on a nonlocal zakharov equation this article deals with the cauchy problem for a nonlocal zakharov equation we will first recall the physical motivation due to v e zakharov of this system then we will study the local cauchy problem for certain initial data and will identify the limit of the solutions when the ion velocity tends to infinity 1 origin of the nonlocal zakharov system the physical theory which follows has been developed by v e zakharov5 6 to describe the langmuir oscillations in plasma physics we assume that 1 the plasma is sufficiently uniform 2 the magnetic field is sufficiently weak 3 the nonlinearity level is not too high 4 there are no transverse high frequency electromagnetic waves we consider the following equations that model the phenomena linearized hydrodynamical equations t ne div n0 n ve 0 t ve 3v 2 te n n e me e

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

instability of standing waves for klein gordon zakharov equations with different propagation speeds in three space dimensions abstract this paper is concerned with the standing wave for klein gordon zakharov equations with different propagation speeds in three space dimensions the existence of standing wave with the ground state is established by applying an intricate variational argument and the instability of the standing wave is shown by applying pagne and sattinger s potential well argument and levine s concavity method

remarks on the blow up rate for critical nonlinear schrodinger equation with harmonic potential abstract we consider the blow up solutions of the cauchy problem for the critical nonlinear schrodinger equation with a harmonic potential which models the attractive bose einstein condensate we establish the sharp lower and upper bounds of blow up rate as t t blow up time which improve the result of q liu y zhou j zhang upper and lower bound of the blow up rate for nonlinear schrodinger equation with a harmonic potential appl math comput 172 2006 1121 1132

an example of blow up for the complex kdv equation and existence beyond the blow up the kdv equation with smooth complex initial data that blows up in a finite amount of time is studied first examples are given using the elliptic solutions of airault mckean and moser second the kdv flow of spectral coordinates is described showing that blow up occurs when eigenvalues run to infinity in finite time but eigenvalue collisions are harmless and do not cause a singularity eigenvalue collisions and blow up are studied numerically finally the solutions are continued through the singularity by continuing the coordinate flows on the appropriate riemann surface

existence and multiplicity of normalized solutions for the nonlinear kirchhoff type problems abstract in this paper we prove the existence and multiplicity results of solutions with prescribed l 2 norm for a class of kirchhoff type problems a b r 3 u 2 d x u u f u in r 3 where a b 0 are constants r and f c r r to obtain such solutions we look into critical points of the energy functional e b u a 2 r 3 u 2 b 4 r 3 u 2 2 r 3 f u constrained on the l 2 spheres s c u h 1 r 3 u 2 2 c here c 0 and f s 0 s f t d t under some mild assumptions on f we show that critical points of e b unbounded from below on s c exist for c 0 in addition we establish the existence of infinitely many radial critical points u n b of e b on s c provided that f is odd finally the asymptotic behavior of u n b as b 0 is analyzed these conclusions extend some known ones in previous papers

an a posteriori error estimate for hp adaptive dg methods for convection diffusion problems on anisotropically refined meshes we prove an a posteriori error estimate for hp adaptive discontinuous galerkin methods for the numerical solution of convection diffusion equations on anisotropically refined rectangular elements the estimate yields global upper and lower bounds of the errors measured in terms of a natural norm associated with diffusion and a semi norm associated with convection the anisotropy of the underlying meshes is incorporated in the upper bound through an alignment measure we present a series of numerical experiments to test the feasibility of this approach within a fully automated hp adaptive refinement algorithm

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

nonlinear schrodinger equation with harmonic potential this paper discusses a class of nonlinear schrodinger equation with harmonic potential by constructing a cross constrained variational problem and so called invariant manifolds of the evolution flow we derive a sharp criterion for blow up and global existence of the solutions

reliable anisotropic adaptive discontinuous galerkin method for simplified pn approximations of radiative transfer abstract in this paper we present the analysis for the error estimator for radiative transfer problems presented in giani and seaid 2016 where we showed the capabilities of the error estimator to accurately drive the adaptivity to resolve steep boundary layers which are among the difficulties that most numerical methods fail to resolve accurately we prove reliability for the error estimator in terms of a global upper bound of the error measured in the natural norm we present a series of numerical experiments to test the efficiency of this approach within a fully automated h p adaptive refinement algorithm

upper bound of weak limitation for the gross pitaevskii equation abstract this paper is concerned with the blow up solutions of gross pitaevskii equation we obtain the upper bound of weak limitation for the blow up solutions by using the method of cazenave 2003 3 as well as the concentration compact principle

