on the natural nehari constraint and order the various families of standing waves according to their increasing action thus determining a nonlinear spectrum of the problem finally we show that the ground state is orbitally stable for every allowed omega if the nonlinearity is subcritical or critical for supercritical nonlinearities there exists an omega frac alpha 2 n 2 such that the ground state is orbitally stable for frac alpha 2 n 2 omega omega action minimization requires technical subtleties in particular an adaptation of decreasing rearrangements on a star graph is developed with a generalization of the p olya szeg h o inequality

random sequential addition a distribution function approach random sequential addition rsa of hard objects is an irreversible process defined by three rules objects are introduced on a surface or ad dimensional volume randomly and sequentially two objects cannot overlap and once inserted an object is clamped in its position the configurations generated by an rsa can be characterized in the macroscopic limit by a unique set of distribution functions and a density we show that these nonequilibrium rsa configurations can be described in a manner which in many respects parallels the usual statistical mechanical treatment of equilibrium configurations kirkwood salsburg like hierarchies for the distribution functions zero separation theorems diagrammatic expansions and approximate equations for the pair distribution function approximate descriptions valid for low to intermediate densities can be combined with exact results already derived for higher densities close to the jamming limit of the process similarities and differences between the equilibrium and the rsaconfigurations are emphasized finally the potential application of rsa processes to the study of glassy phases is discussed

statistical mechanics of interfering links we consider the statistical mechanics of interfering transmissions in a wireless communications protocol in this case a connection between two nodes requires all other nodes within communication distance of the given two nodes to remain quiet on the given channel this leads to an interesting problem of dimers on a lattice with a restriction that no two dimers can overlap or be nearest neighbors we consider both an equilibrium and a nonequilibrium greedy dynamics for the links the equilibrium properties of the model are found to exhibit an interesting spin glass transition at maximum density on certain lattices while the greedy construction is related to the problem of random sequential adsorption

saturated random packing built of arbitrary polygons under random sequential adsorption protocol random packings and their properties are a popular and active field of research numerical algorithms that can efficiently generate them are useful tools in their study this paper focuses on random packings produced according to the random sequential adsorption rsa protocol developing the idea presented by g zhang phys rev e 97 043311 2018 2470 004510 1103 physreve 97 043311 where saturated random packings built of regular polygons were studied we create an algorithm that generates strictly saturated packings built of any polygons then the algorithm was used to determine the packing fractions for arbitrary triangles the highest mean packing density 0 552814 0 000063 was observed for triangles of side lengths 0 63 1 1 additionally microstructural properties of such packings kinetics of their growth as well as distributions of saturated packing fractions and the number of rsa iterations needed to reach saturation were analyzed

precise algorithm to generate random sequential addition of hard hyperspheres at saturation the study of the packing of hard hyperspheres in d dimensional euclidean space mathbb r d has been a topic of great interest in statistical mechanics and condensed matter theory while the densest known packings are ordered in sufficiently low dimensions it has been suggested that in sufficiently large dimensions the densest packings might be disordered the random sequential addition rsa time dependent packing process in which congruent hard hyperspheres are randomly and sequentially placed into a system without interparticle overlap is a useful packing model to study disorder in high dimensions of particular interest is the infinite time saturation limit in which the available space for another sphere tends to zero however the associated saturation density has been determined in all previous investigations by extrapolating the density results for nearly saturated configurations to the saturation limit which necessarily introduces numerical uncertainties we have refined an algorithm devised by us s torquato o u uche and f h stillinger phys rev e 74 061308 2006 to generate rsa packings of identical hyperspheres the improved algorithm produce such packings that are guaranteed to contain no available space in a large simulation box using finite computational time with heretofore unattained precision and across the widest range of dimensions 2 ensuremath le d ensuremath le 8 we have also calculated the packing and covering densities pair correlation function g 2 r and structure factor s k of the saturated rsa configurations as the space dimension increases we find that pair correlations markedly diminish consistent with a recently proposed decorrelation principle and the degree of hyperuniformity suppression of infinite wavelength density fluctuations increases we have also calculated the void exclusion probability in order to compute the so called quantizer error of the rsa packings which is related to the second moment of inertia of the average voronoi cell our algorithm is easily generalizable to generate saturated rsa packings of nonspherical particles

