A FINITE DIFFERENCE SCHEME FOR THE GENERALIZED NONLINEAR SCHRDINGER EQUATION WITH VARIABLE COEFFICIENTS

1. Generalized Nonlinear Schrsdinger EquationThe Schr6dinger equation has been extensively used in physics research, particularlyin the modeling of nonlinear dispersion waves [8]. Numerical methods for solving theSchr6dinger equation have been discussed in the literature. In this article, we considera generalized nonlinear Schr6dinger equation with variable coefficientsi: ~ g(A(x)Z) iF(t)u B(x) lulp~' u = 0, iZ ~ ~l, P > 1, (1)where u(x, 0) ~ of (x). The coefficients A(x), F(t) and, …     

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF THREE-DIMENSIONAL NONLINEAR SCHRDINGER EQUATION WITH WEAKLY DAMPED

The three-dimensional nonlinear Schrdinger equation with weakly damped that pos-sesses a global attractor are considered.The dynamical properties of the discrete dynamicalsystem which generate by a class of finite difference scheme are analysed.The existence ofglobal attractor is proved for the discrete dynamical system.     

CONCENTRATION PHENOMENA TO THE NONLINEAR SCHRDINGER EQUATION WITH HARMONIC POTENTIAL IN GENERAL DATA

We analyze the blowup problems to the nonlinear Schrodinger equation with har-monic potential. This equation always models the Bose-Einstein condensation in lower dimensions. It is known that the mass of the blowup solutions from radially symmet-ric initial data can concentrate on the point of blowup. In this paper based on the refined compactness lemma, we extend the result to general data.     

HARNACK’S INEQUALITY FOR GENERALIZED SUBELLIPTIC SCHRDINGER OPERATORS

We prove a uniform Harnack inequality for nonnegative solutions of ΔGu Gμu = 0, where ΔG is a sublaplacian, μ is a non-negative Radon measure and satisfying scale-invariant Kato condition.     

ON THE CONCENTRATION PROPERTIES FOR THE NONLINEAR SCHRDINGER EQUATION WITH A STARK POTENTIAL

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.     

EXISTENCE TIME FOR THE SEMILINEAR SCHR?DINGER EQUATION

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ ,ｗｅｗｉｌｌｃｏｎｓｉｄｅｒｔｈｅｓｅｍｉｌｉｎｅａｒＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎｉｎｏｎｅｓｐａｃｅｄｉｍｅｎｓｉｏｎｏｆｔｈｅｔｙｐｅｕｔ-ｉｕｘｘ =Ｆ(ｕ) .　 (ｘ ,ｔ)∈Ｒ×Ｒ+,( 1 )ｕ(ｘ ,0 ) =ｕ0 ,　ｘ ∈Ｒ ,( 2 )ｗｈｅｒｅｕ =ｕ(ｘ ,ｔ)ｉｓｃｏｍｐｌｅｘ ｖａｌｕｅｄｆｕｎｃｔｉｏｎ ,ａｎｄＦｉｓａｓｍｏｏｔｈｆｕｎｃｔｉｏｎｏｆｕｓｕｃｈｔｈａｔ｜Ｆ(ｕ) ｜ =Ｏ( ｜ｕ｜α+1)ｆｏｒ |ｕ|ｓｕｆｆｉｃｉｅｎｔｌｙｓｍａｌｌａｎｄ…     

EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHRDINGER SYSTEM

We study the existence and stability of the standing waves of two coupled Schrdinger equations with potentials |x|bi(bi ∈ R, i = 1, 2). Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the Schrdinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1= b2= 2.     

GLOBAL WEAK SOLUTION TO THE NONLINEAR SCHRDINGER EQUATIONS WITH DERIVATIVE

In this paper, we study the nonlinear Schr¨odinger equations with derivative. By using the Gal¨erkin method and a priori estimates, we obtain the global existence of the weak solution.     

Stability of solutions for nonlinear Schrdinger equations in critical spaces

We consider the Cauchy problem for nonlinear Schrdinger equation iut + Δu = ±|u|pu,4/d< p <4 /d-2 in high dimensions d 6. We prove the stability of solutions in the critical space H˙xsp , where sp = d/2-p/2 .     

Existence of Generalized Heteroclinic Solutions of the Coupled Schrdinger System under a Small Perturbation

The following coupled Schrdinger system with a small perturbation uxx + u- u3+ βuv2+ f(, u, ux, v, vx) = 0 in R,vxx- v + v3+ βu2v + g(, u, ux, v, vx) = 0 in R is considered, where β and are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution(called the generalized heteroclinic solution thereafter).     

