small solutions of nonlinear schrodinger equations near first excited states abstract consider a nonlinear schrodinger equation in r 3 whose linear part has three or more eigenvalues satisfying some resonance conditions solutions which are initially small in h 1 l 1 r 3 and inside a neighborhood of the first excited state family are shown to converge to either a first excited state or a ground state at time infinity an essential part of our analysis is on the linear and nonlinear estimates near nonlinear excited states around which the linearized operators have eigenvalues with nonzero real parts and their corresponding eigenfunctions are not uniformly localized in space

backward bifurcation analysis of epidemiological model with partial immunity this paper presents a two stage sis epidemiological model in animal population with bovine tuberculosis btb in african buffalo as a guiding example the proposed model is rigorously analyzed the analysis reveals that the model exhibits the phenomenon of backward bifurcation where a stable disease free equilibrium dfe coexists with a stable endemic equilibrium ee when the associated reproduction number r v is less than unity it is shown under two special cases of the presented model that this phenomenon of backward bifurcation does not arise depending on vaccination coverage and efficacy of vaccine numerical simulations of the model show that the use of an imperfect vaccine can lead to effective control of the disease if the vaccination coverage and the efficacy of vaccine are high enough

on transitions to stationary states in one dimensional nonlinear wave equations we consider the long time asymptotics of solutions to one dimensional nonlinear wave equations which are infinite dimensional hamiltonian systems we assume that the nonlinear term is concentrated at a finite segment of the line we prove long time convergence to stationary states for all finite energy solutions in the frechet topology defined by local energy seminorms this means that the set of stationary states is a point attractor for the systems in the frechet topology the investigation is inspired by n bohr s postulate on the transitions between stationary states in quantum systems

on global well posedness for klein gordon equation with concentrated nonlinearity abstract we prove global well posedness for the 3d klein gordon equation with a concentrated nonlinearity

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

dynamics of an age structured two strain model for malaria transmission abstract a new age structured deterministic model for assessing the impact of anti malaria drugs on the transmission dynamics of malaria is designed and qualitatively analysed the resulting two strain age structured model undergoes backward bifurcation which arises due to malaria induced mortality in humans conditions for the existence of unique resistant strain only and low endemicity equilibria are derived for special cases it is shown for the case when treatment does not cause drug resistance that the disease free equilibrium of the wild strain only component of the model is globally asymptotically stable whenever the associated reproduction number of the model is less than unity similar result is established for the resistant strain only component of the model for this case numerical simulations of the model for the case when treatment does not cause drug resistance show that the model undergoes competitive exclusion where the malaria strain with the higher reproduction number drives the other to extinction

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

solitary wave dynamics in an external potential we study the behavior of solitary wave solutions of some generalized nonlinear schrodinger equations with an external potential the equations have the feature that in the absence of the external potential they have solutions describing inertial motions of stable solitary waves we consider solutions of the equations with a non vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that over a large interval of time they describe a solitary wave whose center of mass motion is a solution of newton s equations of motion for a point particle in the given external potential up to small corrections corresponding to radiation damping

causes of backward bifurcations in some epidemiological models abstract the phenomenon of backward bifurcation in disease transmission models where a stable endemic equilibrium co exists with a stable disease free equilibrium when the associated reproduction number is less than unity has been observed in a number of disease transmission models the epidemiological consequence of backward bifurcation is that the classical requirement of the reproduction number being less than unity becomes only a necessary but not sufficient for disease elimination hence the presence of this phenomenon in the transmission dynamics of a disease makes its effective control in the community difficult this paper addresses the problem of finding the causes of backward bifurcation in some standard deterministic models for the spread of some emerging and re emerging diseases it contains a brief review of some common causes as well as some new causes of backward bifurcation in some standard disease transmission models it is shown that in addition to the usual causes such as the use of imperfect vaccine and exogenous re infection in tb disease a number of other biological or epidemiological mechanisms such as vaccine derived immunity waning at a slower rate than natural immunity disease induced mortality in vector borne diseases and differential susceptibility in risk structured models could also cause backward bifurcation in disease transmission models

