multiple existence of sign changing solutions for coupled nonlinear schrodinger equations abstract in this paper we consider the multiple existence of sign changing solutions of coupled nonlinear schrodinger equations p u 1 u 1 u 3 v 2 u in r 3 v 2 v 2 v 3 u 2 v in r 3 where 1 2 1 2 are positive and 0

orthogonal polynomials in transport theories the asymptotical k co behaviour of zeros of the polynomials g v encoun tered in the treatment of direct and inverse problems of scattering in neutron transport as well as radiative transfer theories is investigated in terms of the amplitude kjk of the kth legendre polynomial needed in the expansion of the scattering function the parameters gi describe the anisotropy of scattering of the medium considered in particular it is shown that the asymptotical density of zeros of the polynomials gp v is an inverted semicircle for the anisotropic non multiplying scattering medium

optimal liouville type theorems for noncooperative elliptic schrodinger systems and applications we study multi component elliptic schrodinger systems arising in nonlinear optics and bose einstein condensation phenomena we prove new liouville type nonexistence theorems as well as a priori bounds decay and singularity estimates this is shown under an optimal sobolev growth restriction on the nonlinearities thus improving on recent results of dancer et al and of tavares et al these systems are of non cooperative form and hence cannot be tackled by maximum principle methods such as moving planes instead we rely on a delicate combination of rellich pohozaev type identities sobolev and interpolation inequalities on sn 1 and feedback and measure arguments we also extend our results to a rather general class of gradient type systems

infinitely many sign changing solutions for system of p laplace equations in rn abstract in this paper we consider the system of p laplacian equations in r n p u j j x u j p 2 u j i 1 k i j u i q u j q 2 u j in r n u j x 0 as x j 1 k where 1 p n max 2 p 2 q p n p n p i j are constants i j 1 k j j 0 j 1 2 k i j 0 i j i j 1 k j j 1 k are the potential functions and satisfy suitable decay assumptions the existence of infinitely many sign changing solutions is proved by the truncation method and by the concentration compactness analysis on the approximating solutions

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

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

multiple existence of solutions for coupled nonlinear schrodinger equations abstract in this paper we consider the multiple existence of solutions of the coupled nonlinear schrodinger equations p u 1 u u 3 v 2 u in r 3 v 2 v v 3 u 2 v in r 3 where 1 2 and 0

weak concentration and wave operator for a 3d coupled nonlinear schrodinger system reported in this paper are results concerning the cauchy problem and the dynamics for a cubic nonlinear schrodinger system arising in nonlinear optics a sharp criterion is given concerned with the dichotomy global existence versus finite time blow up when a radial solution blows up in finite time we prove the concentration in the critical lebesgue space sufficient condition for the scattering and the construction of the wave operator in the energy space is also provided

bifurcation diagrams of coupled schrodinger equations radially symmetric solutions of many important systems of partial differential equations can be reduced to systems of special ordinary differential equations a numerical solver for initial value problems for such systems is developed based on matlab and numerical bifurcation diagrams are obtained according to the behavior of the solutions various bifurcation diagrams of coupled schrodinger equations from nonlinear physics are obtained which suggests the uniqueness of the ground state solution

existence of multiple solutions to a class of nonlinear schrodinger system with external sources terms abstract we study a class of nonlinear schrodinger system with external sources terms as perturbations in order to obtain existence of multiple solutions this system arises from bose einstein condensates etc as these external sources terms are positive functions and small in some sense we use nehari manifold to get the existence of a positive ground state solution and a positive bound state solution

solutions with a prescribed number of zeros for nonlinear schrodinger systems abstract we find an interval of some parameters where there exists a sign changing solution of nonlinear schrodinger systems having a prescribed number of zeros

