conditional stability of solitary waves propagating in elastic rods abstract the problem of dynamic stability of solitary wave solutions of the equations describing bending oscillations of inextensible elastic rods is treated the governing equations describe sufficiently large displacements though we are restricted to small strains we show that under the condition of well posedness of the initial value problem in some specific sense the family of solitary wave solutions of the mentioned equations is stable for the planar elastic rod the framework of the analysis is largely based on the spectral properties of the linearized hamiltonian h we show that for planar oscillations h is positively semidefinite subject to a certain constraint which implies the stability

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

stability and instability of periodic travelling wave solutions for the critical korteweg de vries and nonlinear schrodinger equations abstract in this paper we establish new results about the existence stability and instability of periodic travelling wave solutions related to the critical korteweg de vries equation u t 5 u 4 u x u x x x 0 and the critical nonlinear schrodinger equation i v t v x x v 4 v 0 the periodic travelling wave solutions obtained in our study tend to the classical solitary wave solutions in the infinite wavelength scenario the stability approach is based on the theory developed by angulo natali in j angulo f natali positivity properties of the fourier transform and the stability of periodic travelling wave solutions siam j math anal 40 2008 1123 1151 for positive periodic travelling wave solutions associated to dispersive evolution equations of korteweg de vries type the instability approach is based on an extension to the periodic setting of arguments found in bona souganidis strauss j l bona p e souganidis w a strauss stability and instability of solitary waves of korteweg de vries type proc roy soc london ser a 411 1987 395 412 regarding the critical schrodinger equation stability instability theories similar to the critical korteweg de vries equation are obtained by using the classical grillakis shatah strauss theory in m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry ii j funct anal 94 1990 308 348 m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry i j funct anal 74 1987 160 197 the arguments presented in this investigation have prospects for the study of the stability of periodic travelling wave solutions of other nonlinear evolution equations

stability and asymptotic stability in the energy space of the sum of n solitons for subcritical gkdv equations we prove in this paper the stability and asymptotic stability in h1 of a decoupled sum of n solitons for the subcritical generalized kdv equations the proof of the stability result is based on energy arguments and monotonicity of the local l2 norm note that the result is new even for p 2 the kdv equation the asymptotic stability result then follows directly from a rigidity theorem in 16

hylomorphic solitons for the benjamin ono and the fractional kdv equations abstract this paper concerns with the existence of solitons namely stable solitary waves for the benjamin ono and the fractional kdv equations

stability of semiclassical bound states of nonlinear schrodinger equations with potentials in this paper we study the lyapunov stabilities of some semiclassical bound states of the nonhomogeneous nonlinear schrodinger equation we prove that among those bound states those which are concentrated near local minima respectively maxima of the potentialv are stable respectively unstable we also prove that those bound states are positive if is sufficiently small

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

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

conditional stability theorem for the one dimensional klein gordon equation the paper addresses the conditional non linear stability of the steady state solutions of the one dimensional klein gordon equation for large time we explicitly construct the center stable manifold for the steady state solutions using the modulation method of soffer and weinstein and strichartz type estimates the main difficulty in the one dimensional case is that the required decay of the klein gordon semigroup does not follow from strichartz estimates alone we resolve this issue by proving an additional weighted decay estimate and further refinement of the function spaces which allows us to close the argument in spaces with very little time decay