minimal mass blow up solutions for the l2 critical nls with inverse square potential abstract we study minimal mass blow up solutions of the focusing l 2 critical nonlinear schrodinger equation with inverse square potential i t u u c x 2 u u 4 n u 0 with n 3 and 0 c n 2 2 4 we first prove a sharp global well posedness result all h 1 solutions with a mass i e l 2 norm strictly below that of the ground states are global note that unlike the equation in free space we do not know if the ground state is unique in the presence of the inverse square potential nevertheless all ground states have the same minimal mass we then construct and classify finite time blow up solutions at the minimal mass threshold up to the symmetries of the equation every such solution is a pseudo conformal transformation of a ground state solution

universal enveloping algebras of poisson hopf algebras abstract for a poisson algebra a by exploring its relation with lie rinehart algebras we prove a poincare birkhoff witt theorem for its universal enveloping algebra a e some general properties of the universal enveloping algebras of poisson hopf algebras are studied given a poisson hopf algebra b we give the necessary and sufficient conditions for a poisson polynomial algebra b x p to be a poisson hopf algebra we also prove a structure theorem for b e when b is a pointed poisson hopf algebra namely b e is isomorphic to b h b the crossed product of b and h b where h b is the quotient hopf algebra b e b e b

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

on the blow up phenomenon of the critical nonlinear schrodinger equation abstract in this paper we consider the blow up phenomenon of critical nonlinear schrodinger equations in dimension 1d and 2d we define the minimal mass as the l 2 norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism asymptotic compactness properties and l 2 concentration are proved the proof relies on linear and nonlinear profile decompositions

lieb thirring constant on the sphere and on the torus abstract we prove on the sphere s 2 and on the torus t 2 the lieb thirring inequalities with improved constants for orthonormal families of scalar and vector functions

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

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

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

remarks on global existence for the supercritical nonlinear schrodinger equation with a harmonic potential abstract this paper is concerned with the nonlinear schrodinger equation with a harmonic potential which describes the attractive bose einstein condensate under the magnetic trap by combining the best constant of gagliardo nirenberg s inequality with the characteristic of this equation we derive out a global existence condition for the supercritical equation which coincides with the critical case

supersonic limit for the zakharov rubenchik system abstract we study the asymptotic behavior of the solutions of zakharov rubenchik system when appropriate parameters go to zero namely we state weak and strong convergence results of these solutions to solutions of zakharov system the proof of the weak limit is a classical argument in the theory of compactness whose main ingredient is the aubin lions theorem and the ascoli theorem strong limits are conveniently treated by decomposing the nonlinearities and using the strichartz estimates associated with the group of the schrodinger equation and the wave group

supercritical geometric optics for nonlinear schrodinger equations we consider the small time semi classical limit for nonlinear schrodinger equations with defocusing smooth nonlinearity for a super cubic nonlinearity the limiting system is not directly hyperbolic due to the presence of vacuum to overcome this issue we introduce new unknown functions which are defined nonlinearly in terms of the wave function itself this approach provides a local version of the modulated energy functional introduced by y brenier the system we obtain is hyperbolic symmetric and the justification of wkb analysis follows

the global extension problem crossed products and co flag non commutative poisson algebras let p be a poisson algebra e a vector space and e p an epimorphism of vector spaces with v ker the global extension problem asks for the classification of all poisson algebra structures that can be defined on e such that e p becomes a morphism of poisson algebras from a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component all such poisson algebra structures on e are classified by an explicitly constructed classifying set gph2 p v which is the coproduct of all non abelian cohomological objects ph2 p v v v which are the classifying sets for all extensions of p by v v v the second classical poisson cohomology group h2 p v appears as the most elementary piece among all components of gph2 p v several examples are provided in the case of metabelian poisson algebras or co flag poisson algebras over p the latter being poisson algebras q which admit a finite chain of epimorphisms of poisson algebras pn q npn 1 p1 1p0 p such that dim ker i 1 for all i 1 n