multi bump bound states for a system of nonlinear schrodinger equations abstract motivated by problems arising in nonlinear optics and bose einstein condensates we consider in r n with n 3 the following system of coupled schrodinger equations u i v i x u i u i j 1 d i j u j 2 u i 0 lim x u i x 0 i 1 d where 0 is a parameter i j j i are positive constants and v i non negative given potentials we assume that the interior of i 1 d v i 1 0 admits m connected components 1 m which are of class c 1 and isolated in each v i 1 0 for each non empty j 1 m we prove that the system admits for any large a multi bump solution u r n r d which is small in r n j j j and on each j j j close in h 1 norm to a least energy solution of the limit problem u i u i j 1 d i j u j 2 i 1 d subjected to homogeneous dirichlet boundary condition an explicit condition on the matrix i j is given to ensure our solutions have at least two positive components

a minimal energy tracking method for non radially symmetric solutions of coupled nonlinear schrodinger equations we aim at developing methods to track minimal energy solutions of time independent m component coupled discrete nonlinear schrodinger dnls equations we first propose a method to find energy minimizers of the 1 component dnls equation and use it as the initial point of the m component dnls equations in a continuation scheme we then show that the change of local optimality occurs only at the bifurcation points the fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve by combining all these techniques with a parameter switching scheme we successfully compute a non radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly

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

existence and asymptotic behavior of vector solutions for coupled nonlinear kirchhoff type systems abstract this paper deals with the following linearly coupled nonlinear kirchhoff type system a 1 b 1 r 3 u 2 d x u 1 u f u v in r 3 a 2 b 2 r 3 v 2 d x v 2 v g v u in r 3 u v h 1 r 3 where a i 0 b i 0 i 0 are constants for i 1 2 0 is a parameter and f g c r r under the general berestycki lions type assumptions on f and g we establish the existence of positive vector solutions and positive vector ground state solutions respectively by using variational methods we also study the asymptotic behavior of these solutions as 0

etude de processus generalisant l aire de levy given ann dimensional brownian motionb t b 1 t b n t we consider the stochastic process n dimensional levy s stochastic area l n t sum limits sigma varepsilon sigma int limits 0 t ldots int limits 0 t 2 b sigma 1 t 1 db sigma 2 t 1 ldots db sigma n t n 1 were e is the signature of the permutation we show that this process can be explicitly expressed as a functionalf of 2 dimensional ordinary levy s stochastic areas f is calculated then evaluation of the characteristic function ofl 3 t follows then an iterated logarithm theorem is proved for l n t

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

profiles and quantization of the blow up mass for critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window for a suitable class of initial data in the energy space h1 we prove that the solution splits in two parts the first part corresponds to the singular part and accumulates a quantized amount of l2 mass at the blow up point the second part corresponds to the regular part and has a strong l2 limit at blow up time

positive solutions for a weakly coupled nonlinear schrodinger system abstract existence of a nontrivial solution is established via variational methods for a system of weakly coupled nonlinear schrodinger equations the main goal is to obtain a positive solution of minimal action if possible with all vector components not identically zero generalizations for nonautonomous systems are considered

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

special set and solutions of coupled nonlinear schrodinger equations a special set of n coupled nonlinear schrodinger equations which consists of 2n interaction types each of which is characterized by a specific array of interaction parameters is presented it is shown that every lame function of order n n is a solution for one or more components for one or more interaction types of this special set simple rules that relate interaction and solution types and interesting features of these relationships are presented

least energy solitary waves for a system of nonlinear schrodinger equations in mathbb r n in this paper we consider systems of coupled schrodinger equations which appear in nonlinear optics the problem has been considered mostly in the one dimensional case here we make a rigorous study of the existence of least energy standing waves solitons in higher dimensions we give conditions on the parameters of the system under which it possesses a solution with least energy among all multi component solutions conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie

stationary solitons of a three wave model generated by type ii second harmonic generation in quadratic media abstract in this paper we prove the uniqueness of stationary standing wave solutions of an optical model generated by type ii second harmonic generation shg with behaviors tending to zero at infinity under certain conditions on parameters in addition we provide the same issues for the dirichlet boundary value problems on the ball centered at the origin a classification of solutions for radial case is also established