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

self similarity and asymptotic stability for coupled nonlinear schrodinger equations in high dimensions abstract this paper is concerned with systems of coupled schrodinger equations with polynomial nonlinearities and dimension n 1 we show the existence of global self similar solutions and prove that they are asymptotically stable in a framework based on weak l p spaces whose elements have local finite l 2 mass the radial symmetry of the solutions is also addressed

corrected mean field model for random sequential adsorption on random geometric graphs a notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non overlapping congruent spheres in the d dimensional euclidean space with d ge 2 spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres due to spatial correlations characterizing the fraction of accepted spheres remains largely intractable we study this fraction by taking a novel approach that compares random sequential adsorption in euclidean space to the nearest neighbor blocking on a sequence of clustered random graphs this random network model can be thought of as a corrected mean field model for the interaction graph between the attempted spheres using functional limit theorems we characterize the fraction of accepted spheres and its fluctuations

exact three wave solutions for the kp equation a new type of three wave solution periodic two solitary wave solutions for 1 2 d kadomtsev petviashvili kp equation is obtained using the extended three soliton method and with the help of maple

construction of the thermodynamic jamming limit for the parking process and other exclusion schemes on mathbb z d we provide an explicit construction of the thermodynamic jamming limit for the parking process and other finite range exclusion schemes on mathbb z d by means thereof a strong law of large numbers for occupation densities is accomplished and amongst other results the so called super exponential i e gamma decay of pair correlation functions is established

soliton solutions of higher order generalized derivative nonlinear schrodinger equation the lax pair and hirota s bilinear form of higher order generalized derivative nonlinear schrodinger equation are given the expression of n soliton solutions are obtained through hirota s standard procedure

coupled nonlinear schrodinger equations arising in the study of monomode step index optical fibers in this paper it is shown that nonlinear propagation in a step index monomode optical fiber is not generally governed by the nonlinear schrodinger equation even when the fiber is axially symmetric it is governed by a coupled pair of nonlinear partial differential equations that includes the nonlinear schrodinger equations only as a special case three simple types of solution to the coupled system are analyzed and the corresponding field patterns are interpreted one case shows that for uniform wavetrains nonlinearity not only alters the phase speed but also causes the field pattern in an elliptically polarized mode to rotate gradually about the fiber axis the other two cases each allow the system to be reduced to a single nonlinear schrodinger equation so showing two distinct situations in which solutions have soliton properties

asymptotic stability of small gap solitons in nonlinear dirac equations we prove dispersive decay estimates for the one dimensional dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear dirac equations with quintic and higher order nonlinear terms

orbital stability of spatially synchronized solitary waves of an m coupled nonlinear schrodinger system in this paper we investigate the orbital stability of solitary wave solutions for an m coupled nonlinear schrodinger system i tuj 2 x2uj i 1mbijui2uj 0 j 1 m where m 2 uj are complex valued functions of x t 2 bjj j 1 2 m and bij i j are positive coupling constants satisfying bij bji it will be shown that spatially synchronized solitary wave solutions of the m coupled nonlinear schrodinger system exist and are orbitally stable here by synchronized solutions we mean solutions in which the components are proportional to one another our results completely settle the question on the existence and stability of synchronized solitary waves for the m coupled system while only partial results were known in the literature for the cases of m 3 heretofore furthermore the conditions imposed on the symmetric matrix b bij satisfied here are both sufficient and necessary for the m coupled nonlinear schrodinger system to admit synchronized ground state solutions

precise algorithm to generate random sequential adsorption of hard polygons at saturation random sequential adsorption rsa is a time dependent packing process in which particles of certain shapes are randomly and sequentially placed into an empty space without overlap in the infinite time limit the density approaches a saturation limit although this limit has attracted particular research interest the majority of past studies could only probe this limit by extrapolation we have previously found an algorithm to reach this limit using finite computational time for spherical particles and could thus determine the saturation density of spheres with high accuracy in this paper we generalize this algorithm to generate saturated rsa packings of two dimensional polygons we also calculate the saturation density for regular polygons of three to ten sides and obtain results that are consistent with previous extrapolation based studies

stable standing waves of nonlinear klein gordon equations in this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear klein gordon equations

geometry of random sequential adsorption by sequentially adding line segments to a line or disks to a surface at random positions without overlaps we obtain configurations of the one and two dimensional random sequential adsorption rsa problem we have simulated the one and two dimensional problem with periodic boundary condition the one dimensional simulations are compared with the exact analytical solutions to give an estimate of the accuracy of the simulation in two dimensions the geometrical properties of the rsa configuration are discussed and in addition known results of the rsa process are reproduced various statistical distributions of the voronoi dirichlet vd network corresponding to the rsa disk configuration are analyzed in order to characterize pores in the rsa configuration we introduce circular holes there is a direct correspondence between vertices of the vd network and these holes and also between direct indirect geometrical neighbors and these holes the hole size distribution is found to be a parabola we also find general relations that connect the asymptotic behavior of the surface coverage the correlation function and the hole size distribution

