the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

quasilinear elliptic equations involving the n laplacian with critical exponential growth in rn abstract this paper shows the existence of nontrivial solutions for the quasilinear equations of the form n u v x u n 2 u n u 2 u h u in r n where n is the n laplacian operator v is a continuous function bounded from below away from zero and h u is a continuous function having critical exponential growth

product integration in the presence of a singularity this paper considers the approximate evaluation of int a b k x f x dx where k is a fixed lebesgue integrable function by quadrature rules of the form sigma i 0 m n w in f x in normally rules of this kind are used only for smooth functions f and any singularities are incorporated into k here however we allow f to have a singularity either at an endpoint or in the interior general convergence properties are established for a wide class of product integration rules more detailed results are established for the class of rules which are exact if f belongs to a specified family of piecewise polynomials

on the local well posedness of quasilinear schrodinger equations in arbitrary space dimension abstract a class of quasilinear schrodinger equations is studied which contain strongly singular nonlinearities including derivatives of second order such equations have been derived as models of several physical phenomena the results of this paper apply to the superfluid film equation in fluid mechanics the local well posedness for smooth solutions of the cauchy problem is proved in arbitrary space dimension without any smallness assumption on the initial value this improves results in the literature which cover the one dimensional case only almost global well posedness results are obtained as well the proof combines nash moser techniques on implicit function theorems in frechet spaces with hyperbolic semigroup theory including evolution systems using the intersection of all sobolev spaces as basic function space the nondissipativity is overcome by a transformation procedure yielding the evolution operator and a priori estimates of higher order sobolev norms for the linearized problem here a certain trace has to vanish in view of apparent integrability conditions the transformations and the regularity theory are substantially different from the easier one dimensional case in particular continuity estimates are proved for the operator norms of the linear evolution operator in negative sobolev spaces

soliton solutions for quasilinear schrodinger equations the critical exponential case abstract quasilinear elliptic equations in r 2 of second order with critical exponential growth are considered by using a change of variable the quasilinear equations are reduced to semilinear equations whose respective associated functionals are well defined in h 1 r 2 and satisfy the geometric hypotheses of the mountain pass theorem using this fact we obtain a cerami sequence converging weakly to a solution v in the proof that v is nontrivial the main tool is the concentration compactness principle p l lions the concentration compactness principle in the calculus of variations the locally compact case part i and ii ann inst h poincare anal non lineaire 1 1984 109 145 223 283 combined with test functions connected with optimal trudinger moser inequality

associated generalized jacobi functions and polynomials abstract pointwise estimates are obtained for associated functions and polynomials corresponding to generalized jacobi weights these estimates hold uniformly in the interval of orthogonality

on the painleve classification of partial differential equations semilinear second order partial differential equations in two independent variables with rational right hand side are investigated it is found that the painleve admissible pdes can be transformed to a form with maximally three poles a complete list of pdes that pass the test for the painleve property is given

on the convergence of product formulas for the numerical evaluation of derivatives of cauchy principal value integrals product rules of interpolatory type on the zeros of generalized smooth jacobi polynomials for the numerical approximation of cauchy principal value integrals and their derivatives are considered convergence results for these rules are given and some estimates of the remainder are established in relation to the regularity of the density function

numerical integration in the presence of an interior singularity abstract the convergence of sequences of integration rules applied to an improperly integrable function with an interior singularity is investigated the rules studied include the gauss jacobi rules and the interpolatory integration rules based on the zeros of 1 x 2 m p n 2m x m 0 1 for certain values of and the results are then applied to the study of the convergence of hunter s method for cauchy principal value integrals

positive solutions for schrodinger poisson equations with a critical exponent abstract in this paper we establish the existence of a positive solution of the schrodinger poisson equations with a critical sobolev exponent the methods used here are based on the concentration compactness principle of p l lions and methods of brezis and nirenberg

existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in rn abstract mountain pass in a suitable orlicz space is employed to prove the existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in r n these equations contain strongly singular nonlinearities which include derivatives of the second order such equations have been studied as models of several physical phenomena the nonlinearity here corresponds to the superfluid film equation in plasma physics