two coupled nonlinear schrodinger equations involving a non constant coupling coefficient abstract in this paper we study the existence and multiplicity of vector positive solutions for two coupled nonlinear schrodinger equations involving a non constant coupling coefficient

multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent abstract this paper deals with the following system linearly coupled by nonlinear elliptic equations u 1 u u 2 2 u v x v 2 v v 2 2 v u x u v 0 on here is a smooth bounded domain in r n n 3 1 2 1 are constants 1 is the first eigenvalue of h 0 1 2 2 n n 2 is the sobolev critical exponent and r is a coupling parameter by variational method we prove that this system has a positive ground state solution for some 0 via a perturbation argument we show that this system also admits a positive higher energy solution when is small moreover the asymptotic behaviors of the positive ground state and higher energy solutions as 0 are analyzed

on the classification of standing wave solutions to a coupled schrodinger system in this paper we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear schrodinger equation we establish the existence and uniqueness of solutions to the dirichlet problem on the ball and entire space as the parameters within certain regions in addition a complete structure of different types of solutions for the radial case is also provided

existence and symmetry of positive ground states for a doubly critical schrodinger system n n 2 and 1 1 satisfying 2 this problem is related to coupled nonlinear schrodinger equations with critical exponent for bose einstein condensate for different ranges of n and 0 we obtain positive ground state solutions via some quite different methods which are all radially symmetric it turns out that the least energy level depends heavily on the relations among and 2 besides for sufficiently small 0 positive solutions are also obtained via a variational perturbation approach note that the palais smale condition can not hold for any positive energy level which makes the study via variational methods rather complicated

a liouville s theorem for a fractional elliptic system with indefinite nonlinearities abstract in this paper we consider a fractional elliptic system with indefinite nonlinearities we prove that every positive bounded solution of the system is monotone increasing along x 1 direction by using the direct method of moving planes based on this monotonicity we obtain the nonexistence of positive solutions and a liouville type results for the fractional elliptic system as an application we get the liouville type results for the fractional elliptic system with cubic nonlinearities

standing waves of a weakly coupled schrodinger system with distinct potential functions abstract the standing wave solutions of a weakly coupled nonlinear schrodinger system with distinct trapping potential functions in r n 1 n 3 are considered this type of system arises from models in bose einstein condensates theory and nonlinear optics the existence of a positive ground state solution is shown when the coupling constant is larger than a sharp threshold value which is explicitly defined in terms of potential functions and system parameters it is also shown that such solutions concentrate near the minimum points of potential functions and multiple positive concentration solutions exist when the topological structure of the set of minimum points satisfies certain condition variational approach is used for the existence and concentration of positive solutions

multiple bound states of nonlinear schrodinger systems this paper is concerned with existence of bound states for schrodinger systems which have appeared as several models from mathematical physics we establish multiplicity results of bound states for both small and large interactions this is done by different approaches depending upon the sizes of the interaction parameters in the systems for small interactions we give a new approach to deal with multiple bound states the novelty of our approach lies in establishing a certain type of invariant sets of the associated gradient flows for large interactions we use a minimax procedure to distinguish solutions by analyzing their morse indices

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

new generating functions for gegenbauer polynomials a family of new generating functions for the gegenbauer polynomials is presented this work is based upon the elementary manipulation of series and is motivated by the recent appearance of these polynomials in certain aspects of applied mathematics