a coupled momentum method for modeling blood flow in three dimensional deformable arteries abstract blood velocity and pressure fields in large arteries are greatly influenced by the deformability of the vessel moreover wave propagation phenomena in the cardiovascular system can only be described considering wall deformability since blood is usually described as an incompressible fluid however computational methods for simulating blood flow in three dimensional models of arteries have either considered a rigid wall assumption for the vessel or significantly simplified or reduced geometries computing blood flow in deformable domains using standard techniques like the ale method remains a formidable problem for large realistic anatomic and physiologic models of the cardiovascular system we have developed a new method to simulate blood flow in three dimensional deformable models of arteries the method couples the equations of the deformation of the vessel wall at the variational level as a boundary condition for the fluid domain we consider a strong coupling of the degrees of freedom of the fluid and the solid domains and a linear membrane model enhanced with transverse shear for the vessel wall the effect of the vessel wall boundary is therefore added in a monolithic way to the fluid equations resulting in a remarkably robust scheme we present here the mathematical formulation of the method and discuss issues related to the fluid solid coupling membrane formulation time integration method and boundary and initial conditions implementation is discussed and results with simple geometries as well as large subject specific models are presented

an instability mechanism for radially symmetric standing waves of a nonlinear schrodinger equation abstract a condition is proved for the spectrum of nonlinear schrodinger equations linearised at a standing wave to have a positive eigenvalue the standing waves considered are radially symmetric ones in higher space dimensions the instability result is applied to show that if there are multiple asymptotically positive nondegenerate waves with a fixed number of zeroes then there is an unstable one the techniques used are dynamical systems arguments and involve a shooting argument in the space of lagrangian planes

an asymptotic approach to solitary wave instability and critical collapse in long wave kdv type evolution equations abstract instability development and critical collapse of solitary waves are considered in the framework of generalized korteweg de vries kdv equations in one and two dimensions an analytical theory of the solitary wave dynamics and generation of radiation is constructed for the critical case when the solitary waves are weakly unstable characteristic types of the global essentially nonlinear evolution of the unstable solitary waves are analyzed for some typical generalized kdv equations the scalling laws of the self similar wave field transformation are found analytically for the power like kdv equation in the critical case p 4 the asymptotic approach is also developed for the modified zakharov kuznetsov equation in two dimensions and the rate of the singularity formation is found to be smaller than in one dimension due to diffractive wave effects

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

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

stability for line solitary waves of zakharov kuznetsov equation abstract in this paper we consider the stability for line solitary waves of the two dimensional zakharov kuznetsov equation on r t l which is one of a high dimensional generalization of korteweg de vries equation where t l is the torus with the 2 l period the orbital and asymptotic stability of the one soliton of korteweg de vries equation on the energy space was proved by benjamin 2 pego and weinstein 41 and martel and merle 30 we regard the one soliton of korteweg de vries equation as a line solitary wave of zakharov kuznetsov equation on r t l we prove the stability and the transverse instability of the line solitary waves of zakharov kuznetsov equation by applying the method of evans function and the argument of rousset and tzvetkov 44 moreover we prove the asymptotic stability for orbitally stable line solitary waves of zakharov kuznetsov equation by using the argument of martel and merle 30 31 32 and a liouville type theorem if l is the critical period with respect to a line solitary wave the line solitary wave is orbitally stable however since this line solitary wave is a bifurcation point of the stationary equation the linearized operator of the stationary equation is degenerate because of the degeneracy of the linearized operator we can not show the liouville type theorem for the line solitary wave by using the usual virial type estimate to show the liouville type theorem for the line solitary wave we modify a virial type estimate

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

pointwise error estimates for a streamline diffusion finite element scheme pointwise error estimates for a streamline diffusion scheme for solving a model convection dominated singularly perturbed convection diffusion problem are given these estimates improve pointwise error estimates obtained by johnson et al 5

the blow up dynamic and upper bound on the blow up rate for critical nonlinear schrodinger equation we consider the critical nonlinear schr odinger equation iut u u 4 n u with initial condition u 0 x u0 in dimension n 1 for u0 h1 local existence in the time of solutions on an interval 0 t is known and there exist finite time blow up solutions that is u0 such that limt t 1

a modified diffusion coefficient technique for the convection diffusion equation a new modified diffusion coefficient mdc technique for solving convection diffusion equation is proposed the galerkin finite element discretization process is applied on the modified equation rather than the original one for a class of one dimensional convection diffusion equations we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size then prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and or equation coefficients the application of the derived analytic formula of mdc is extended for other classes of convection diffusion equations where the analytic formula is computed locally within each element according to the mesh size and the values of associated coefficients in each direction the numerical results of the proposed approach for two dimensional variable coefficients with boundary layers convection dominated problems show stable and accurate solutions even on coarse grids accordingly multigrid based solvers retain their efficient convergence rates

