Existence of Generalized Heteroclinic Solutions of the Coupled Schr¨odinger System under a Small Perturbation

The following coupled Schrodinger system with a small perturbation
is considered, where β and ε 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）.     

一类非线性薛定谔方程的多解的存在性

本文讨论了如下一类非线性薛定谔方程：-△u＋V（x）u=f（u）,x∈R^N,在H^1（R^N）中无穷多解的存在性,其中N≥3,V（x）是RN上的实值连续函数并且满足对（A）x∈R^N,V（z）≥V0＞0.     

应用全新G′/（G+G′）展开方法求解广义非线性Schr?dinger方程和耦合非线性Schr?dinger方程组

研究了一种全新的G′/（G+G′）展开方法,并应用这种方法讨论了广义非线性Schr?dinger方程和一类耦合非线性Schr?dinger方程组新形式的精确解,包括双曲余切函数解、余切函数解和有理函数解.全新G′/（G+G′）展开方法不但直接而有效地求出方程的新精确解,而且扩大了解的范围,这种新方法对于研究偏微分方程具有广泛的应用意义.     

R3中带势项的非线性Schr(o)dinger方程的L2集中现象

该文主要讨论带调和势项的非线性Schr(o)dinger方程的L2集中现象;同时也对破裂点的分布进行了估计,并且给出了整体适定性的条件.     

指数型增长的四阶Schr(o)dinger方程的能量散射理论

研究在给定初值充分小的条件下,通过Banach不动点理论来证明指数型增长的四维四阶非线性Schrdinger方程的整体适定性,同时得到整体时空模有界,从而得到散射.     

Periodic Solutions of Schr(o)dinger Flow from S3 to S2

This paper deals with the periodic solutions of Schr(o)dinger flow from S3 to S2. It is shown that the periodic solution is related to the variation of some functional and there exist S1-invariant critical points to this functional. The proof makes use of the Principle of Symmetric Criticality of Palais.     

非线性Schr(o)dinger方程的整体吸引子

本文讨论了一类带调和势|x|2的非线性Schr(o)dinger方程解的长时间行为,证明了整体吸引子的存在性.     

非线性薛定谔方程组的散射

本文主要考虑如下非线性薛定谔方程组的柯西问题:{-iu1t=△u1-μ|u1 |p1u1--α |u1 | q1-2 |u2 |q2u1,(x,t)∈RN×(0,T),-iu2t=△u2-ν |u2 |p2u2-β|u1|q1|u2 | q2-2u2, (x,t)∈RN×(0,T),u1 (x,0)=φ(x),u2(x,0)=φ2(x), x∈RN,其中μ,ν,α,β＞0,q1+q2=p3+2,且α/q1=β/q2=b.本文主要研究一些渐近性质,并分别在Sobolev空间、Σ空间及L2(RN)中建立散射理论,这里三={u∈H1(RN),|x|u∈L2 (RN)}.     

Strichartz Estimates for Schr(o)dinger Equations with Non-degenerate Coefficients

In the present paper, the full range Strichartz estimates for homogeneous Schr(o)dinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained.     

抛物型薛定谔方程的最优正则性估计(英文)

本文通过特征化抛物型薛定谔算子t-Δ+V的域,探讨了抛物型薛定谔方程u/t-Δu+Vu=f在Orlicz空间中的最优正则性估计,并且对空间的维数没有限制.     