THE INVARIANT MANIFOLDS FOR A PERTURBED QUINTIC-CUBIC SCHRDINGER EQUATION

In this paper, the authors apply the analytical method to establish the persistence of invariant manifolds for certain perturbation of the quintic-cubic Schrodinger equation under even periodic boundary conditions.     

Compressible Limit of the Nonlinear Schrdinger Equation with Different-Degree Small Parameter Nonlinearities

The authors study the compressible limit of the nonlinear Schrdinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singularities in the limit system.On the one hand,the existence and uniqueness of the classical solution are proved for the dispersive perturbation of the quasi-linear symmetric system corresponding to the initial value problem of the above nonlinear Schrdinger equation.On the other hand,in the limi...     

Wellposedness of some quasi-linear Schrdinger equations

This article is devoted to the study of a quasilinear Schrdinger equation coupled with an elliptic equation on the metric g. We first prove that, in this context, the propagation of regularity holds which ensures local wellposedness for initial data small enough in˙H1/2 and belonging to the Besov space˙B3/22,1. In a second step, we establish Strichartz estimates for time dependent rough metrics to obtain a lower bound of the time existence which only involves the˙B1+ε2,∞norm on the initial data.     

A NONCONFORMING QUADRILATERAL FINITE ELEMENT APPROXIMATION TO NONLINEAR SCHRDINGER EQUATION

In this article, a nonconforming quadrilateral element(named modified quasiWilson element) is applied to solve the nonlinear schr¨odinger equation(NLSE). On the basis of a special character of this element, that is, its consistency error is of order O(h~3) for broken H1-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover,the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.     

Existence and Concentration of Bound States of a Class of Nonlinear Schrdinger Equations in R~2 with Potential Tending to Zero at Infinity

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schrdinger equation -ε2 Δuε + V(x)uε = K(x)|uε|p-2 uεeα0 |uε|γ,uε0, uε∈H 1(R2),where 2 p ∞, α0 0, 0 γ 2. When the potential function V (x) decays at infinity like (1 + |x|)-α with 0 α≤ 2 and K(x) 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H1 (R2 )-solution uε exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schrdinger equation-Δu + V (ξ)u = K(ξ)|u| p-2 ue α0 |u|γ has local minimum points. Furthermore, the concentration property of uε is also established as ε tends to zero.     

The behavior of attractors for damped Schrdinger-Boussinesq equation

IntroductionIn the laser and plasma physics, the Schrsdinger-Boussinesq system has been raised. InRely. [1,2], the authors studied the echtence of the global solution of initial boundary conditionfor the system. Here we consider the behavior of attractors for this type of equationswhere a, 7 and A are positive constants, the eXternal forces g and h are given, and j(n) is asufficiently smooth real function with j(0) = 0. We first prove that the dissipative SchrsdingerBoussinesq system posses…     

Analytic Solutions of Klein-gordon Equation and Generalized Schrdinger Equation

Inmanyapproximationcases,wecansumlotsofphysicalphenomenonsuptoKlein_gor donequationutt- (uxx+uyy) +α2 u +g(uu )u =0 ,(1 )whereg(z)isafunctionofzandu iscojugatecomplexnumberofu .ManyscholarshavebeeninterestedinanalyticsolutionofEq .(1 ) .Sinceitisestablihed .Papers [1 ,2 ]and [3]viewedrespecrtivelyaccuratesolutionandanalyticsolutionofEq .(1 )wheng(z) =βz .Inpa per [4] ,weobtainedaclassofanalyticsolutionofEq .(1 )wheng(z) =βz1 /k,k∈R+ andaclassofanalyticsolutionofgeneralizedSchrodingerequ…     

BIFURCATION RESULTS FOR A SEMILINEAR SCHRDINGER EQUATION WITH INDEFINITE LINEAR PART

This paper studies the following semilinear Schrodinger problem It is proven that there exists a bifurcation branch of solutions for the above problem, when g(x) can possibly vanish except for a bounded domain Ω RN.     

非线性扰动耦合Schrdinger系统激波的近似解法

研究了一类非线性扰动耦合Schrdinger系统.利用精确解与近似解相关联的特殊技巧,首先讨论了对应典型的耦合系统,利用投射法得到了精确的激波行波解.再利用近似方法得到了扰动耦合Schrdinger系统的行波渐近解.     

Non-Nehari manifold method for asymptotically periodic Schrdinger equations

We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1-periodic in each of x1, x2,..., x N and sup[σ(-△ + V0) ∩(-∞, 0)] 0 inf[σ(-△ +V0)∩(0, ∞)], V1∈ C(RN) and lim|x|→∞V1(x) = 0. Inspired by previous work of Li et al.(2006), Pankov(2005)and Szulkin and Weth(2009), we develop a more direct approach to generalize the main result of Szulkin and Weth(2009) by removing the "strictly increasing" condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.