on the dynamics of solitons in the nonlinear schrodinger equation we study the behavior of the soliton solutions of the equation i frac partial psi partial t frac 1 2m delta psi frac 1 2 w varepsilon prime psi v x psi where w varepsilon prime is a suitable nonlinear term which is singular for varepsilon 0 we use the strong nonlinearity to obtain results on existence shape stability and dynamics of the soliton the main result of this paper theorem 1 shows that for varepsilon rightarrow 0 the orbit of our soliton approaches the orbit of a classical particle in a potential v x

a solution to the focusing 3d nls that blows up on a contracting sphere we rigorously construct radial h 1 solutions to the 3d cubic focusing nls equation i t 2j j 2 0 that blow up along a contracting sphere with blow up time set to t 0 the solutions concentrate on a sphere at radius t 1 3 but focus towards this sphere at the faster rate t 2 3 such dynamics were originally proposed heuristically by degtyarev zakharov rudakov 2 in 1975 and independently later in holmer roudenko 5 in 2007 where it was demonstrated to be consistent with all conservation laws of this equation in the latter paper it was proposed as a solution that would yield divergence of the l 3 norm within the wide radius kr u t k 1 2 l2 but not within the tight radius kr u t k 2 l2

stable solutions of the logarithmic schrodinger equation on considere l equation de schrodinger suivante u t u ulog u 2 0 dans r r n on donne des resultats sur le comportement asymptotique des solutions

random sequential adsorption of lines and ellipses infinitely many lines of zero thickness may be adsorbed onto a plane without overlap and it is shown that the number of lines adsorbed should scale as 13 where is the number of adsorption trials per unit area of surface this is confirmed by numerical simulation the adsorption of ellipses with major and minor axes 2a 2b respectively is also studied the area coverage theta approaches its maximum value theta max as alpha as to infinity where alpha is typically 1 3 as b a decreases from unity theta max first increases to a maximum value 0 58 or 0 01 when b approximately 0 5 and then decreases a feature not noted in previous work

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

on the car parking problem the radial distribution functions of randomly distributed one dimensional spheres are studied the differences between equilibrium fluids and sequentially constructed systems are discussed in terms of the corresponding configuration spaces and numerical simulations it is shown that for the sequentially constructed systems which correspond to the car parking problem a memory effect occurs such that the sequential spheres are temporally distinguishable exact solutions for some simple cases and numerical simulations for large systems are shown discussed and compared with the equilibrium fluid results

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

pair correlation function in random sequential adsorption processes an analytical expression of the pair distribution function is derived for the car parking problem and for the random sequential adsorption of k mers onto a one dimensional lattice both on a lattice and in the continuum limit super exponential decay of the distribution function is observed a comparison of the spatial correlations with those at equilibrium demonstrates the influence of the irreversibility of the rsa process

random packing of regular polygons and star polygons on a flat two dimensional surface random packing of unoriented regular polygons and star polygons on a two dimensional flat continuous surface is studied numerically using random sequential adsorption algorithm obtained results are analyzed to determine the saturated random packing ratio as well as its density autocorrelation function additionally the kinetics of packing growth and available surface function are measured in general stars give lower packing ratios than polygons but when the number of vertexes is large enough both shapes approach disks and therefore properties of their packing reproduce already known results for disks

precise algorithms to compute surface correlation functions of two phase heterogeneous media and their applications the quantitative characterization of the microstructure of random heterogeneous media in d dimensional euclidean space r d via a variety of n point correlation functions is of great importance since the respective infinite set determines the effective physical properties of the media in particular surface surface f ss and surface void f sv correlation functions obtainable from radiation scattering experiments contain crucial interfacial information that enables one to estimate transport properties of the media e g the mean survival time and fluid permeability and complements the information content of the conventional two point correlation function however the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use we first present a concise derivation of the small r behaviors of these functions which are linked to the mean curvature of the system then we demonstrate that one can reduce the computational complexity of the problem without sacrificing accuracy by extracting the necessary interfacial information from a cut of the d dimensional statistically homogeneous and isotropic system with an infinitely long line accordingly we devise algorithms based on this idea and test them for two phase media in continuous and discrete spaces specifically for the exact benchmark model of overlapping spheres we find excellent agreement between numerical and exact results we compute surface correlation functions and corresponding local surface area variances for a variety of other model microstructures including hard spheres in equilibrium decorated stealthy patterns as well as snapshots of evolving pattern formation processes e g spinodal decomposition it is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures

