高阶Chen-Lee-Liu方程在半直线上的初边值问题

该文运用Fokas方法分析了高阶Chen-Lee-Liu方程在半直线上的初边值问题,证明了高阶Chen-Lee-Liu方程初边值问题的解可以用复λ平面上的矩阵Riemann-Hilbert问题的形式解唯一表示.     

一类新的矩阵型修正Korteweg-de Vries方程的Riemann-Hilbert方法与精确解北大核心CSCD

在本文中,一类新的矩阵型修正Korteweg-de Vries(简记为mmKdV)方程被首次通过RiemannHilbert方法研究,而且,这一方程可通过选取特殊的势矩阵来降阶为我们熟知的耦合型修正Kortewegde Vries方程.从方程对应的Lax对的谱分析入手,作者成功地建立了方程对应的Riemann-Hilbert问题.在无反射势的特殊条件下,mmKdV方程的精确解可由Riemann-Hilbert问题的解给出.而且,基于特殊势矩阵所对应的特殊对称性,作者可以对原有的孤子解进行分类,从而得到一些有趣的解的现象,比如呼吸孤子、钟形孤子等.     

超AKNS方程新的可积分解

该文介绍从3×3矩阵形式超谱问题出发, 构造新高阶矩阵形式超谱问题的方法.以超AKNS方程为例, 作者构造了5×5矩阵形式的超AKNS谱问题并且运用双非线性化方法,给出了超AKNS方程的新约束, 得到该约束下超AKNS方程新的可积分解.     

快速衰减初值的修正Korteweg-de Vries方程的孤子解:Riemann-Hilbert方法

本文借助于Riemann-Hilbert (RH)问题研究修正Korteweg-de Vries (mKdV)方程,给出一种有效方法来获得快速衰减初值空间下的孤子解.在正散射过程建立Jost函数和散射矩阵重要性质来构建一个合适的RH问题,进而建立mKdV方程的解和RH问题解之间的关系.在反问题过程中,考虑了两类散射数据,包括简单零点和二阶零点,以及求解相应的RH问题,成功构建在这两种情形下mKdV方程的显示解.最后,结合具体参数,详细分析了几类孤子解的传播行为.     

四元数分析中正则函数与非齐次n阶方程(■~n F)/(■z~n)=f的某些边值问题

主要讨论了四元数空间中正则函数与非齐次n阶方程(■~n F)/(■z~n)=f在超球上的Dirichlet问题和双圆柱上具有任意整数指标的Riemann-Hilbert问题,给出了可解条件和解的积分表示式.     

四元数分析中正则函数与非齐次n阶程(e)nF/(e)(z)n=f的某些边值问题

主要讨论了四元数空间中正则函数与非齐次n阶方程(e)nw/(e)(z)n=f在超球上的Dirichlet问题和双圆柱上具有任意整数指标的Riemann-Hilbert问题,给出了可解条件和解的积分表示式.     

求解动力学问题的普遍方法

本文阐明了标准化的Routh方程[1]的应用,给出了求解系统动力学问题的约束反力及运动状态变化的普遍方法,并给出了相应的矩阵方程。     

一族耦合Kaup-Newell方程及其相伴可积分解

通过介绍一个含四个位势的4×4矩阵谱问题,得到一个新的非线性发展方程族,其中较有意义的一个方程是耦合Kaup-Newell方程.利用迹恒等式,得到了它的双哈密顿结构.在某个约束条件下,通过特征值问题的非线性化方法,得到了Liouville意义下耦合Kaup-Newell方程新的可积分解.     

应用广义Laguerre函数的Navier-Stokes方程外部问题混合谱方法

研究应用广义Laguerre函数的四阶非线性偏微分方程外部问题混合谱方法.构造了圆外Navier-Stokes方程流函数形式的混合谱方法,数值结果显示了该方法在空间方向的谱精度.     

与含有三个位势矩阵谱问题相联系的非线性发展方程族及其Hamilton结构

从含有三个位势的4×4矩阵谱问题出发,导出两类非线性发展方程.然后利用迹公式,给出了这两类方程的广义Hamilton结构.     