finite difference method for generalized zakharov equations abstract a conservative difference scheme is presented for the initial boundary value problem for generalized zakharov equations the scheme canbe implicit or semiexplicit depending on the choice of a parameter on thebasis of a priori estimates and an inequality about norms convergence of thedifference solution is proved in order 0 h2 t2 which is better than previous results introductionthe zakharov equations 20 1 1 iet exx ne 0 1 2 ntt n e 2 xx 0describe the propagation of langmuir waves in plasmas here the complexunknown function e is the slowly varying envelope of the highly oscillatoryelectric field and the unknown real function n denotes the fluctuation of the ion density about its equilibrium value the global existence of a weak solution for the zakharov equations in one dimension is proved in 19 and existence and uniqueness of a smooth solutionfor the equations are obtained provided smooth initial data are prescribed numerical methods for the zakharov equations are studied only in 5 9 10

a uniformly accurate ua multiscale time integrator fourier pseudospectral method for the klein gordon schrodinger equations in the nonrelativistic limit regime a multiscale time integrator fourier pseudospectral mti fp method is proposed and analyzed for solving the klein gordon schrodinger kgs equations in the nonrelativistic limit regime with a dimensionless parameter 0 varepsilon le 1 0 1 which is inversely proportional to the speed of light in fact the solution of the kgs equations propagates waves with wavelength at o varepsilon 2 o 2 and o 1 in time and space respectively when 0 varepsilon ll 1 0 1 which brings significantly numerical burden in practical computation the mti fp method is designed by adapting a multiscale decomposition by frequency of the solution at each time step and applying the fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives respectively we rigorously establish two independent error bounds for the mti fp at o tau 2 varepsilon 2 h m 0 o 2 2 hm0 and o varepsilon 2 h m 0 o 2 hm0 for varepsilon in 0 1 0 1 with tau time step size h mesh size and m 0 ge 4 m0 4 an integer depending on the regularity of the solution which imply that the mti fp converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau o for varepsilon in 0 1 0 1 in addition the mti fp method converges optimally with quadratic convergence rate at o tau 2 o 2 in the regime when 0 tau lesssim varepsilon 2 0 2 and the error is at o varepsilon 2 o 2 independent of tau in the regime when 0 varepsilon lesssim tau 1 2 0 1 2 thus the meshing strategy requirement or varepsilon scalability of the mti fp is tau o 1 o 1 and h o 1 h o 1 for 0 varepsilon ll 1 0 1 which is significantly better than that of classical methods numerical results demonstrate that our error bounds are optimal and sharp finally the mti fp method is applied to study numerically convergence rates of the kgs equations to its limiting models in the nonrelativistic limit regime

universal enveloping algebras and universal derivations of poisson algebras abstract let k be an arbitrary field of characteristic 0 it is shown that for any n 1 the universal enveloping algebras of the poisson symplectic algebra p n k and the weyl algebra a n k are isomorphic and the canonical isomorphism between them easily leads to the moyal product a basis of the universal enveloping algebra p e of a free poisson algebra p k x 1 x n is constructed and it is proved that the left dependence of a finite number of elements of p e over p e is algorithmically recognizable we describe the poisson dependence of any two elements of a free poisson algebra in characteristic 0 in the language of universal derivatives the fox derivatives on free poisson algebras are defined and it is proved that an analogue of the jacobian conjecture for two generated free poisson algebras is equivalent to the two dimensional classical jacobian conjecture a new proof of the tameness of automorphisms of two generated free poisson algebras is also given

nonlinear scattering with nonlocal interaction we consider the scattering problem for the hartree type equation in n withn 2 i frac partial u partial t frac 1 2 delta u v u 2 u where v x sum limits j 1 2 lambda j x gamma j lambda 1 lambda 2 ne 0 0 lambda j in mathbb r gamma j 0 and denotes the convolution in n we prove the existence of wave operators inh0 k l2 n x k l2 n for any positive integerk under the assumption 1 1 2 2 this is an optimal result in the sense that the existence of wave operators breaks down if min 1 2 1 the case where 1 1 2 2 is also treated according to the sign of 2

two asymptotic problems for a singular nonlinear schrodinger system in this paper we continue our investigation of the relation between various systems that can be derived from the klein gordon zakharov system in the high frequency and subsonic limits in this paper we start from the singular nonlinear schr odinger system which was derived in a previous work and derive the classical nonlinear schr odinger system in two different limit cases these two nonlinear schr odinger systems have different coefficients in the nonlinearity