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

spike layer solutions to singularly perturbed semilinear systems of coupled schrodinger equations abstract let be a bounded domain in r n n 3 we are concerned with the interaction and the configuration of spikes in a double condensate by analyzing the least energy solutions of the following two couple schrodinger equations in se e 2 u u 1 u 3 u v 2 e 2 v v 2 v 3 u 2 v u 0 v 0 where 1 2 are positive constants we prove that under neumann or dirichlet boundary conditions for any e 0 when min 1 2 or max 1 2 system se has a least energy solution u e v e and when min 1 2 max 1 2 system se has no solution suppose p e q e are the local maximum points of u e v e respectively then under neumann boundary conditions as e small enough both of p e q e locate on the boundary of furthermore when 0 p e q e e 0 as e 0 and for n 2 and n 3 p e q e converge to the same point on the boundary which is the maximum point of mean curvature of the boundary however when 0 p e q e e as e 0 and suppose p e p and q e q then for n 2 and n 3 p q must be the maximum points of the mean curvature on the boundary and p q might be a same point if the mean curvature of the boundary has only one maximum point under dirichlet boundary conditions we can prove that as long as the least energy solution se exists the same asymptotic behavior of the least energy solution u e v e holds as described in lin and wei 2005 10 for 0 or for 0 thus our results are an extension of the results in lin and wei 2005 10

new existence and symmetry results for least energy positive solutions of schrodinger systems with mixed competition and cooperation terms abstract in this paper we focus on existence and symmetry properties of solutions to the cubic schrodinger system u i i u i j 1 d i j u j 2 u i in r n i 1 d where d 2 i i i 0 i j j i r for j i n 2 3 the underlying domain is either bounded or the whole space and u i h 0 1 or u i h rad 1 r n respectively we establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients as well as in the purely cooperative case

sign changing solutions for coupled nonlinear schrodinger equations with critical growth abstract we consider the following nonlinear schrodinger system with critical growth u j j u j i 1 k i j u i 2 2 u j 2 2 2 u j in u j 0 on j 1 k where is a bounded smooth domain in r n 2 2 n n 2 0 j 1 j 1 k 1 is the first eigenvalue of with zero dirichlet boundary condition we consider the repulsive case namely j j 0 j 1 k i j j i 0 i j i j 1 k the existence of infinitely many sign changing solutions as bound states is proved provided n 7 by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions

a parametrix for step two hypoelliptic diffusion equations in this paper i construct a parametrix for the hypoelliptic diffusion equations 0 ot l u 0 where l ea 1 9a2 and where the ga are vector fields which satisfy the property that they together with all of the commutators ga gbi for a b are at each point linearly independent and span the tangent space

orthogonal polynomials in neutron transport theory the asymptotic average properties of zeros of the polynomials gkm x which play a fundamental role in neutron transport and radiative transfer theories are investigated analytically in terms of the angular expansion coefficients wk of the scattering kernel for three wide classes of scattering models in particular it is found that the scattering models of eccleston mccormick shultis et al and henyey greenstein belong in one of the above mentioned classes and their associated polynomials gkm x have the same asymptotic density of zeros

positive solutions to some systems of coupled nonlinear schrodinger equations abstract we study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time independent schrodinger equations u j j u j j u j 3 k 1 k j n j k u j u k 2 u j w 1 2 r n j 1 n where n 1 2 3 j j 0 for j 1 n the coupling parameters j k k j r for j k 1 n j k precisely we prove the existence of nonnegative bound state solutions for suitable conditions on the coupling factors additionally with more restrictive conditions on the coupled parameters we show that the bound states founded are positive

sharp thresholds of blow up and global existence for the coupled nonlinear schrodinger system in this paper we establish two new types of invariant sets for the coupled nonlinear schrodinger system in the euclidean n space rn and derive two sharp thresholds of blow up and global existence for its solutions some analogous results for the nonlinear schrodinger system posed on the hyperbolic space hn and on the standard 2 sphere s2 are also presented our arguments and constructions are improvements of some previous works on this direction at the end we give some heuristic analysis about the strong instability of the solitary waves the relation between the two types of thresholds is a very interesting problem and we leave it as an open problem for further study