Riemann‐Hilbert problems and soliton solutions of a multicomponent mKdV system and its reduction

An arbitrary order matrix spectral problem is introduced and its associated multicomponent AKNS integrable hierarchy is constructed. Based on this matrix spectral problem, a kind of Riemann‐Hilbert problems is formulated for a multicomponent mKdV system in the resulting AKNS integrable hierarchy. Through special corresponding Riemann‐Hilbert problems with an identity jump matrix, soliton solutions to the presented multicomponent mKdV system are explicitly worked out. A specific reduction of the multicomponent mKdV system is made, together with its reduced Lax pair and soliton solutions.     

RIEMANN-HILBERT PROBLEMS OF A SIX-COMPONENT MKDV SYSTEM AND ITS SOLITON SOLUTIONS

Based on a 4 x 4 matrix spectral problem, an AKNS soliton hierarchy with six potentials is generated. Associated with this spectral problem, a kind of Riemann-Hilbert problems is formulated for a six-component system of mKdV equations in the resulting AKNS hierarchy. Soliton solutions to the considered system of coupled mKdV equations are computed, through a reduced Riemann-Hilbert problem where an identity jump matrix is taken.     

A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation

In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann–Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracy of the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional order Van der Pol oscillator (VPO) equation.     

A perturbation method for optimizing matrix stability

We present the first practical perturbation method for optimizing matrix stability using spectral abscissa minimization. Using perturbation theory for a matrix with simple eigenvalues and coupling this with linear programming, we successively reduce the spectral abscissa of a matrix until it reaches a local minimum. Optimality conditions for a local minimizer of the spectral abscissa are provided and proved for both the affine matrix problem and the output feedback control problem. Experiments show that this novel perturbation method is efficient, especially for a matrix with the majority of whose eigenvalues are already located in the left half of the complex plane. Moreover, unlike most available methods, the method does not require the introduction of Lyapunov variables. The method is illustrated for a small size matrix from an affine matrix problem and is then applied to large matrices actually arising from more sophisticated control problems used in the design of the Boeing 767 jet and a nuclear powered turbo-generator.     

具有中心对称结构的非负张量的谱半径(英文)

The matrix with centrosymmetric structure is an important kind of structured matrices with many applications in various physics and engineering problems.The eigenvalue problem for a centrosymmetric matrix can be reduced to the other eigenvalue problems of lower order.We introduce the centrosymmetric structure to the tensor field and focus on the spectral radius of this particular tensor.Then we show that properties of centrosymmetric matrices hold true for tensors situation.     

Two families of Liouville integrable lattice equations

By virtue of zero curvature representations, we are successful to generate the Lax representations of two hierarchies of discrete lattice equations respectively, which are derived from two new and interesting 3 × 3 matrix spectral problems. Moreover, by using the trace identity, the bi-Hamiltonian structures of the above systems are given, and it is shown that they are integrable in the Liouville sense. Finally, infinitely many conservation laws for the second hierarchy of lattice equations are given by a direct method.     

Integrability of Generalized (Matrix) Ernst Equations in String Theory

We elucidate the integrability structures of the matrix generalizations of the Ernst equation for Hermitian or complex symmetric (d×d)-matrix Ernst potentials. These equations arise in string theory as the equations of motion for the truncated bosonic parts of the low-energy effective action for the respective dilaton and d×d matrix of moduli fields or for a string gravity model with a scalar (dilaton) field, a U(1) gauge vector field, and an antisymmetric 3-form field, all depending on only two space-time coordinates. We construct the corresponding spectral problems based on the overdetermined 2d×2d linear systems with a spectral parameter and the universal (i.e., solution-independent) structures of the canonical Jordan forms of their matrix coefficients. The additionally imposed existence conditions for each of these systems of two matrix integrals with appropriate symmetries provide specific (coset) structures of the related matrix variables. We prove that these spectral problems are equivalent to the original field equations, and we envisage an approach for constructing multiparametric families of their solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 214–225, August, 2005.     

Inverse problems for the matrix Sturm–Liouville equation with a Bessel-type singularity

The matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. We consider inverse problems by the Weyl matrix and by the spectral data for this equation. Constructive solutions, based on the method of spectral mappings, are obtained for these inverse problems.     

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.     

Matrix Riccati equations and matrix Sturm-Liouville problems

The asymptotic behavior of determinants of unitary solutions of matrix Riccati differential equations containing a large parameter is determined. The result leads to theorems on existence and asymptotic distribution of eigenvalues of indefinite matrix Sturm-Liouville problems.