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

completable filiform lie algebras we determine the solvable complete lie algebras whose nilradical is isomorphic to a filiform lie algebra moreover we show that for any positive integer n there exists a solvable complete lie algebra whose second cohomology group with values in the adjoint module has dimension at least n

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

threshold of global existence for 3 d attractive nonlinear schrodinger equations under confined potentials abstract based on a nonlinear critical index a threshold of global existence on the l2 norm of initial data for 3 d attractive nonlinear schrodinger equations under confined potentials is derived for numerical verification a newly developed radial basis function method is applied in this paper to solve the related nonlinear scalar field equation in r3

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

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

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

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

invariants of solvable rigid lie algebras up to dimension 8 the invariants of all complex solvable rigid lie algebras up to dimension 8 are computed moreover we show for rank 1 solvable algebras some criteria to deduce the non existence of nontrivial invariants or the existence of fundamental sets of invariants formed by rational functions of the casimir invariants of the associated nilradical

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

deformation par quantification et rigidite des algebres enveloppantes abstract we call a finite dimensional complex lie algebra g strongly rigid if its universal enveloping algebra u g is rigid as an associative algebra i e every formal associative deformation is equivalent to the trivial deformation in quantum group theory this phenomenon is well known to be the case for all complex semisimple lie algebras we show that a strongly rigid lie algebra over a field of characteristic 0 has to be rigid as lie algebra and that in addition its second scalar cohomology group has to vanish which excludes nilpotent lie algebras of dimension greater or equal than two moreover using kontsevitch s theory of deformation quantization we show that every polynomial deformation of the linear poisson structure on g which induces a nonzero cohomology class of a complex lie algebra g leads to a nontrivial deformation of u g hence every poisson structure on a vector space which is zero at some point and whose linear part is a strongly rigid lie algebra is therefore formally linearizable in the sense of a weinstein finally we provide examples of rigid lie algebras which are not strongly rigid and give a classification of all strongly rigid lie algebras up to dimension 6

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

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

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 rigid leibniz algebra with non trivial hl2 abstract in this article we generalize richardson s example of a rigid lie algebra with non trivial h 2 to the leibniz setting namely we consider the hemisemidirect product h of a semidirect product lie algebra m k g of a simple lie algebra g with some non trivial irreducible g module m k with a non trivial irreducible g module i l then for g s l 2 c we take m k resp i l to be the standard irreducible s l 2 c module of dimension k 1 resp l 1 assume k 2 5 is an odd integer and l 2 is odd then we show that the leibniz algebra h is geometrically rigid and has non trivial h l 2 with adjoint coefficients we close the article with an appendix where we record further results on the question whether h 2 g g 0 implies h l 2 g g 0

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

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

sharp condition of global existence for nonlinear schrodinger equation with a harmonic potential this paper discusses nonlinear schrodinger equation with a harmonic potential by constructing a different cross constrained variational problem and the so called invariant sets we derive a new threshold for blow up and global existence of solutions

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 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

cauchy problem and ehrenfest s law of nonlinear schrodinger equations with potentials abstract in this paper we prove the long time existence of the flow in q h for the nonlinear schrodinger equation in r n i t 1 2 v p 1 1 p 4 n where h 1 2 v and q h is the form domain of h when the potential v is bounded below and satisfies the condition that d v is bounded for all 2 we also show that the nonlinear analogue of the classical ehrenfest s law for the linear schrodinger equation holds for the above equation

new examples of rank one solvable real rigid lie algebras possessing a nonvanishing chevalley cohomology abstract the class of rank one solvable lie algebras possessing a maximal torus t with eigenvalue spectrum spec t 1 4 5 n 2 is studied in the context of rigidity it is shown that from the value n 18 three isomorphism classes of rigid lie algebras exist two of them being algebraically rigid and the third being geometrically rigid with a two dimensional cohomology space h 2 g g

advances in modeling model discrepancy comment on wu and browne 2015 wu and browne psychometrika 79 2015 have proposed an innovative approach to modeling discrepancy between a covariance structure model and the population that the model is intended to represent their contribution is related to ongoing developments in the field of uncertainty quantification uq on modeling and quantifying effects of model discrepancy we provide an overview of basic principles of uq and some relevant developments and we examine the wu browne work in that context we view the wu browne contribution as a seminal development providing a foundation for further work on the critical problem of model discrepancy in statistical modeling in psychological research

a sharp threshold of blow up for coupled nonlinear schrodinger equations this paper is concerned with the coupled supercritical nonlinear schrodinger equations which have applications in many physical problems especially in nonlinear optics two types of new invariant evolution flows are established a sharp threshold of blow up and global existence of solutions for the equations is derived it is shown that the main result obtained includes parts of those presented by fanelli and montefusco 2007 j phys a math theor 40 14139 50x