positive least energy solutions and phase separation for coupled schrodinger equations with critical exponent in this paper we study the following coupled schrodinger system which can be seen as a critically coupled perturbed brezis nirenberg problem left begin array ll delta u lambda 1 u mu 1 u 3 beta uv 2 quad x in omega delta v lambda 2 v mu 2 v 3 beta vu 2 quad x in omega u geqq 0 v geqq 0 rm in omega quad u v 0 quad rm on partial omega end array right here omega subset mathbb r 4 is a smooth bounded domain lambda 1 omega 0 and beta neq 0 where lambda 1 omega is the first eigenvalue of with the dirichlet boundary condition note that the nonlinearity and the coupling terms are both critical in dimension 4 that is frac 2n n 2 4 when n 4 we show that this critical system has a positive least energy solution for negative positive small and positive large for the case in which lambda 1 lambda 2 we obtain the uniqueness of positive least energy solutions we also study the limit behavior of the least energy solutions in the repulsive case beta to infty and phase separation is expected these seem to be the first results for this schrodinger system in the critical case

levy s stochastic area formula in higher dimensions let wt w t 1 w t 2 w t d d 2 be a d dimensional standand brownian motion and let a t be a bounded measurable function from into the space of dxd skew symmetric matrices and x t into d we are concerned with a class of stochastic processes l t a x a particular example of which is p l vy s stochastic area open image in new window

existence and nonexistence of bound state solutions for schrodinger systems with linear and nonlinear couplings abstract we study the schrodinger systems with linear and nonlinear coupling terms doubly coupled nonlinear schrodinger system for short which arise naturally in nonlinear optics and in the hartree fock theory for bose einstein condensates among other physical problems first for small linear coupling constant we get existence of a nontrivial bound state solution to the system via perturbation method furthermore we prove each component of the bound state solution is nonnegative by energy estimate second we establish a version of pohozaev nehari identity and prove a nonexistence result for the more general system when the spatial dimension n 4

positive ground state solutions and multiple nontrivial solutions for coupled critical elliptic systems in this study we consider the following coupled elliptic system with a sobolev critical exponent p u1 1u1 1u1p1 1 1u12 1 u12 2 1u22 2 x u2 2u2 2u2p2 1 2u22 1 u12 2u22 2 1 x u1 u2 0in u1 u2 0on where rn is a bounded smooth domain n 5 2 0 for j 1 2 and 1 is the first eigenvalue of with the dirichlet boundary condition we demonstrate the existence of a positive ground state solution for problem p when the coupling parameter 1 2 under some other conditions we show the nonexistence of positive solutions for p when n 3 we also construct multiple nontrivial solutions and sign changing solutions for the following system u1 1u1 1u13 u22u1 x u2 2u2 2u23 u12u2 x u1 u2 0on where rn is a bounded smooth domain and n 4

two component bose einstein condensates abstract recently coupled systems of nonlinear schrodinger equations have been used extensively to describe bose einstein condensates in this paper we study the structure of vortices of the coupled nonlinear equations for two component bose einstein condensates bec in a three dimensional space we show that vortices is 1 rectifiable set and give its mean curvature in particular we show that large interspecies scattering length causes vortices for two component bec

blowup and solitary wave solutions with ring profiles of two component nonlinear schrodinger systems abstract blowup ring profiles have been investigated by finding non vortex blowup solutions of nonlinear schrodinger equations nlses cf fibich et al 2005 7 and fibich et al 2007 8 however those solutions have infinite l 2 norm so one may not maintain the ring profile all the way up to the singularity to find h 1 non vortex blowup solutions with ring profiles we study the blowup solutions of two component systems of nlses with nonlinear coefficients and j j 1 2 when 0 and 1 2 0 the two component system can be transformed into a multi scale system with fast and slow variables which may produce h 1 blowup solutions with non vortex ring profiles we use the localized energy method with symmetry reduction to construct these solutions rigorously on the other hand these solutions may describe steady non vortex bright ring solitons various types of ring profiles including m ring and ring ring profiles are presented by numerical solutions