on asymptotic stability of stationary solutions to nonlinear wave and klein gordon equations we consider nonlinear wave and klein gordon equations with general nonlinear terms localized in space conditions are found which provide asymptotic stability of stationary solutions in local energy norms these conditions are formulated in terms of spectral properties of the schrodinger operator corresponding to the linearized problem they are natural extensions to partial differential equations of the known lyapunov condition for the nonlinear wave equation in three dimensional space we find asymptotic expansions as t of the solutions which are close enough to a stationary asymptotically stable solution

lp continuity of wave operators in z abstract we recover for discrete schrodinger operators on the lattice z stronger analogues of the results by weder r weder the w k p continuity of the schrodinger wave operators on the line comm math phys 208 1999 507 520 and by d ancona and fanelli p d ancona l fanelli l p boundedness of the wave operator for the one dimensional schrodinger operator comm math phys 268 2006 415 438 on r

stability of the line soliton of the kp ii equation under periodic transverse perturbations we prove the nonlinear stability of the kdv solitary waves considered as solutions of the kp ii equation with respect to periodic transverse perturbations our proof uses a miura transform which sends the solutions of an mkp ii equation to solutions of the kp ii equation

bifurcation analysis of a generalized impulsive kolmogorov model with applications to pest and disease control state dependent impulsive dynamical systems have been widely applied to model and qualitatively analyze discontinuous and state dependent managements of pests and various infectious diseases rich

on asymptotic stability in 3d of kinks for the model we add to a kink which is a 1 dimensional structure two transversal directions we then check its asymptotic stability with respect to compactly supported perturbations in 3d and a time evolution under a nonlinear wave equation nlw the problem is inspired by work by jack xin on asymptotic stability in dimension larger than 1 of fronts for reaction diffusion equations the proof involves a separation of variables the transversal variables are treated as in work on nonlinear klein gordon equation nlkg originating from klainerman and from shatah in a particular elaboration due to delort et al the longitudinal variable is treated by means of a result by weder on dispersion for schrodinger operators in id

multichannel nonlinear scattering for nonintegrable equations ii the case of anisotropic potentials and data abstract the nonlinear scattering and stability results of soffer and weinstein comm math phys 1990 are extended to the case of anisotropic potentials and data the range of nonlinearities for which the theory is shown to be valid is also extended considerably

the w k p continuity of the schrodinger wave operators on the line we prove that the wave operators for the schrodinger equation on the line are continuous on the sobolev spaces w k p 1 p moreover if the potential is exceptional and where f 1 x 0 is a jost solution at zero energy the wave operators are continuous on w k 1 and on w k

on asymptotic stability of moving kink for relativistic ginzburg landau equation we prove the asymptotic stability of kink for the nonlinear relativistic wave equations of the ginzburg landau type in one space dimension for any odd initial condition in a small neighborhood of the kink the solution asymptotically in time is the sum of the kink and dispersive part described by the free klein gordon equation the remainder converges to zero in a global norm

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

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

global attractor for a nonlinear oscillator coupled to the klein gordon field the long time asymptotics is analyzed for all finite energy solutions to a model mathbf u 1 invariant nonlinear klein gordon equation in one dimension with the nonlinearity concentrated at a single point each finite energy solution converges as t to the set of all nonlinear eigenfunctions of the form x e i t the global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation

invariant manifolds for a class of dispersive hamiltonian partial differential equations we construct an invariant manifold of periodic orbits for a class of non linear schrodinger equations using standard ideas of the theory of center manifolds we rederive the results of soffer and weinstein comm math phys 133 119 146 1997 j differential equations98 376 390 1992 on the large time asymptotics of small solutions scattering theory

the unit condition and global existence for a class of nonlinear klein gordon equations abstract global existence for a special class of nonlinear klein gordon equations in two space dimensions is proved in particular it is shown that for small initial data the equation tt u u u u 2 has a global solution

on stabilization of string nonlinear oscillator interaction abstract in the present paper we consider a system of equations that describes the interaction of a nonlinear oscillator with an infinite string the main result is the stabilization roughly speaking each finite energy solution to the system tends to a stationary solution as t and similarly as t the proof uses the description of a reversible system by an irreversible the limit stationary solutions corresponding to t may be different and arbitrary the result gives a mathematical model of transitions to stationary states in reversible systems these transitions are similar to bohr ones such transitions are impossible for finite dimensional hamiltonian systems and for linear autonomous shrodinger equations the paper contains the complete exposition and an extension of the author s recent results