quasilinear elliptic equations with critical growth via perturbation method abstract we consider a class of quasilinear schrodinger equations which include the modified nonlinear schrodinger equations a new perturbation approach is used to treat the critical exponent case giving new existence results

gauss type quadratures for weakly singular integrals and their application to fredholm integral equations of the second kind in this paper we establish gauss type quadrature formulas for weakly singular integrals an application of the quadrature scheme is given to obtain numerical solutions of the weakly singular fredholm integral equation of the second kind we call this method a discrete product integration method since the weights involved in the standard product integration method are computed numerically

bifurcations for quasilinear schrodinger equations i abstract this paper is concerned with bifurcation problems for quasilinear schrodinger type equations such as the modified nonlinear schrodinger equations which have appeared as several models from mathematical physics we give a new approach for dealing with bifurcation problems of nonlinear potential operators the novelty of our approach lies in establishing multiple bifurcations without making a reduction to finite dimensional problems the finite dimensional reduction procedures that have been used in the past work for semilinear problems but are not suitable for quasilinear problems we give a unified approach requiring much less regularity of the problems as applications we provide multiple bifurcation results for the modified nonlinear schrodinger equations

on a nonhomogeneous and singular quasilinear equation involving critical growth in r2 abstract this paper establishes sufficient conditions for the existence and multiplicity of solutions for nonhomogeneous and singular quasilinear equations of the form u v x u u 2 u g x u x a h x in r 2 where a 0 2 v x is a continuous positive potential bounded away from zero and which can be large at infinity the nonlinearity g x s is allowed to enjoy the critical exponential growth with respect to the trudinger moser inequality and the nonhomogeneous term h belongs to l q r 2 for some q 1 2 by combining variational arguments in a nonstandard orlicz space context with a singular version of the trudinger moser inequality we obtain the existence of two distinct solutions when h q is sufficiently small schrodinger equations of this type have been studied as models of several physical phenomena

rates of convergence of gaussian quadrature for singular integrands the authors obtain the rates of convergence or divergence of gaussian quadrature on functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a bounded smooth weight function on 1 1 the error in n point gaussian quadrature of f x ix y1 1 is o n 2 28 if y 1 and o n if y e 1 1 provided we avoid the singularity if we ignore the singularityy the error is o n1 23 log n 8 loglog n 8t 1 for almost all choices of y these assertions are sharp with respect to order

existence and multiplicity of non trivial solutions for a class of modified schrodinger poisson systems abstract in the present paper we study the following modified schrodinger poisson system p u v x u u 1 2 u u 2 u p 1 u h x in r 3 u 2 in r 3 where 4 p 1 12 h l 2 r 3 and h 0 this model describes the interaction of a charge particle with an electro magnetic field under some hypotheses on v x we prove the existence of a non trivial ground state solution and two non trivial ground state solutions for the system p

stable evaluations of fractional derivative of the muntz legendre polynomials and application to fractional differential equations abstract the aim of this paper is to present efficient and stable methods to compute caputo fractional derivative cfd of the muntz legendre polynomials based on three term recurrence relations and gauss jacobi quadrature rules this approach with collocation method at chebyshev gauss lobatto points has been applied for solving linear and nonlinear fractional multi order differential equations fdes described in caputo sense the main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations in this work for the first time we present the new rates of convergence for projection error which are more accurate than the rate presented by shen and wang in shen and wang 2016 moreover we present convergence rate for spectral collocation method for linear fdes with initial value on a finite interval and endpoint singularities also we propose an error analysis for jacobi gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper finally the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples

high energy solutions for the superlinear schrodinger maxwell equations abstract in this paper we study the existence of infinitely many large energy solutions for the superlinear schrodinger maxwell equations u v x u u f x u in r 3 u 2 in r 3 via the fountain theorem in critical point theory

time dependent dissipation in nonlinear schrodinger systems a coupled nonlinear schrodinger poisson equation is considered which contains a time dependent dissipation function as a specific model of dissipation effects in nonlinear quantum transport theory and other areas the wigner poisson equation associated with this system is derived using conservation and quasiconservation laws and certain growth assumptions for the nonlinearities and the dissipation function global existence of solutions to the cauchy problem of the time dependent schrodinger poisson system is shown both for small attractive case or arbitrary data repulsive case

