Spectral Analysis of a Class of Non-Self-Adjoint Differential Operator Pencils with a Generalized Function

We investigate the spectrum and solve the inverse problem for a pencil of non-self-adjoint second-order differential operators with a generalized function in the space L₂(−∞, +∞). __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 102–107, October, 2005.     

Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H^2,2(?) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H^2,2(?) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.     

具有未知源的某些非线性伪抛物型方程的反问题

考 虑具有未 知源项的 某些非 线性伪 抛物 型方程 的反演 问题． 首先 将伪抛 物型 方程初 边值问 题化为非线 性发展方 程 Ｃｏｕｃｈ ｙ 问题,然 后,利用半 群理论,论 证发展 方程反问 题解的存 在唯一 性,最后, 利用不 动点方法得到 伪抛物型方程反 问题的可解性     

矩阵方程X+A~*X~(-n)A=P的Hermite正定解及其扰动分析

考虑非线性矩阵方程X A~*X~(-n)A=P,其中A是m阶非奇异复矩阵,P是m阶Hermite正定矩阵.本文利用不动点理论讨论了该方程Hermite正定解的存在性及包含区间,给出了极大解的性质及求极大,极小解的迭代算法.研究了极大解的扰动问题,利用微分等方法获得了两个新的一阶扰动界,并给出数值例子对所得结果进行了比较说明.     

分块矩阵的Moore-Penrose逆

该文研究了两类3×3分块矩阵M1=AOOBCODEF,M2=ABCDEFGHK的Moore-Penrose逆的表达式,并给出了表达式成立时的条件.     

秩约束下矩阵方程AX=B的反对称解及其最佳逼近

本文研究了秩约束下矩阵方程AX=B的反对称解问题.利用矩阵秩的方法,获得了矩阵方程AX=B有最大秩和最小秩解的充分必要条件以及定秩解的表达式,同时对于最小秩解的解集合,得到了最佳逼近解.     

Boundary blow-up solutions for a class of quasilinear elliptic equations

For a given bounded domain Ω in R ^N with smooth boundary ?Ω, we give sufficient conditions on f so that the m-Laplacian equation △ _m u = f(x, u, ?u) admits a boundary blow-up solution u ∈ W ^1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle.     

关于KdV方程孤子解的研究

KdV方程的多孤子解很难直接验证，本文通过证明GLM反散射变换方程导出的Jost解满足两个Lax方程的方法，解决了这个问题．     

一类具一阶奇性核的奇异积分方程

当(?)是复平面C上的光滑封闭曲线,k（z）是在(?)所围成的有界闭区域上连续.在其内部解析的函数时.借助于奇异积分算子的广义逆.讨论了具一阶奇性核的正则型奇异积分方程:　在H类中的求解问题.作为应用,作者给出了当k（z）是一类有理函数时的具体解法,从而统一并推广了 Cauchy核和Hilbert核奇异积分方程的经典结果.     

抽象空间中非线性算子方程变号解的存在性研究

利用Banach空间中的锥理论和不动点定理讨论了非线性算子方程变号解的存在性,给出了E_u_0空间下非线性算子方程变号解至少有一个变号解、一个正解和一个负解的条件,并讨论了仅通过一个上解条件得出非线性算子方程变号解的存在性定理.     

矩阵方程的通解及其结构

本文导出了齐次矩阵方程的基础解阵形式通解 ,还得出了非齐次矩阵方程具有广义逆矩阵形式通解的一个充分必要条件 .     

On uniqueness for an inverse problem in inhomogeneous elasticity

We consider the scattering of time-harmonic elastic waves inan isotropic medium which is characterized by constant Lamécoefficients and an inhomogeneous mass density. We prove thata knowledge of the far-field pattern for all incident planewaves and a single frequency provides sufficient informationto determine the mass density uniquely. To this end we constructa periodic Faddeev-type fundamental solution for the Navierequation and derive the existence of complex geometrical opticssolutions to the Navier equation.     

一类非线性Dirac方程组

本文寻求方程组 i sum from j=0 to 3 r~j _jΨ+L(Ψ■)Ψ=0的某些含两个零分量的解.     

Piecewise constant positive potentials with practically the same fixed energy phase shifts

It has recently been shown that spherically symmetric potentials of finite range are uniquely determined by the part of their phase shifts at a fixed energy level k² > 0. However, numerical experiments show that two quite different potentials can produce almost identical phase shifts. It has been guessed by physicists that such examples are possible only for “less physical” oscillating and changing sign potentials. In this note it is shown that the above guess is incorrect: we give examples of four positive spherically symmetric compactly supported quite different potentials having practically identical phase shifts. The note also describes a hybrid stochastic deterministic method for global minimization for the construction of these potentials.     

Spectral Theory of the Nonstationary Schrodinger Equation with a Two-Dimensionally Perturbed Arbitrary One-Dimensional Potential

We consider the nonstationary Schrodinger equation with the potential being a perturbation of a generic one-dimensional potential by means of a decaying two-dimensional function in the framework of the extended resolvent approach. We give the corresponding modification of the Jost and advanced/retarded solutions and spectral data and present relations between them. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 257–276, August, 2005.