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

numerical study of blow up and dispersive shocks in solutions to generalized korteweg de vries equations abstract we present a detailed numerical study of solutions to general korteweg de vries equations with critical and supercritical nonlinearity both in the context of dispersive shocks and blow up we study the stability of solitons and show that they are unstable against being radiated away and blow up in the l 2 critical case the blow up mechanism by martel merle and raphael can be numerically identified in the limit of small dispersion it is shown that a dispersive shock always appears before an eventual blow up in the latter case always the first soliton to appear will blow up it is shown that the same type of blow up as for the perturbations of the soliton can be observed which indicates that the theory by martel merle and raphael is also applicable to initial data with a mass much larger than the soliton mass we study the scaling of the blow up time t in dependence of the small dispersion parameter and find an exponential dependence t and that there is a minimal blow up time t 0 greater than the critical time of the corresponding hopf solution for 0 to study the cases with blow up in detail we apply the first dynamic rescaling for generalized korteweg de vries equations this allows to identify the type of the singularity

on the cauchy problem for the nonlinear schrodinger equations with time dependent linear loss gain abstract this paper is devoted to the cauchy problem for the nonlinear schrodinger equation with time dependent loss gain which reads i u t u u u i a t u 0 this equation appears in the recent studies of bose einstein condensates and optical systems we obtain some global existence and blow up results which depend on the size of the loss gain coefficient in particular we prove the global existence for the energy critical nonlinearity by scaling and compactness arguments we also discuss asymptotic profiles and concentration properties of blow up solutions

standing waves for a generalized davey stewartson system in this paper we establish the existence of non trivial solutions for a semi linear elliptic partial differential equation with a non local term this result allows us to prove the existence of standing wave ground state solutions for a generalized davey stewartson system a sharp upper bound is also obtained on the size of the initial values for which solutions exist globally

on the convergence of the solution for a generalized zakharov system in this paper we study the limit behavior of the smooth solution for the generalized zakharov system with a magnetic field which describes the spontaneous generation of a magnetic field in a cold plasma by showing uniform bounds of solutions for this system with suitable initial data we prove that solution of the generalized zakharov system converges to the solution of the nonlinear schrodinger equation with a magnetic field

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

exact solutions of the gross pitaevskii equation in periodic potential in the presence of external source exact periodic solutions solitonlike solutions singular solitary and singular trigonometric wave solutions of the time dependent gross pitaevskii equation gpe with elliptic function potential in the presence of external source are analyzed a simple approach that applies equally to both attractive and repulsive time dependent gpe and allows one to find an extensive list of explicit periodic solutions of the gpe in terms of the jacobian elliptic functions is developed in the limit as the elliptic modulus tends to unity or to zero the linear solutions in either the jacobian elliptic cosine or the jacobian elliptic function of third order give solitonlike solutions while the rational solutions in these elliptic functions lead to singular solitary or trigonometric wave solutions the stability of these solutions is investigated numerically

classification of connected hopf algebras of dimension p 3 i abstract let p be a prime and k be an algebraically closed field of characteristic p in this paper we provide the classification of connected hopf algebras of dimension p 3 except for the case when the primitive space of the hopf algebra is a two dimensional abelian restricted lie algebra each isomorphism class is presented by generators x y z with relations and hopf algebra structures let be the multiplicative group of p 2 p 1 th roots of unity when the primitive space is one dimensional and p is odd there is an infinite family of isomorphism classes which is naturally parameterized by a k 1

simplified p n approximations to the equations of radiative heat transfer and applications simplified pn spn approximations to the equations of radiative heat transfer are derived for optically thick diffusive systems and appropriate boundary conditions are formulated the spn equations are derived by an asymptotic analysis while the boundary conditions are obtained from a variational analysis numerical comparisons of the spn equilibrium diffusion and full transport equations are presented to demonstrate the accuracy and efficiency of the spn approximations

virial type blow up solutions for the zakharov system with magnetic field in a cold plasma abstract in this paper we study blow up solutions of virial type to the zakharov system with magnetic field in a cold plasma in r n n 2 3 after obtaining some a priori estimates on those terms generated by the magnetic field we obtain a virial type blow up result to the system under consideration the result suggests that the magnetic field in a cold plasma doesn t affect the virial type blow up character of the zakharov system