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

solitary and self similar solutions of two component system of nonlinear schrodinger equations conventionally to learn wave collapse and optical turbulence one must study finite time blow up solutions of one component self focusing nonlinear schrodinger equations nlse here we consider simultaneous blow up solutions of two component systems of self focusing nlse by studying the associated self similar solutions we prove two components of solutions blow up at the same time these self similar solutions may come from solitary wave solutions with multi bumps forming abundant geometric patterns which cannot be found in one component self focusing nlse our results may provide the first step to investigate optical turbulence in two component systems of nlse

semitrivial vs fully nontrivial ground states in cooperative cubic schrodinger systems with d 3 equations abstract in this work we consider the weakly coupled schrodinger cubic system u i i u i i u i 3 u i j i b i j u j 2 u i h 1 r n r i 1 d where 1 n 3 i i 0 and b i j b j i 0 for i j this system admits semitrivial solutions that is solutions u u 1 u d with null components we provide optimal qualitative conditions on the parameters i i and b i j under which the ground state solutions have all components nontrivial or conversely are semitrivial this question had been clarified only in the d 2 equations case for d 3 equations prior to the present paper only very restrictive results were known namely when the above system was a small perturbation of the super symmetrical case i and b i j b we treat the general case uncovering in particular a much more complex and richer structure with respect to the d 2 case

a new set of orthogonal polynomials for use in neutron transport theory the proper set of orthogonal polynomials to use in approximation work depends on the weight function of the problem the author introduces a new set of polynomials orthogonal with respect to a simple closed form approximate representation of the weight function of case s half range neutron transport theory 1967 the accuracy of this approximate weight function is shown to be very good and the construction procedure of the new set of orthogonal polynomials ci mu c or mu or 1 is demonstrated

an approximation of small time probability density functions in a general jump diffusion model we propose a method for approximating probability density functions related to multidimensional jump diffusion processes for small time horizons a closed form approximation of the characteristic function is derived based on the ito taylor expansion the probability density function is then approximated numerically by inverting the characteristic function using fast fourier transform as application we consider a general stochastic volatility model which involves time state dependent drift and diffusion functions as well as jump components we test our approach under the heston model and the bates model and show that our method provides accurate approximations

orthogonal polynomials and differential equations in neutron transport and radiative transfer theories abstract there is a set of orthogonal polynomials gn x which plays a relevant role in the treatment of the case of anisotropic scattering in neutron transport and radiative transfer theories they appear also in the spherical harmonics treatment of the isotropic scattering these polynomials are orthogonal with respect to a weight function which is continuous in the interval 1 1 and has a finite number of symmetric dirac masses although some other structural properties of these polynomials e g the three term recurrence relation as well as some properties of their zeros have been published much more need to be known in particular neither the second order differential equation nor the density of zeros i e the number of zeros per unit of interval of the polynomial gn x have been found here we obtain the second order differential equation in the case that these polynomials are hypergeometric so leaving open the general case furthermore the exact expressions of the moments around the origin of the density of zeros of gn x are given in the general case the asymptotic density of zeros is also pointed out finally these polynomials are shown to belong to the nevai s class

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

segregated and synchronized vector solutions for nonlinear schrodinger systems we consider the following nonlinear schrodinger system in mathbb r 3 left begin array ll delta u p x u mu u 2 u beta v 2u quad x in mathbb r 3 delta v q x v nu v 2 v beta u 2v quad x in mathbb r 3 end array right where p r and q r are positive radial potentials mu 0 nu 0 and beta in mathbb r is a coupling constant this type of system arises in particular in models in bose einstein condensates theory

symmetric ground state solutions of m coupled nonlinear schrodinger equations abstract we prove the existence of radial and radially decreasing ground states of an m coupled nonlinear schrodinger equation with a general nonlinearity

on strongly indefinite systems involving the fractional laplacian abstract this paper is devoted to study strongly indefinite systems involving the fractional laplacian on bounded domains explicitly we obtain existence and non existence results a p r i o r i estimates of gidas spruck type and a symmetry result in addition we give a different proof for the a p r i o r i estimate for nonlinear elliptic problems with the fractional laplacian obtained in cabre and tan 2010 and tan 2013