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

stability in h 1 of the sum of k solitary waves for some nonlinear schrodinger equations in this article we consider nonlinear schrodinger nls equations in rd for d 1 2 and 3 we consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable let rk t x be k solitary wave solutions of the equation with different speeds v1 v2 vk provided that the relative speeds of the solitary waves vk vk 1 are large enough and that no interaction of two solitary waves takes place for positive time we prove that the sum of the rk t is stable for t 0 in some suitable sense in h1 to prove this result we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear schrodinger equation this property is similar to the l2 monotonicity property that has been proved by martel and merle for the generalized korteweg de vries gkdv equations see 12 lem 16 proof of prop 6 and that was used to prove the stability of the sum of k solitons of the gkdv equations by the authors of the present article see 15 th 1 i

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

nonlinear stability of periodic traveling wave solutions to the schrodinger and the modified korteweg de vries equations abstract this work is concerned with stability properties of periodic traveling waves solutions of the focusing schrodinger equation i u t u x x u 2 u 0 posed in r and the modified korteweg de vries equation u t 3 u 2 u x u x x x 0 posed in r our principal goal in this paper is the study of positive periodic wave solutions of the equation 3 0 called dnoidal waves a proof of the existence of a smooth curve of solutions with a fixed fundamental period l 2 2 l 2 h per 0 l is given it is also shown that these solutions are nonlinearly stable in the energy space h per 1 0 l and unstable by perturbations with period 2l in the case of the schrodinger equation

stability of solitary waves for a nonlinear dispersive system in a critical case we study a nonlinear dispersive system with critical power nonlinearities building on the works of weinstein and angulo bona linares and scialom a conditional stability modulo the group of translations in space and dilations is proved for a solitary wave solution

a three dimensional computational method for blood flow in the heart 1 immersed elastic fibers in a viscous incompressible fluid abstract this paper describes the numerical solution of the 3 dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers implementation details such as vectorization and the efficient use of external memory are discussed the method is applied to the damped vibrations of a fiber wound toroidal tube and empirical evidence of convergence is presented

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

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

self focusing on bounded domains the critical nonlinear schrodinger equation nls on bounded domains models the propagation of cw laser beams in hollow core fibers unlike the nls on unbounded domains which models propagation in bulk media the ground state waveguide solutions are stable and the condition of critical power for singularity formation is generically sharp

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

stability of the solitary wave solutions to a coupled bbm system abstract in this work we present a stability criteria for the solitary wave solutions to a bbm system that contains coupled nonlinear terms using the idea by bona chen and karakashian 5 and exploiting the accurate point spectrum information of the associated schrodinger operator we improve the stability results previously gotten by pereira 15

stability of the solitary manifold of the perturbed sine gordon equation abstract we study the perturbed sine gordon equation t t x x sin f e x where f is of differentiability class c n in e and the first k derivatives vanish at e 0 i e e l f 0 0 for 0 l k we construct implicitly a virtual solitary manifold by deformation of the classical solitary manifold in n iteration steps our main result establishes that the initial value problem with an appropriate initial state e n close to the virtual solitary manifold has a unique solution which follows up to time 1 c e k 1 2 and errors of order e n a trajectory on the virtual solitary manifold the trajectory on the virtual solitary manifold is described by two parameters which satisfy a system of odes in contrast to previous works our stability result yields arbitrarily high accuracy as long as the perturbation f is sufficiently often differentiable

