P耗散算子的研究

本文利用广义半内积空间及广义不定度规空间,解决Ｂａｎａｃｈ空间中Ｐ耗散算子的极大耗散扩张表示,并得到很有意义的应用．     

具有耗散项的非线性Schrodinger方程的精确包络波解

In this paper, weconsider the evolution of a soliton when dissipative lose exists. By means of non-perturbed method, an exact envelope wave solution of nonlimear Schroedinger equation with dissipative term is obtained. It is shown that when Г=γ0/(1 2γot), the solution given here still maintains the hyperbolic secant profile.     

p耗散算子的扩张

该文研究Ｂａｎａｃｈ空间中ｐ耗散算子的极大耗散扩张．通过建立自然边界空间，将ｐ耗散算子的极大耗散扩张与自然边界空间中极大负子空间—一对应，并给出表达式．利用这一工作本文给出一类Ｓｃｈｒｏｄｉｎｇｅｒ算子的扩张应用．     

无穷维线性空间中的非游荡算子

本文研究无穷维线性空间一类具有混沌特性的线性算子；非游荡算子。这是无穷维线性混沌系统中一类有广泛意义的算子，同时本文给出了非游荡算子紧集上的超循环分解。     

二阶耗散动力系统的降维对解长期行为的误差估计

基于非线性动力学理论,对一类高维二阶耗散自治动力系统的降维及其对解的长期行为的影响进行了理论分析．该分析将方程的解投影到控制方程的线性算子的特征向量所张成的完备空间中,并在相空间中引入一距离的概念,方便地解决了缩减后系统与原始系统解之间的误差或距离的描述．基于此距离定义,首先,分析了由于高阶模态的截取对解的长期行为的影响,并推导出了相应的误差估计,该估计表明由于降维对系统长期行为的影响不仅与系统的高阶子空间中的固有频率和阻尼比乘积的最小值有关,并且与高阶子空间中的某一最大固有频率有关．然后,将一般的模态截取视为对原系统的解的一个扰动,对一些文献中由于降维程度的不同而造成解的拓扑性质发生变化的现象进行了定性的解释．     

无穷远点处与Schrodinger算子相关的极细集

本文首先得到了一些新的关于锥中无穷远点处与Schrodinger算子相关极细集的判定准则,其证明是基于对带有修改测度的Green-Sch位势在无穷远点处渐近行为的估计.接着,刻画了这类极细集的几何性质.最后,通过一个反例来说明,所得几何性质的逆命题并不成立.     

无穷维系统中算子Riccati方程解的注记[英文]

本文采用正交投影技巧研究无穷维系统中算子Riccati方程的解,利用有限维空间中一序列来逼近该算子Riccati方程的解.并给出一个数值例子来说明我们的结论.     

关于Kuramoto-Sivashinsky方程的非线性Galerkin方法之注记*

本文致力于讨论求解Kuramoto-Sivashinsky方程的非线性Galerkin方法,我们采用了s_m个小尺度分量作反馈,并给出了收敛性结果,分析了误差估计．结论表明我们的修正方法是十分有效的．     

关于无穷维算子的有限维最佳逼近

A method of approaching to the infinite-dimensional linear operators by the finite-dimensional operators is discussed. It is shown that,for every infinite-dimensional operator A and every natural number n,there exists an n-dimensional optimal approximation to A. The norm error is found and the necessary and sufficient condition for such n-dimensional optimal approximations to be unique is obtained.     

Hilbert空间中耗散算子的性质研究

文章研究了Hilbert空间中耗散算子的性质,并给出此类算子成为C0压缩半群无穷小生成元的一些条件。     

Solvability of Boundary Value Problems for Operator-Differential Equations of Mixed Type: The Degenerate Case

We obtain some results on solvability of boundary value problems for the equation B u _t –L u =f (t(0,)), where B and L are selfadjoint and dissipative operators defined in a Hilbert space E. The kernel of B may be nontrivial and L is uniformly dissipative on a subspace M of finite codimension.     

一类非线性算子方程拟Galerkin逼近意义下的可解性

对非线性算子方程 x+KF x=y,本文构造了一种新的不同于 Galerkin逼近的另一种逼近方程 ,并证明了在此逼近意义下上述方程是逼近可解的     

Discrete Spectrum of the Schrodinger Operator Perturbed by a Narrowly Supported Potential

We study asymptotic properties of the discrete spectrum of the Schrodinger operator perturbed by a narrowly supported potential. The first terms of the asymptotic expansions in the small parameter equal to the width of the support of the potential are constructed for the eigenvalues and the corresponding eigenfunctions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 372–384, December, 2005.     

Perturbation Theory of Resonances and Embedded Eigenvalues of the Schrodinger Operator For a Crystal Film

We obtain formulas for resonances and eigenvalues embedded in the continuous spectrum that are similar to formulas in the standard perturbation theory. We prove that although the imaginary part of the first-order correction to the eigenvalue embedded in the continuous spectrum is zero, the perturbed eigenfunction, as a rule, ceases to be square-summable.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 417–430, June, 2005.     

On the existence of full dimensional KAM torus for fractional nonlinear Schrodinger equation

In this paper,\ we study fractional nonlinear Schrodinger equation (FNLS) with periodic boundary condition $$ \textbf{i}u_{t}=-(-\Delta)^{s_{0}} u-V*u-\epsilon f(x)|u|^4u,\ ~~x\in \mathbb{T}, ~~t\in \mathbb{R}, ~~s_{0}\in (\frac12,1),~~~~~~~~~~~~~~~~~~~~~~~~~~~~(0.1) $$ where $(-\Delta)^{s_{0}}$ is the Riesz fractional differentiation defined in [21] and $V*$ is the Fourier multiplier defined by $\widehat{V*u}(n)=V_n\widehat{u}(n),\ V_n\in\left[-1,1\right],$ and $f(x)$ is Gevrey smooth. We prove that for $0\leq|\epsilon|\ll1$ and appropriate $V$,\ the equation (0.1) admits a full dimensional KAM torus in the Gevrey space satisfying $ \frac12e^{-rn^{\theta}}\leq \left|q_n\right|\leq 2e^{-rn^{\theta}}, \theta\in (0,1),$ which generalizes the results given by [8-10] to fractional nonlinear Schrodinger equation.     

The existence of global solutions for the fourth-order nonlinear Schrodinger equations

In this paper, the problem of a class of multidimensional fourth-order nonlinear Schr\"{o}dinger equation including the derivatives of the unknown function in the nonlinear term is studied, and the existence of global weak solutions of nonlinear Schr\"{o}dinger equation is proved by the Galerkin method according to the different values of $\lambda$.     

On a Class of Operator Equations

We assume that E₁ and E₂ are Banach spaces, a:E₁ E₂ is a continuous linear surjective operator, f:E₁ E₂ is a nonlinear completely continuous operator. In this paper, we study existence problems for the equation a(x)=f(x) and estimate the topological dimension dim of the set of solutions.