multi peak solutions to coupled schrodinger systems with neumann boundary conditions abstract in this paper we are concerned with the following two coupled schrodinger systems in a bounded domain r n n 2 3 with neumann boundary conditions e 2 u u 1 u 3 u v 2 e 2 v v 2 v 3 u 2 v u 0 v 0 u n 0 v n 0 on suppose the mean curvature h p of the boundary has several local minimums or local maximums we obtain the existence of solutions with multi peaks to the system with all peaks being on the boundary and all peaks locate either near the local maxima or near the local minima of the mean curvature at the boundary of the domain

on global solutions to the initial boundary value problem for the damped nonlinear schrodinger equations where a 0 and q is a bounded domain in rn with smooth boundary x when a 0 undamped case the global existence of smooth solutions to the initial boundary value problem l 3 has been proved only when n 2 and f lu12 w lulp with p 2 brezis and gallouet cl with 2 i p i3 m tsutsumi 9 higher dimensional cases are very difficult pribl s and remain unproved as far as the author knows if we add a linear damping term iau a 0 then the mass ilull iztr1 and the energy ivu x t l dx j f lu x t l dx decay exponentially as t cc as is seen below which may assert the global existence of small smooth solutions even if the nonlinear term f lul u does not have a proper sign it is our purpose to establish the global existence and decay estimates of smooth solutions of the problem l 3 with small data there are a number of works on the nonlinear evolution equations see 3 5 71 and their references in the following spirit energy and decay estimates give global existence of solutions if the damping term is not added there is no hope that the decay estimates of solutions in some f q p 2 2 can be obtained since

blow up behavior of ground states for a nonlinear schrodinger system with attractive and repulsive interactions abstract we consider a nonlinear schrodinger system arising in a two component bose einstein condensate bec with attractive intraspecies interactions and repulsive interspecies interactions in r 2 we get ground states of this system by solving a constrained minimization problem for some kinds of trapping potentials we prove that the minimization problem has a minimizer if and only if the attractive interaction strength a i i 1 2 of each component of the bec system is strictly less than a threshold a furthermore as a 1 a 2 a a the asymptotical behavior for the minimizers of the minimization problem is discussed our results show that each component of the bec system concentrates at a global minimum of the associated trapping potential

multiplicity of localized solutions of nonlinear schrodinger systems for infinite attractive case abstract in this paper we consider the following nonlinear schrodinger system 2 u v 1 x u u v 2 in r n u h 1 r n 2 v v 2 x v u 2 v in r n v h 1 r n where n 3 0 1 v i c 1 r n r 0 inf x r n v i x sup x r n v i x i 1 2 and there exists a bounded open set r n with smooth boundary such that inf x min v 1 x v 2 x 0 we prove the existence of a given number of semi positive solutions localized in here we say u v is semi positive if either u 0 or v 0

note on ground states of a nonlinear schrodinger system abstract we give a sufficient condition for the existence of positive radial ground states of the time independent schrodinger system u 1 u 1 1 u 1 3 12 u 2 2 u 1 13 u 3 2 u 1 in r n u 2 u 2 12 u 1 2 u 2 2 u 2 3 23 u 3 2 u 2 in r n u 3 u 3 13 u 1 2 u 3 23 u 2 2 u 3 3 u 3 3 in r n u 1 x 0 u 2 x 0 u 3 x 0 as x where n 1 2 3 0 j 0 and i j 0 i j for i j 1 2 3 and in some special cases our conditions are also necessary

infinitely many solutions for the perturbed bose einstein condensates system abstract we consider the perturbed nonlinear elliptic system as following u j j u j i 1 2 i j u i 2 u j j x u j h 0 1 j 1 2 where is a bounded smooth domain in r n n 3 and j 1 1 is the first eigenvalue of with the dirichlet boundary condition the perturbation terms j l 2 for j 1 2 assume that the matrix i j i j 1 2 is either positive definite when n 1 2 or anti symmetric when n 3 we show the existence of infinitely many solutions to this system the results are also true for the m coupled system i e i j 1 m m 2