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

solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities we study the nonlinear schrodinger nls generalized korteweg de vries gkdv and generalized benjamin ono equations gbo with critical power nonlinearities the ground state solitary wave is conditionally stable modulo the group of translations in space and phase and dilations this is a natural extension of the result for subcritical nonlinearities where solitary waves are stable modulo translations alone to a case where solutions may blow up we also present results on global existence for gbo and the structure of singularities for certain solutions of gbo which are analogues of previous results by the author for nls and gkdv

orbital stability of one parameter periodic traveling waves for dispersive equations and applications abstract this paper establishes sufficient conditions for the orbital stability of one parameter spatially periodic traveling wave solutions for one dimensional dispersive equations our method of proof combines known techniques with some new ideas as a consequence of our result we give several applications for well known dispersive equations extension of the theory to regularized equations is also established

study of cardiovascular function using a coupled left ventricle and systemic circulation model to gain insight into cardio arterial interactions a coupled left ventricle systemic artery lv sa model is developed that incorporates a three dimensional finite strain left ventricle lv and a physiologically based one dimensional model for the systemic arteries sa the coupling of the lv model and the sa model is achieved by matching the pressure and the flow rate at the aortic root i e the sa model feeds back the pressure as a boundary condition to the lv model and the aortic flow rate from the lv model is used as the input for the sa model the governing equations of the coupled system are solved using a combined immersed boundary finite element ib fe method and a lax wendroff scheme a baseline case using physiological measurements of healthy subjects and four exemplar cases based on different physiological and pathological scenarios are studied using the lv sa model the results of the baseline case agree well with published experimental data the four exemplar cases predict varied pathological responses of the cardiovascular system which are also supported by clinical observations the new model can be used to gain insight into cardio arterial interactions across a range of clinical applications

global well posedness and scattering for nonlinear schrodinger equations with combined nonlinearities in the radial case we consider the cauchy problem for the nonlinear schrodinger equation with combined nonlinearities one of which is defocusing mass critical and the other is focusing energy critical or energy subcritical the threshold is given by means of variational argument we establish the profile decomposition in h1 rd and then utilize the concentration compactness method to show the global wellposedness and scattering versus blowup in h1 rd below the threshold for radial data when d 4

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 asymptotic stability of moving ground states of the nonlinear schrodinger equation we extend to the case of moving solitons the result on asymptotic stability of ground states of the nls obtained by the author in cu1 for technical reasons we consider only smooth solutions the proof is similar to the earlier paper however now the flows required for the darboux theorem and the birkhoff normal forms instead of falling within the framework of standard theory of ode s are related to quasilinear hyperbolic symmetric systems it is also not obvious that darboux theorem can be applied since we need to compare two symplectic forms in a neighborhood of the ground states not in h r but rather in the space where also the variance is bounded but the nls does not preserve small neighborhoods of the ground states in

shishkin mesh simulation a new stabilization technique for convection diffusion problems abstract a new stabilization procedure is presented it is based on a simulation of the interaction between the coarse and fine parts of a shishkin mesh but can be applied on coarse and irregular meshes and on domains with nontrivial geometries the technique which does not require adjusting any parameter can be applied to different stabilized and non stabilized methods numerical experiments show it to obtain oscillation free approximations on problems with boundary and internal layers on uniform and nonuniform meshes and on domains with curved boundaries

mapping between generalized nonlinear schrodinger equations and neutral scalar field theories and new solutions of the cubic quintic nls equation we highlight an interesting mapping between the moving breather solutions of the generalized nonlinear schrodinger nls equations and the static solutions of neutral scalar field theories using this connection we then obtain several new moving breather solutions of the cubic quintic nls equation both with and without uniform phase in space the stability of some stationary solutions is investigated numerically and the results are confirmed via dynamical evolution

asymptotic stability of ground states in 2d nonlinear schrodinger equation including subcritical cases abstract we consider a class of nonlinear schrodinger equations in two space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l 2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a carefully chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