backward bifurcation and stability analysis of a network based sis epidemic model with saturated treatment function abstract in this paper we present a study on a network based sis epidemic model with a saturated treatment function to characterize the saturation phenomenon of limited medical resources in this model we first obtain a threshold value r 0 which is the threshold condition for the stability of the disease free equilibrium we show that a backward bifurcation occurs under certain conditions more precisely the saturated treatment function leads to a such bifurcation in this case r 0 1 is not sufficient to eradicate the disease from the population furthermore we also study the stability of the endemic equilibrium and the corresponding stability condition is given numerical experiments are conducted and their results validate the theoretical results

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

weighted energy decay for 1d wave equation we obtain a dispersive long time decay in weighted energy norms for solutions to the 1d wave equation with generic potential the decay extends the results obtained by murata for the 1d schrodinger equation

on the variational structure of breather solutions i sine gordon equation abstract in this paper we describe stability properties of the sine gordon breather solution these properties are first described by suitable variational elliptic equations which also implies that the stability problem reduces in some sense to i the study of the spectrum of explicit linear systems and ii the understanding of how bad directions if any can be controlled using low regularity conservation laws then we discuss spectral properties of a fourth order linear matrix system using numerical methods we confirm that all spectral assumptions leading to the h 2 h 1 stability of sg breathers are numerically satisfied even in the ultra relativistic singular regime

multidimensional stability of planar travelling waves the multidimensional stability of planar travelling waves for systems of reaction diffusion equations is considered in the case that the diffusion matrix is the identity it is shown that if the wave is exponentially orbitally stable in one space dimension then it is stable for x rn n 2 furthermore it is shown that the perturbation of the wave decays like t n 1 4 as t the result is proved via an application of linear semigroup theory

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

on the asymptotic stability of solitary waves in the discrete schrodinger equation coupled to a nonlinear oscillator abstract the long time asymptotics is analyzed for finite energy solutions of the 1d discrete schrodinger equation coupled to a nonlinear oscillator the coupled system is invariant with respect to the phase rotation group u 1 for initial states close to a solitary wave the solution converges to a sum of another solitary wave and a dispersive wave which is a solution to the free schrodinger equation the proofs use the strategy of buslaev perelman v s buslaev g s perelman on the stability of solitary waves for nonlinear schrodinger equations amer math soc trans 2 164 1995 75 98 the linearization of the dynamics on the solitary manifold the symplectic orthogonal projection method of majorants etc

re infection induced backward bifurcation in the transmission dynamics of chlamydia trachomatis a new two group deterministic model for chlamydia trachomatis is designed and analyzed to gain insights into its transmission dynamics the model is shown to exhibit the phenomenon of backward bifurcation where a stable disease free equilibrium dfe co exists with one or more stable endemic equilibria when the associated reproduction number is less than unity it is further shown that the backward bifurcation dynamic is caused by the re infection of individuals who recovered from the disease the epidemiological implication of this result is that the classical requirement of the reproduction number being less than unity becomes only a necessary but not sufficient condition for disease elimination the basic model is extended to incorporate the use of treatment for infectious individuals including those who show disease symptoms and those who do not rigorous analysis of the treatment model reveals that the use of treatment could have positive or negative population level impact depending on the sign of a certain epidemiological threshold the treatment model is used to evaluate various treatment strategies namely treating every infected individual showing symptoms of chlamydia universal strategy treating only infectious males showing chlamydia symptoms male only strategy and treating only infectious females showing symptoms of chlamydia female only strategy numerical simulations show that the implementation of the male only or female only strategy can induce an indirect benefit of saving new cases of chlamydia infection in the opposite sex further the universal strategy gives the highest reduction in the cumulative number of new cases of infection