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

on the classification of rigid lie algebras after having given the classification of solvable rigid lie algebras of low dimensions we study the general case concerning rigid lie algebras whose nilradical is filiform and present their classification

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

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

poincare and sl 2 algebras of order 3 in this paper we initiate a general classification for lie algebras of order 3 and we give all lie algebras of order 3 based on sl 2 c and iso 1 3 the poincare algebra in four dimensions we then set the basis of the theory of the deformations in the gerstenhaber sense and contractions for lie algebras of order 3

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

theory of singular vortex solutions of the nonlinear schrodinger equation we present a systematic study of singular vortex solutions of the critical and supercritical two dimensional nonlinear schrodinger equation in particular we study the critical power for collapse and the asymptotic blowup profile of singular vortices

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

cohomologically rigid solvable lie algebras with a nilradical of arbitrary characteristic sequence it is shown that for a finite dimensional solvable rigid lie algebra r its rank is upper bounded by the length of the characteristic sequence c n of its nilradical n for any characteristic sequence c n 1 n k 1 it is proved that there exists at least a solvable lie algebra re the nilradical of which has this characteristic sequence and that satisfies the conditions h p r c r c 0 for p 3

poisson algebras in terms of non associative algebras poisson algebra is usually defined to be a commutative algebra together with a lie bracket and these operations are required to satisfy the leibniz rule we describe poisson structures in terms of a single bilinear operation this enables us to explore poisson algebras in the realm of non associative algebras we study their algebraic and cohomological properties their deformations as non associative algebras and settle the classification problem in low dimensions

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 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

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

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

asymptotic properties of standing waves for maxwell schrodinger poisson system abstract in this paper we study the asymptotic properties of minimizers for a class of constraint minimization problems derived from the maxwell schrodinger poisson system u u 2 x 1 u u 2 p u p u 0 x r 3 on the l 2 spheres a u h 1 r 3 r 3 u 2 d x where p 0 let q 2 3 2 2 and q 2 3 is the unique up to translations positive radial solution of 3 p 2 u 2 p 2 u u 2 p u 0 in r 3 with p 2 3 we prove that if 3 2 then minimizers are relatively compact in a as p 2 3 on the contrary if 3 2 by directly using asymptotic analysis we prove that all minimizers must blow up and give the detailed asymptotic behavior of minimizers

7 dimensional nilpotent lie algebras all 7 dimensional nilpotent lie algebras over c are determined by elementary methods a multiplication table is given for each isomorphism class distinguishing features are given proving that the algebras are pairwise nonisomorphic moduli are given for the infinite families which are indexed by the value of a complex parameter

classification of solvable real rigid lie algebras with a nilradical of dimension n 6 the complete classification of real solvable rigid lie algebras possessing a nilradical of dimension at most six is given eleven new isomorphism classes of indecomposable algebras are obtained it is further shown that the resulting solvable lie algebras have a vanishing second chevalley cohomology group thus correspond to algebraically rigid lie algebras

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

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

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

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

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

bound state for the fractional schrodinger equation with unbounded potential this paper considers the fractional schrodinger equation with unbounded potential we prove the existence of ground state by lagrange multiplier method the standing wave with prescribed frequency is also obtained by nehari s manifold approach in the last part we obtain that the standing wave is bound state

solving partial differential equations by collocation using radial basis functions after a series of application papers have proven the approach to be numerically effective this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with possibly radial basis functions

quantifying adventitious error in a covariance structure as a random effect we present an approach to quantifying errors in covariance structures in which adventitious error identified as the process underlying the discrepancy between the population and the structured model is explicitly modeled as a random effect with a distribution and the dispersion parameter of this distribution to be estimated gives a measure of misspecification analytical properties of the resultant procedure are investigated and the measure of misspecification is found to be related to the root mean square error of approximation an algorithm is developed for numerical implementation of the procedure the consistency and asymptotic sampling distributions of the estimators are established under a new asymptotic paradigm and an assumption weaker than the standard pitman drift assumption simulations validate the asymptotic sampling distributions and demonstrate the importance of accounting for the variations in the parameter estimates due to adventitious error two examples are also given as illustrations

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

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

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