multidimensional modelling for the carotid artery blood flow abstract in this work a multidimensional 3d 1d fem model of the whole arterial tree is implemented it comprises a 3d compliant model of the carotid bifurcation coupled with a 1d model for the remaining part of the arterial tree with this approach difficulties arising from the treatment of boundary conditions for the 3d model are naturally handled the navier stokes equations are used as the governing equations for the blood flow while an elastic compliant model is implemented for the arterial wall also the a l e formulation is considered within the 3d blood regions taking into account the domain deformations produced by the moving artery wall this 3d model is complemented with a 1d model of the entire arterial tree in order to appropriately set inflow and outflow boundary conditions for the former the reduced 1d model solves the momentum and continuity equations in compliant tubes so as to reproduce the propagation of the pressure pulse in the arterial network also a volumetric flow rate is imposed at the inlet to model the systolic work of the heart the peripheral arteriole beds are simulated with the well known lumped windkessel model a standard geometry of the carotid bifurcation is discretized with p1 bubble p1 tetrahedral elements the obtained results properly reproduce the general flow patterns reported in the literature very good agreement between the outcomes of a pure 1d model and those of the combined multidimensional model was obtained it is worth noting that this kind of model may provide useful information to gain understanding in the genesis and development of arterial diseases

asymptotic stability of ground states in 3d nonlinear schrodinger equation including subcritical cases we consider a class of nonlinear schrodinger equation in three space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a careful chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

on weak interaction between a ground state and a non trapping potential abstract we show that ground states of the nls moving at nonzero speed are asymptotically stable if they either stay far from the potential or the potential is small or the ground state has large speed we search an effective hamiltonian using the birkhoff normal forms argument in 11 treating the potential as a perturbation the so called fermi golden rule which is used to describe the decay to 0 of the internal discrete modes of the ground state is similar to that in 12 the continuous modes dispersion requires the theory in 40 on charge transfer models

on the multi symplectic structure of boussinesq type systems i derivation and mathematical properties the boussinesq equations are known since the end of the xixst century however the proliferation of various boussinesq type systems started only in the second half of the xxst century today they come under various flavours depending on the goals of the modeller at the beginning of the xxist century an effort to classify such systems at least for even bottoms was undertaken and developed according to both different physical regimes and mathematical properties with special emphasis in this last sense on the existence of symmetry groups and their connection to conserved quantities of particular interest are those systems admitting a symplectic structure with the subsequent preservation of the total energy represented by the hamiltonian in the present paper a family of boussinesq type systems with multi symplectic structure is introduced some properties of the new systems are analyzed their relation with already known boussinesq models the identification of those systems with additional hamiltonian structure as well as other mathematical features like well posedness and existence of different types of solitary wave solutions the consistency of multi symplectic systems with the full euler equations is also discussed

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

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

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

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

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

numerical crosswind smear in the streamline diffusion method abstract the streamline diffusion method is a popular element method for approximating convection dominated transport equations the solution to such equations may develop boundary or internal layers in the crosswind direction orthogonal to the flow direction and this results in numerical crosswind layers it is known that the size of these layers is o h 1 2 or perhaps o h 3 4 h being the size of the mesh used in the numerical computation but in practice o h is observed if the gridlines of the method are aligned with the streamlines of the problem this connection is studied and an explanation is given