existence of non trivial solution for a class of modified schrodinger poisson equations via perturbation method abstract in this paper by using the methods of perturbation and the mountain pass theorem we prove the existence of non trivial solution to a class of modified schrodinger poisson equations

numerical approximation to the fractional derivative operator in this paper we consider the numerical approximation of a alpha a by contour integral we are mainly interested to the case of a a representing the discretization of the first derivative by means of a backward differentiation formula and 0 alpha 1 0 1 the computation of the contour integral yields a rational approximation to a alpha a which can be used to define k k step formulas for the numerical integration of fractional differential equations

the painleve classification of partial differential equations iii rational equations with several poles semilinear second order partial differential equations in two independent variables with rational right hand sides are investigated it is found that the painleve admissible partial differential equations can be transformed to the form with maximally three poles and equations with two and three poles are classified results of a series of papers on the painleve classification are summarized

on a class of periodic quasilinear schrodinger equations involving critical growth in r2 abstract we consider the equation u v x u k 2 u 2 u g x u u 0 x r 2 where v r 2 r and g r 2 r r are two continuous 1 periodic functions and k is a positive constant also we assume g behaves like exp u 4 as u we prove the existence of at least one weak solution u h 1 r 2 with u 2 h 1 r 2 the mountain pass in a suitable orlicz space together with the trudinger moser inequality are employed to establish this result such equations arise when one seeks for standing wave solutions for the corresponding quasilinear schrodinger equations schrodinger equations of this type have been studied as models of several physical phenomena the nonlinearity here corresponds to the superfluid film equation in plasma physics

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

on the convergence rate of clenshaw curtis quadrature for integrals with algebraic endpoint singularities abstract in this paper we are concerned with clenshaw curtis quadrature for integrals with algebraic endpoint singularities an asymptotic error expansion and convergence rate are derived by combining a delicate analysis of the chebyshev coefficients of functions with algebraic endpoint singularities and the aliasing formula of chebyshev polynomials numerical examples are provided to confirm our analysis

convergence of product integration rules for functions with interior and end point singularities over bounded and unbounded intervals the convergence of product integration rules based on gaussian quadrature points is investigated for functions with interior and endpoint singularities over bounded and unbounded intervals the investigation is based on a new convergence result for lagrangian interpolation and gaussian quadrature of singular integrands

rates of convergence of gauss lobattom and radau integration rules for singular integrands rates of convergence or divergence are obtained in the application of gauss lobatto and radau integration rules to functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a generalized jacobi weight function on 1 1 the error in applying an n point rule to f x ix vi 6 is o n 2 28 if y 1 and o n 1 6 if y e 1 1 providedwe avoid the singularity if we ignore the singularity y the error is o n 1 28 log n 6 log log n 1 e for almost all choices of y these assertions are sharp with respect to order 1 introduction this paper is a sequel to that of lubinsky and myself 2 on the rates of convergence of gauss integration rules for singular integrands in this paper we extend the results in 2 in several directions on the one hand we extend the results for gauss integration rules to results for lobatto and radau rules this extension follows easily from the generalized markov stieltjes inequality for lobatto and radau rules given in lemma 3 2 and from a representation of the coefficients or weights or cotes numbers in the lobatto and radau rules in terms of the coefficients in the gauss rule with respect to a related weight function a second extension is to the special case of gauss jacobi integration rules as well as lobatto jacobi and radau jacobi rules for certain values of the parameters defining the jacobi weight function

existence and multiplicity of non trivial solutions for a class of modified schrodinger equation with non coercive potential in the present paper we study the following modified nonlinear schrodinger equation u v x u 12u u 2 u p 1u h x x r n p where n 3 4 0 under some hypotheses on v we obtain the existence and multiplicity of non trivial solutions