Classification of Positive Ground State Solutions with Different Morse Indices for Nonlinear N-Coupled Schr(o)dinger System

In this paper,we study the following N-coupled nonlinear Schr(o)dinger sys-tem{-△uj+ uj =μju3j + ∑i≠jβi,ju3iuj,in Rn,uj＞0 in Rn,uj(x)→0 as |x|→+∞,j=1,…,N,where n ≤ 3,N ≥ 3,μj ＞ 0,βi,j =βj,i ＞ 0 are constants and βj,j =μj,j =1,…,N.There have been intensive studies for the system on existence/non-existence and clas-sification of ground state solutions when N =2.However fewer results about the classification of ground state solution are available for N ≥ 3.In this paper,we first give a complete classification result on ground state solutions with Morse indices 1,2 or 3 for three-coupled Schr(o)dinger system.Then we generalize our results to N-coupled Schr(o)dinger system for ground state solutions with Morse indices 1 and N.We show that any positive ground state solutions with Morse index 1 or Morse index N must be the form of (d1w,d2w,…,dNw) under suitable conditions,where w is the unique positive ground state solution of certain equation.Finally,we generalize our results to fractional N-coupled Schr(o)dinger system.     

广义的带导数非线性薛定谔方程的有理解

广义带导数非线性薛定谔方程是与Kaup-Newell谱问题相联系的一个非线性发展方程,方程可在合适的条件方程下,利用Wronsiki技巧,寻找广义双Wronsikian形式的一般解,进而得到其孤子解和有理解.     

On the Well-Posedness for Stochastic Schr(o)dinger Equations with Quadratic Potential

The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schr(o)dinger equations.The local and global well-posedness are proved with values in the space ∑(Rn) =｛f ∈ H1(Rn),｜ · ｜f ∈ L2(Rn)}.When the nonlinearity is focusing and L2-supercritical,the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential.Especially for the repulsive case,the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time.Thus,compared with the deterministic equation for the repulsive case,the blow-up condition is stronger on average,and depends on the regularity of the noise.If φ =0,our results coincide with the ones for the deterministic equation.     

Compressible Limit of the Nonlinear Schr(o)dinger Equation with Different-Degree Small Parameter Nonlinearities

The authors study the compressible limit of the nonlinear Schr(o)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(o)dinger equation. On the other hand, in the limit system,it is shown that the density converges to the solution of the compressible Euler equation and the validity of the WKB expansion is justified.     

新的变系数可积耦合非线性Schr(o)dinger方程及其孤子解

基于延拓结构和Hirota双线性方法研究了广义的变系数耦合非线性Schrdinger方程.首先导出了3组新的变系数可积耦合非线性Schrdinger方程及其线性谱问题(Lax对),然后利用Hirota双线性方法给出了它们的单、双向量孤子解.这些向量孤子解在光孤子通讯中有重要的应用.     

非线性Schr(o)dinger-Boussinesq方程组的新精确解

通过使用改进的F-展开法得到了Schr(o)dinger-Boussinesq方程组具有Jacobi椭圆函数的新精确解. 同时在一些特殊的情况下, 也得到了一些新的孤立波解.     

高阶非线性薛定谔方程的精确解

通过修正的映射方法和推广的映射方法，我们得到了高阶非线性薛定谔方程新的精确解，它们是两个不同的雅可比椭圆函数的线性组合．并研究了在极限情况下高阶非线性薛定谔方程的解．     

基于广义高阶非线性薛定谔方程深海内波传播的数值模拟

基于与实际海洋背景参数相关的广义高阶非线性薛定谔方程,首先讨论了不同的海洋环境参数对方程的非线性项和频散项的影响;然后通过有限差分算子给出了方程的二阶三层数值差分格式,并且分析了该差分格式的稳定性与精度阶;最后又通过得到的差分格式数值模拟了不同的海洋环境参数下深海内波的传播情况,结果显示:内波由深海向浅海的传播过程中,随着总水深的变化,发生了分裂现象,并且密度差之比越大,波的分裂速度越快.     

一类非线性Schr(o)dinger方程的非协调有限元分析

在半离散格式下研究一类带幂次非线性项的Schr(o)dinger方程的非协调矩形EQrot1元方法.直接利用插值技巧和该单元的两个特殊性质(相容误差比插值误差高一阶及其插值算子与传统的Ritz投影是一致的),给出相应的收敛性分析及误差估计.