the instability of bourgain wang solutions for the l2 critical nls we consider the two dimensional l 2 critical nonlinear schr odinger equation i partial tu delta u uu 2 0 in their pioneering 1997 work bourgain and wang have constructed smooth solutions which blow up in finite time t nabla u t l 2 sim 1 over t t and which display some decoupling between the regular and the singular part of the solution at blow up time we prove that this dynamic is unstable more precisely we show that any such solution with small super critical l 2 mass lies on the boundary of both h 1 open sets of global solutions that scatter forward and backwards in time and solutions that blow up in finite time on the right in the log log regime exhibited in work by f merle and p rapha e l we moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in f merle s 1992 it comm pure appl math paper in the critical mass case

a small initial data criterion of global existence for the damped nonlinear schrodinger equation this paper is concerned with the damped nonlinear schrodinger equation through analyzing the characteristics of the equation and the effect of the damping on the global existence we construct a variational problem then combining the variational problem we establish a crucial invariant evolution flow to derive an explicit and computed criterion to answer how small are the initial data such that the solutions of the system globally exist moreover the small initial data criterion can be applied in the nonlinear schrodinger equation with any positive damped parameter

a sharp condition for scattering of the radial 3d cubic nonlinear schrodinger equation we consider the problem of identifying sharp criteria under which radial h 1 finite energy solutions to the focusing 3d cubic nonlinear schrodinger equation nls i t u u u 2 u 0 scatter i e approach the solution to a linear schrodinger equation as t the criteria is expressed in terms of the scale invariant quantities u 0 l 2 nabla u 0 l 2 and m u e u where u 0 denotes the initial data and m u and e u denote the conserved in time mass and energy of the corresponding solution u t the focusing nls possesses a soliton solution e it q x where q is the ground state solution to a nonlinear elliptic equation and we prove that if m u e u q l 2 nabla q l 2 then the solution blows up in finite time the technique employed is parallel to that employed by kenig merle 17 in their study of the energy critical nls

efficient numerical methods for radiation in gas turbines numerical methods for radiative heat transfer equations coupled to a temperature equation are considered efficient solution methods and approximate equations for this system are investigated and a comparative numerical study of the different approximations is given the approximate equations considered in this paper include moment methods and diffusive approximations fast iterative solvers for the problem like multilevel methods with suitable preconditioning are considered in detail numerical experiments and comparisons in different space dimensions and for various physical situations are presented

on the nonlinear schrodinger limit of the magnetic zakharov system abstract in this paper we study the limit behavior of a smooth solution for the magnetic type zakharov system as the parameters tend to infinity we prove that solutions of the magnetic zakharov system converge to the solution of the nonlinear schrodinger equation moreover the detailed convergence rate is also investigated

on the role of quadratic oscillations in nonlinear schrodinger equations ii the l 2 critical case we consider a nonlinear semi classical schrodinger equation for which quadratic oscillations lead to focusing at one point described by a nonlinear scattering operator the relevance of the nonlinearity was discussed by r carles c fermanian kammerer and i gallagher for l 2 supercritical power like nonlinearities and more general initial data the present results concern the l 2 critical case in space dimensions 1 and 2 we describe the set of non linearizable data which is larger due to the conformal invariance as an application we precise a result by f merle and l vega concerning finite time blow up for the critical schrodinger equation the proof relies on linear and nonlinear profile decompositions

global existence and blow up of solutions for the inhomogeneous nonlinear schrodinger equation in r2 abstract this paper discusses a class of inhomogeneous nonlinear schrodinger equation i t u t x u t x v x u t x p 1 u t x u 0 x u 0 x where t x r r 2 v x satisfies some assumptions by a constrained variational problem we firstly define some cross constrained invariant sets for the inhomogeneous nonlinear schrodinger equation then we obtain some sharp conditions for global existence and blow up of solutions as a consequence it is shown that the solution is globally well posed in h r 1 r 2 with the h 1 norm of the initial data u 0 which is dominated by the minimal value of the constrained variational problem

on the existence and uniqueness of smooth solution for a generalized zakharov equation abstract the paper deals with the existence and uniqueness of smooth solution for a generalized zakharov equation we establish local in time existence and uniqueness in the case of dimension d 2 3 moreover by using the conservation laws and brezis gallouet inequality the solution can be extended globally in time in two dimensional case for small initial data besides we also prove global existence of smooth solution in one spatial dimension without any small assumption for initial data

singular solutions of the subcritical nonlinear schrodinger equation abstract we show that the subcritical d dimensional nonlinear schrodinger equation i t 2 0 where 1 d 2 admits smooth solutions that become singular in l p for p p where p d d 1 since lim d 2 p 2 these solutions can collapse at any 2 p and in particular for p 2 2

optimal lower bound estimates for the blow up rate for the zakharov system in a nonhomogeneous medium abstract in this paper we obtain lower bound estimates for the blow up rate of finite time blow up solutions to the cauchy problem for the zakharov system in a nonhomogeneous medium in two space dimensions by introducing suitable scale transformations of space and time and the use of compactness arguments we derive an optimal lower bound estimate in the energy space h 2 r 2 l 2 r 2 h 1 r 2 for the blow up rate for t near the finite blow up time t also we give an application to the virial identity for the zakharov system under study