on small energy stabilization in the nlkg with a trapping potential abstract we consider a nonlinear klein gordon equation nlkg with short range potential with eigenvalues and show that in the contest of complex valued solutions the small standing waves are attractors for small solutions of the nlkg this extends the results already known for the nonlinear schrodinger equation and for the nonlinear dirac equation in addition this extends a result of bambusi and cuccagna which in turn was an extension of a result by soffer and weinstein which considered only real valued solutions of the nlkg

global existence of small amplitude solutions to nonlinear klein gordon equations in four space time dimensions consider the following class of nonlinear wave equations square text u u text text f u u u n k g where t 2 1 2 2 2 3 2 is the d alembertian of the 4 dimensional minkowski space time and f a smooth function of u u t x and its first and second partial derivates vanishing together with its first derivatives at u u u 0 in other words f is at least quadratic in u u u in a neighborhood of the trivial solution u 0 we subject u to the pure initial value problem text u mkern 1mu varepsilon text f x u t varepsilon g x at t 0 i v p with f g c 0 3 and e 0 a small parameter

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

on the absence of radiationless motion for a rotating classical charge abstract for the abraham lorentz model of a spinning charge a new approach is used to prove that all solutions converge to the set of stationary solutions in the limit t this new method allows one to get rid of the additional assumptions that have been imposed before e g the wiener condition

the hamiltonian structure of the nonlinear schrodinger equation and the asymptotic stability of its ground states in this paper we prove that ground states of the nls which satisfy the sufficient conditions for orbital stability of m weinstein are also asymptotically stable for seemingly generic equations the key issue is to prove that a certain coefficient is non negative because is a square power we assume that the nls has a smooth short range nonlinearity we assume also the presence of a very short range and smooth linear potential to avoid translation invariance the basic idea is to perform a birkhoff normal form argument on the hamiltonian as in a paper by bambusi and cuccagna on the stability of the 0 solution for nlkg but in our case the natural coordinates arising from the linearization are not canonical so we need also to apply the darboux theorem with some care though in order not to destroy some nice features of the initial hamiltonian

dynamical analysis of an age structured tuberculosis mathematical model with ltbi detectivity abstract the age dependent heterogeneity observed in tuberculosis tb epidemiology includes susceptibility infectiousness contact preferences of an individual also the chance of finding a direct route to infectious pulmonary tb ptb of certain vulnerable risk group and the diagnosis effort to detect latent tb individual ltbi are critical factors in tb epidemiology the current investigation proposes a mathematical model based on a set of coupled partial differential equations pde to encounter these vital characteristics of tb transmission the analytical study mainly encompasses well posedness of the pde system the asymptotic behaviour of the model around the disease free equilibrium point p 0 and existence criterion of endemic equilibrium point p a threshold quantity r 0 called basic reproductive number provides the average size of infected population due to a single infectious individual introduced in the naive community the current expression of r 0 offers a notable refinement in basic reproduction number compared to previous estimations also theoretically we observe detectivity of ltbi cases can both increase and decrease the size of r 0 depending upon a parametric condition

global existence of small solutions for quadratic quasilinear klein gordon systems in two space dimensions consider a quasi linear system of two klein gordon equations with masses m1 m2 we prove that when m1a2m2 and m2a2m1 such a system has global solutions for small smooth compactly supported cauchy data this extends a result proved by sunagawa j differential equations 192 2 2003 308 in the semi linear case moreover we show that global existence holds true also when m1 2m2 and a convenient null condition is satisfied by the nonlinearities

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

global existence of small solutions to nonlinear evolution equations abstract we study the global existence and asymptotic behaviour of small solutions of a large class of nonlinear partial differential equations if the nonlinear terms are of high degree the solutions will be asymptotic to solutions of the linear equation

backward bifurcations in simple vaccination models we describe and analyze by elementary means some simple models for disease transmission with vaccination in particular we give conditions for the existence of multiple endemic equilibria and backward bifurcations we extend the results to include models in which the parameters may depend on the level of infection

existence of localized solutions for a classical nonlinear dirac field we prove the existence of stationary states for nonlinear dirac equations of the form i gamma mu partial mu psi m psi f bar psi psi psi 0 we seek solutions which are separable in spherical coordinates and we use a shooting method for solving the associated problem of ordinary differential equations