stability and instability of the kdv solitary wave under the kp i flow we consider the kp i and gkp i equations in mathbb r times mathbb r 2 pi mathbb z we prove that the kdv soliton with subcritical speed 0 c in the spirit of the work by duyckaerts and merle gafa 18 1787 1840 2009 we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity this last result also holds for the gkp i equation

a centre stable manifold for the focussing cubic nls in mathbb r 1 3 consider the focussing cubic nonlinear schrodinger equation in mathbb r 3 i psi t delta psi psi 2 psi quad 0 1 it admits special solutions of the form eit where phi in mathcal s mathbb r 3 is a positive 0 solution of delta phi alpha phi phi 3 quad 0 2 the space of all such solutions together with those obtained from them by rescaling and applying phase and galilean coordinate changes called standing waves is the 8 dimensional manifold that consists of functions of the form e i v cdot gamma phi cdot y alpha we prove that any solution starting sufficiently close to a standing wave in the sigma w 1 2 mathbb r 3 cap x 1 l 2 mathbb r 3 norm and situated on a certain codimension one local lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves furthermore we show that mathcal n is invariant under the hamiltonian flow locally in time and is a centre stable manifold in the sense of bates jones batjon the proof is based on the modulation method introduced by soffer and weinstein for the l2 subcritical case and adapted by schlag to the l2 supercritical case an important part of the proof is the keel tao endpoint strichartz estimate in mathbb r 3 for the nonselfadjoint schrodinger operator obtained by linearizing 0 1 around a standing wave solution all results in this paper depend on the standard spectral assumption that the hamiltonian mathcal h left begin array cc delta 2 phi cdot alpha 2 alpha quad phi cdot alpha 2 phi cdot alpha 2 quad delta 2 phi cdot alpha 2 alpha end array right quad 0 3 has no embedded eigenvalues in the interior of its essential spectrum infty alpha cup alpha infty

elliptic solutions of the defocusing nls equation are stable the stability of the stationary periodic solutions of the integrable one dimensional cubic defocusing nonlinear schrodinger nls equation is reasonably well understood especially for solutions of small amplitude in this paper we exploit the integrability of the nls equation to establish the spectral stability of all such stationary solutions this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem an additional argument using an appropriate krein signature allows us to conclude the nonlinear orbital stability of all stationary solutions of the defocusing nls equation with respect to so called subharmonic perturbations perturbations that have period equal to an integer multiple of the period of the amplitude of the solution all results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase profiles

existence and orbital stability of standing waves for nonlinear schrodinger systems abstract in this paper we investigate the existence of solutions in h 1 r n h 1 r n for nonlinear schrodinger systems of the form u 1 1 u 1 1 u 1 p 1 2 u 1 r 1 u 1 r 1 2 u 1 u 2 r 2 u 2 2 u 2 2 u 2 p 2 2 u 2 r 2 u 1 r 1 u 2 r 2 2 u 2 under the constraints r n u 1 2 d x a 1 0 r n u 2 2 d x a 2 0 here n 1 0 i 0 r i 1 2 p i 2 4 n for i 1 2 and r 1 r 2 2 4 n this problem is motivated by the search of standing waves for an evolution problem appearing in several physical models our solutions are obtained as constrained global minimizers of an associated functional note that in the system 1 and 2 are unknown and will correspond to the lagrange multipliers our main result is the precompactness of the minimizing sequences up to translation assuming the local well posedness of the associated evolution problem we then obtain the orbital stability of the standing waves associated to the set of minimizers