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

normalized ground states for the nls equation with combined nonlinearities abstract we study existence and properties of ground states for the nonlinear schrodinger equation with combined power nonlinearities u u u q 2 u u p 2 u in r n n 1 having prescribed mass r n u 2 a 2 under different assumptions on q p a 0 and r we prove several existence and stability instability results in particular we consider cases when 2 q 2 4 n p 2 q p i e the two nonlinearities have different character with respect to the l 2 critical exponent these cases present substantial differences with respect to purely subcritical or supercritical situations which were already studied in the literature we also give new criteria for global existence and finite time blow up in the associated dispersive equation

existence of self similar blow up solutions for zakhrov equation in dimension two part i we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right we prove the existence of blow up solutions stable self similar blow up solutions for this problem and we study various properties of these solutions

blow up in several points for the davey stewartson system in r2 abstract in this paper we study the blow up solutions for the davey stewartson system in r 2 which appears in the description of the evolution of surface water waves for any given points x 1 x p in r 2 we construct a solution u t which blows up in finite time t exactly in these points in addition we investigate the precise behavior of the solution u t as t t both at the blow up points x 1 x p and in r 2 x 1 x p our result gives a rigorous analysis for the numerical result of besse et al in 2

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

convergence of an energy preserving scheme for the zakharov equations in one space dimension an energy preserving linearly implicit finite difference scheme is presented for approximating solutions to the periodic cauchy problem for the one dimensional zakharov system of two nonlinear partial differential equations first order convergence estimates are obtained in a standard energy norm in terms of the initial errors and the usual discretization errors

blow up results of viriel type for zakharov equations we consider the zakharov equations in n for n 2 n 3 we first establish a viriel identity for such equations and then prove a blow up result for solutions with a negative energy

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

sharp threshold of global existence and instability of standing wave for a davey stewartson system this paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave phi t x e i omega t u x for the system i phi t delta phi a phi p 1 phi b e 1 phi 2 phi 0 quad t geq 0 quad x in bf r n quad quad quad rm ds where a 0 b 0 1 0 by combining the former results

co frobenius hopf algebras and the coradical filtration we prove that a hopf algebra with a finite coradical filtration is co frobenius we also characterize co frobenius hopf algebras with coradical a hopf subalgebra let h be a hopf algebra whose coradical is a hopf algebra let gr h be the associated graded coalgebra and let r be the diagram of h c f 2 then the following are equivalent 1 h is co frobenius 2 gr h is co frobenius 3 r is finite dimensional 4 the coradical filtration of h is finite this theorem allows us to give systematically examples of co frobenius hopf algebras and opens the way to the classification of ample classes of such hopf algebras

on the effect of equivalent constraints on a maximizing problem associated with the sobolev type embeddings in mathbb r n in this paper we consider the attainability of a maximizing problem begin aligned d sup vert u vert h 1 n gamma 1 left vert u vert n n alpha vert u vert p p right end aligned where n ge 2 n 0 0 gamma le n and vert u vert h 1 n gamma left vert u vert n gamma vert nabla u vert n gamma right frac 1 gamma the existence of a maximizer for d is closely related to the exponent gamma in fact we show that the value begin aligned alpha alpha inf vert u vert h 1 n gamma 1 left frac 1 vert u vert n n vert u vert p p right end aligned is a threshold in terms of the attainability of d