nonequilibrium statistical behavior of nonlinear schrodinger equations this paper explores the equilibrium and near equilibrium statistical behavior of nonlinear schrodinger equations through numerical simulations the investigation centers on a class of finite dimensional hamiltonian systems obtained by discretizing nls equations defined on a bounded interval with saturated focusing nonlinearities these nonlinear systems govern a wave dynamics that is nonintegrable and nonsingular and for which typical solutions are composed of large scale coherent structures interacting within a background of small scale fluctuations the equilibrium statistics of such a system is most naturally defined by a gibbs distribution that is canonical in energy and microcanonical in particle number by sampling these equilibrium distributions using a metropolis type algorithm the typical excitations of the system are determined over the full range of system parameters nonequilibrium relaxations from initial perturbations of these equilibrium distributions are then computed by means of linear response theory the computations reveal a surprising dependence of the qualitative properties of relaxation toward equilibrium on the temperature and the nonlinearity on the one hand a simple exponential decay to equilibrium is observed for sufficiently high temperature or sufficiently strong nonlinearity for low temperature and moderate nonlinearity on the other hand the system rapidly adjusts to a metaequilibrium state that equilibrates extremely slowly this two stage thermalization may be related to quasiperiodicity in the underlying dynamics indeed the metaequilibria are found to occur in a range of system parameters for which approximately quasiperiodic solutions are prevalent in equilibrium

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

stabilizing effects of dispersion management a cubic nonlinear schrodinger equation nls with periodically varying dispersion coefficient as it arises in the context of fiber optics communication is considered for sufficiently strong variation corresponding to the so called strong dispersion management regime the equation possesses pulse like solutions which evolve nearly periodically this phenomenon is explained by constructing ground states for the averaged variational principle and justifying the averaging procedure furthermore it is shown that in certain critical cases e g quintic nonlinearity in one dimension and cubic nonlinearity in two dimensions the dispersion management technique stabilizes the pulses which otherwise would be unstable this observation seems to be new and is reminiscent of the well known kapitza s effect of stabilizing the inverted pendulum by rapidly moving its pivot 2001 published by elsevier science b v

on the bound states of the nonlinear schrodinger equation with a linear potential abstract we study the nonlinear schrodinger equation with an attractive linear potential i t v x 2 0 n 2 nls which arises in the mathematical description of phenomena in nonlinear optics and plasma physics nonlinear bound states are finite energy localized solutions which if dynamically stable play an important role in the structure of general solutions of nls we discuss the existence and nonlinear orbital stability of nonlinear ground states of nls in particular if v supports a linear bound state then nls has stable nonlinear ground states in the supercritical and critical cases 2 n where blow up self focusing or collapse can occur this is a phenomenon not present in the case where v 0 in addition islands of stability in the regime of large h1 norm exist

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

linear enhancements of the streamline diffusion method for convection diffusion problems abstract several computationally simple modifications of the streamline diffusion finite element method are developed for linear convection dominated convection diffusion problems in two dimensions numerical experiments show that these modifications yield significantly more accurate results than are attainable from the basic streamline diffusion method

the gardner equation and the l 2 stability of the n soliton solution of the korteweg de vries equation multi soliton solutions of the korteweg de vries equation kdv are shown to be globally l2 stable and asymptotically stable in the sense of martel and merle the proof is surprisingly simple and combines the gardner transform which links the gardner and kdv equations together with the martel merle tsai and martel merle recent results on stability and asymptotic stability in the energy space applied this time to the gardner equation as a by product the results of maddocks sachs and merle vega are improved in several directions

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

maximizers for gagliardo nirenberg inequalities and related non local problems in this paper we study the existence of maximizers for two families of interpolation inequalities namely a generalized gagliardo nirenberg inequality and a new inequality involving the riesz energy two basic tools in our argument are a generalization of lieb s translation lemma and a riesz energy version of the brezis lieb lemma

periodic finite genus solutions of the kdv equation are orbitally stable abstract the stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade the kdv equation is known to have large families of periodic solutions that are parameterized by hyperelliptic riemann surfaces they are generalizations of the famous multi soliton solutions we show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution

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

asymptotic behavior of the least energy solution of a problem with competing powers we consider the problem e2 u uq up 0 in u 0 in u 0 on here is a smooth bounded domain in rn 1 q p n 2n 2 if n 3 and e is a small positive parameter we study the asymptotic behavior of the least energy solution as e goes to zero in the case q nn 2 we show that the limiting behavior is dominated by the singular solution g gq 0 in p g 0 on the reduced energy is of nonlocal type