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

scaling properties of the number of random sequential adsorption iterations needed to generate saturated random packing the properties of the number of iterations in random sequential adsorption protocol needed to generate finite saturated random packing of spherically symmetric shapes were studied numerical results obtained for one two and three dimensional packings were supported by analytical calculations valid for any dimension d it has been shown that the number of iterations needed to generate finite saturated packing is subject to pareto distribution with exponent 1 1 d and the median of this distribution scales with packing size according to the power law characterized by exponent d obtained results can be used in designing effective random sequential adsorption simulations

packing of spheroids in three dimensional space by random sequential addition packings of spheroidal particles with semi axes of length a b b were generated by random sequential addition rsa simulations as the jamming limit is approached the volume fraction occupied by particles tends towards an asymptote which was determined as a function of aspect ratio this asymptote has a local minimum at spheres with local maxima at prolate spheroids and oblate values of agree with results from rsa simulations of sphere packings but lie below volume fractions obtained in simulations of near spheres packed under gravity volume fractions reported for simulations of spheroids packed under gravity vary widely when the aspect ratio is very large or small differences between these results and the predictions of rsa are discussed

optimal error analysis of cranknicolson schemes for a coupled nonlinear schrdinger system in 3d the paper is concerned with the time step condition of the commonly used semi implicit cranknicolson finite difference schemes for a coupled nonlinear schrdinger system in three dimensional space we present the optimal l2 error estimate without any restriction on time step while all previous works require certain time step conditions our approach is based on a rigorous analysis in both real and imaginary parts of the energy estimate inequality of the error function numerical examples for both two dimensional and three dimensional models are investigated and numerical results illustrate our theoretical analysis

stability of solitary waves for the vector nonlinear schrodinger equation in higher order sobolev spaces abstract in this manuscript a sharp form of orbital stability for the schrodinger system also known as the vector nls i t u j 2 x x u j 2 i 1 m u i 2 u j 0 where u j are complex valued functions of x t r 2 j 1 2 m is established in l 2 based sobolev classes of arbitrarily high order the result means practically that not only does the bulk of what emanates from the perturbed solitary wave stay close in shape and propagation and phase speeds to the original solitary wave but emerging residual oscillations must also be very small and not only in the energy norm

orbital stability of solitary waves of a 3 coupled nonlinear schrodinger system abstract in this paper consideration is given to the 3 coupled nonlinear schrodinger system i t u j 2 x 2 u j i 1 3 b i j u i 2 u j 0 where u j are complex valued functions of x t r 2 j 1 2 3 and b i j are positive constants satisfying b i j b j i it will be shown first that if the symmetric matrix b b i j satisfies certain conditions then ground state solutions of the 3 coupled nonlinear schrodinger system exist and moreover they are orbitally stable the theory is then extended to include solitary waves as well in particular it will be shown that when a solitary wave is perturbed the perturbed solution must stay close to a solitary wave profile in which the translation and phase parameters are prescribed functions of time properties of these functions are then studied this is a continuous work of our previous paper where the 2 coupled nonlinear schrodinger system was considered

random sequential adsorption of oriented superdisks in this work we extend recent study of the properties of the dense packing of superdisks by y jiao phys rev lett 100 245504 2008 to the jammed state formed by these objects in random sequential adsorption the superdisks are two dimensional shapes bound by the curves of the form x 2p y 2p 1 with p 0 we use monte carlo simulations and theoretical arguments to establish that p 1 2 is a special point at which the jamming density rhoj p has a discontinuous derivative as a function of p the existence of this point can be also argued for by a phenomenological excluded area argument

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

the regularized benjamin ono and bbm equations well posedness and nonlinear stability abstract nonlinear stability of nonlinear periodic solutions of the regularized benjamin ono equation and the benjamin bona mahony equation with respect to perturbations of the same wavelength is analytically studied these perturbations are shown to be stable we also develop a global well posedness theory for the regularized benjamin ono equation in the periodic and in the line setting in particular we show that the cauchy problem in both periodic and nonperiodic case cannot be solved by an iteration scheme based on the duhamel formula for negative sobolev indices

on the spectral stability of solitary wave solutions of the vector nonlinear schrodinger equation we consider a system of coupled cubic nonlinear schrodinger nls equations where the interaction coefficients jk are real the spectral stability of solitary wave solutions both bright and dark is examined both analytically and numerically our results build on preceding work by nguyen et al and others specifically we present closed form solitary wave solutions with trivial and nontrivial phase profiles their spectral stability is examined analytically by determining the locus of their essential spectrum their full stability spectrum is computed numerically using a large period limit of hill s method we find that all nontrivial phase solutions are unstable while some trivial phase solutions are spectrally stable to our knowledge this paper presents the first investigation of the stability of the solitary waves of the coupled cubic nls equation without the restriction that all components j are proportional to sech