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

equations of langmuir turbulence and nonlinear schrodinger equation smoothness and approximation we consider the following family of systems parametrized by e 0 which describe langmuir s turbulence in the case where initial data are sufficiently small we study the asymptotic behaviour of the solutions ee ne when e goes to zero for d 1 2 3 namely we state convergence results of ee ne to the couple e e 2 where e is the solution of the nonlinear schrodinger equation moreover in three dimensions we show that the smooth solution ee ne is defined on 0 tmax e with tmax e going to infinity as e goes to 0

continuous interior penalty hp finite element methods for advection and advection diffusion equations a continuous interior penalty hp finite element method that penalizes the jump of the discrete solution across mesh interfaces is introduced error estimates are obtained for first order and advection dominated transport operators the analysis relies on three technical results that are of independent interest an hp inverse trace inequality a local discontinuous to continuous hp interpolation result and hp error estimates for continuous l2 orthogonal projections

energy convergence for singular limits of zakharov type systems we prove existence and uniqueness of solutions to the klein gordon zakharov system in the energy space h1 l2 on some time interval which is uniform with respect to two large parameters c and these two parameters correspond to the plasma frequency and the sound speed in the simultaneous high frequency and subsonic limit we recover the nonlinear schrodinger system at the limit we are also able to say more when we take the limits separately

hp adaptive discontinuous galerkin methods for simplified pn approximations of frequency dependent radiative transfer abstract we investigate the performance of a class of h p adaptive discontinuous galerkin methods for the numerical solution of simplified p n approximations of radiative transfer in non gray semitransparent media by introducing an optical scale and using asymptotic expansions in the radiative transfer equation we formulate the simplified p n approximations the optical spectrum is decomposed in frequency bands and the simplified p n equations are solved for each frequency band as a numerical solver for the simplified p n equations we consider a high order discontinuous galerkin method the solver belongs to a class of finite element methods whose approximate solutions are discontinuous across inter element boundaries this property renders the method ideally suited for the h p adaptivity an error estimator is shown to provide reliable and practically useful upper bounds for the numerical errors independent of the optical scales used in the simulations the proposed method is simple fast and highly accurate the performance of the method is analyzed on several applications in frequency dependent radiative transfer the aim of such a method compared to the conventional finite element methods is to solve the simplified p n equations efficiently and with a high level of accuracy on unstructured meshes with different elements the obtained results demonstrate the ability of the proposed method to capture the main radiative features

global well posedness results for the 2d schrodinger debye system abstract in this work we study the well posedness of the schrodinger debye system i t u 1 2 u u v t 0 x r 2 t v v u 2 0 1 u 0 x u 0 x v 0 x v 0 x in the context of sobolev spaces of type h k 1 r 2 h k 2 r 2 where k 1 k 2 are integers we prove some a priori estimates and therefore global well posedness using nonlinear interpolation of operators we prove some global well posedness results when u 0 v 0 h s r 2 h l r 2 where s l r are some convex combinations of k 1 and k 2

existence and nonexistence of extremals for critical adams inequalities in r4 and trudinger moser inequalities in r2 abstract though much progress has been made with respect to the existence of extremals of the critical first order trudinger moser inequalities in w 1 n r n and higher order adams inequalities on finite domain r n whether there exists an extremal function for the critical higher order adams inequalities on the entire space r n still remains open the current paper represents the first attempt in this direction by considering the critical second order adams inequality in the entire space r 4 the classical blow up procedure cannot apply to solving the existence of critical adams type inequality because of the absence of the polya szego type inequality in this paper we develop some new ideas and approaches based on a sharp fourier rearrangement principle see 31 sharp constants of the higher order gagliardo nirenberg inequalities and optimal poly harmonic truncations to study the existence and nonexistence of the maximizers for the adams inequalities in r 4 of the form s sup u h 2 1 r 4 exp 32 2 u 2 1 u 2 d x where 32 2 we establish the existence of the threshold where 32 2 2 b 2 2 and b 2 1 24 2 such that s is attained if 32 2 and is not attained if 32 2 this phenomenon has not been observed before even in the case of first order trudinger moser inequality therefore we also establish the existence and non existence of an extremal function for the trudinger moser inequality on r 2 furthermore the symmetry of the extremal functions can also be deduced through the fourier rearrangement principle

on the dixmier moeglin equivalence for poisson hopf algebras abstract we prove that the poisson version of the dixmier moeglin equivalence holds for cocommutative affine poisson hopf algebras this is a first step towards understanding the symplectic foliation and the representation theory of cocommutative affine poisson hopf algebras our proof makes substantial use of the model theory of fields equipped with finitely many possibly noncommuting derivations as an application we show that the symmetric algebra of a finite dimensional lie algebra equipped with its natural poisson structure satisfies the poisson dixmier moeglin equivalence

concentration properties of blow up solutions and instability results for zakharov equation in dimension two part ii we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star