existence of solutions for quasilinear elliptic equations with hardy potential in this paper we consider the following quasilinear elliptic equation with hardy potential and dirichlet boundary condition i j 1ndj aij x u diu 12 i j 1ndsai j x u diudju x 2u f x u in where n n 3 is a smooth bounded domain di xi dsaij x s saij x s and 0 n 22 2 and x 2 is called the hardy potential by using the perturbation method we prove the existence of infinitely many solutions for the above problem

convergence properties of a class of product formulas for weakly singular integral equations we examine the convergence of product quadrature formulas of interpolatory type based on the zeros of certain generalized jacobi polynomials for the discretization of integrals of the type f k x y f x dx l 1 and f x has algebraic singularities of the form 1 x a a 1 we prove that the uniform rate of convergence of the rules is o m log2 m in the case of the first kernel and o m 2 a2v logim if v 0 for the second where m is the number of points in the quadrature rule

solutions for a quasilinear schrodinger equation a dual approach abstract we consider quasilinear stationary schrodinger equations of the form 1 u u 2 u g x u x r n introducing a change of unknown we transform the search of solutions u x of 1 into the search of solutions v x of the semilinear equation 2 v 1 1 2f 2 v g x f v x r n where f is suitably chosen if v is a classical solution of 2 then u f v is a classical solution of 1 variational methods are then used to obtain various existence results

the blow up problem for a quasilinear schrodinger equation consider the blow up results for w2 2 rn solutions of a quasilinear schrodinger equation iut u u p 2u u 2 u 0 u t 0 u0 x x rn when x u0 l2 rn we show that the w2 2 rn solutions must blow up for any 0 r and some restriction on p we also show that the radial symmetric solutions in w2 2 rn must blow up at finite time without assuming x u0 l2 rn

on nonlinear perturbations of a periodic elliptic problem in r2 involving critical growth we consider the equation u v x u f x u for x r2 where v r2 r is a positive potential bounded away from zero and the nonlinearity f r2 r r behaves like exp u 2 as u we also assume that the potential v x and the nonlinearity f x u are asymptotically periodic at infinity we prove the existence of at least one weak positive solution u h1 r2 by combining the mountain pass theorem with trudinger moser inequality and a version of a result due to lions for critical growth in r2

quasilinear schrodinger equations with singular and vanishing potentials involving exponential critical growth in r2 abstract we study the existence and nonexistence of solution for the following class of quasilinear schrodinger equations u v x u u 2 u q x h u x r 2 u x 0 as x where v and q are potentials that can be singular at the origin unbounded or vanishing at infinity and the nonlinearity h s is allowed to satisfy the exponential critical growth with respect to the trudinger moser inequality by combining variational methods in a suitable weighted orlicz space with a version of the trudinger moser inequality for this space we obtain the existence of a nonnegative ground state solution for this we have used some regularity results and we need to explore a symmetric criticality type argument moreover under some conditions on v q and h s we prove that this problem does not have positive radial solution schrodinger equations of this type have been studied as models of several physical phenomena

mean convergence of lagrange interpolation for freud s weights with application to product integration rules the connection between convergence of product integration rules and mean convergence of lagrange interpolation in l 1 p 00 has been thoroughly analysed by sloan and smith 37 motivated by this connection we investigate mean convergence of lagrange interpolation at the zeros of orthogonal polynomials associated with freud weights on r our results apply to the weights exp x 2 m 2 4 6 and for the hermite weight m 2 extend results of nevai 28 and bonan 2 in at least one direction the results are sharp in l 1 p g 2 as a consequence we can improve results of smith sloan and opie 38 on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the hermite weight in the process we prove a new markov stieltjes inequality for gauss quadrature sums and solve a problem of nevai on how to estimate certain quadrature sums

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals in a previous paper the authors proposed a modified gaussian rule m wf t to compute the integral wf t in the cauchy principal value sense associated with the weightw and they proved the convergence in closed sets contained in the integration interval the main purpose of the present work is to prove uniform convergence of the sequence m wf t on the whole integration interval and to give estimates for the remainder term the same results are shown for particular subsequences of the gaussian rules m wf t for the evaluation of cauchy principal value integrals a result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made