solitary waves and a stability analysis of an equation of short and long dispersive waves abstract a universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of korteweg de vries kdv type is coupled to an equation of nonlinear schrodinger nls type the system has solutions of steady form in which one component is like a solitary wave solution of the kdv equation and the other component is like a ground state solution of the nls equation we study the stability of solitary wave solutions to an equation of short and long waves by using variational methods based on the use of energy momentum functionals and the techniques of convexity type we use the concentration compactness method to prove the existence of solitary waves we prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed

long time oscillation of solutions of nonlinear schrodinger equations near minimal mass ground state abstract in this paper we consider the long time dynamics of radially symmetric solutions of nonlinear schrodinger equations nls having a minimal mass ground state in particular we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time more precisely we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator then we show that the finite dimensional 2 d part approximately obeys newton s equation of motion for a particle in an anharmonic potential well showing that the infinite dimensional part is well separated from the finite dimensional part we will have long time oscillation

random sequential adsorption series expansion and monte carlo simulation abstract random sequential adsorption is an irreversible surface deposition of extended objects in systems with continuous degrees of freedom coverage follows a power law t j ct where the exponent depends on the geometric shape symmetry of the objects lattice models give typically exponential saturation to jamming coverage we discuss how such function t can be computed by series expansions and analyzed with pade approximations we consider the applications of efficient monte carlo computer simulation method event driven method to random sequential adsorptions with high precision and at very long time scale

integrable systems of derivative nonlinear schrodinger type and their multi hamiltonian structure a spectral problem and the associated hierarchy of schrodinger type equations are proposed it is shown that the hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure it is found that several kinds of important equation such as the kaup newell kn equation the chen lee liu cll equation the gerdjikov ivanov gi equation the modified korteweg de vries equation and the sharma tasso olever equation are members in the hierarchy as its special reductions moreover kn cll and gi equations are described by using a unified generalized derivative schrodinger equation involving a parameter and their hamiltonian structure and lax pairs are also given by unified and explicit formulae

the pair correlation function in a randomly sequentially filled one dimensional lattice we study random sequential adsorption of particles on a one dimensional lattice with exclusion of nearest neighbor occupation the fraction of filled sites in the saturated state y 12 1 e 2 was obtained long ago by widom we show that in the saturated state the probability p k of having two sites a distance k apart both occupied is given by p k y 2 12 1 ke 2 p 0 2k 2p 1 k 2p 1 and hence decays faster than exponentially as k

on the solitary wave dynamics under slowly varying medium for nonlinear schrodinger equations we consider the problem of a solitary wave propagation in a slowly varying medium for a variable coefficients nonlinear schrodinger equation we prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media moreover we describe for all time the behavior of these solutions which include refracted and reflected solitary waves depending on the initial energy

random sequential addition of hard spheres in high euclidean dimensions sphere packings in high dimensions have been the subject of recent theoretical interest employing numerical and theoretical methods we investigate the structural characteristics of random sequential addition rsa of congruent spheres in d dimensional euclidean space mathbb r d in the infinite time or saturation limit for the first six space dimensions 1 ensuremath le d ensuremath le 6 specifically we determine the saturation density pair correlation function cumulative coordination number and the structure factor in each of these dimensions we find that for 2 ensuremath le d ensuremath le 6 the saturation density ensuremath phi s scales with dimension as ensuremath phi s c 1 2 d c 2 d 2 d where c 1 0 202 phantom rule 0 2em 0ex 048 and c 2 0 973 phantom rule 0 2em 0ex 872 we also show analytically that the same density scaling is expected to persist in the high dimensional limit albeit with different coefficients a byproduct of this high dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by ensuremath phi s ensuremath ge d 2 1 ensuremath s 0 2 d 1 where s 0 0 1 is the structure factor at k 0 i e infinite wavelength number variance in the high dimensional limit we demonstrate that a pal asti type conjecture the saturation density in mathbb r d is equal to that of the one dimensional problem raised to the d th power cannot be true for rsa hyperspheres we show that the structure factor s k must be analytic at k 0 and that rsa packings for 1 ensuremath le d ensuremath le 6 are nearly hyperuniform consistent with the recent decorrelation principle we find that pair correlations markedly diminish as the space dimension increases up to six we also obtain kissing contact number statistics for saturated rsa configurations on the surface of a d dimensional sphere for dimensions 2 ensuremath le d ensuremath le 5 and compare to the maximal kissing numbers in these dimensions we determine the structure factor exactly for the related ghost rsa packing in mathbb r d and demonstrate that its distance from hyperuniformity increases as the space dimension increases approaching a constant asymptotic value of 1 2 our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions