Isospectral and Nonisospectral Lattice Hierarchies Associated with a Discrete Spectral Problem and Their Infinitely Many Conservation Laws

在这份报纸，基于分离零个弯曲代表， isospectral 和 nonisospectral 格子层次被建议。借助于解决相应分离光谱方程，我们表明存在无穷地为这的许多能量守恒定律二个层次并且获得相应保存密度和联系流动的公式。

Almost Regularity Conditions of Spectral Problems for a Second Order Equation

The asymptotic distributions are exactly solved for linearly independent solutions considering problem of the second order and for the coefficients of asymptotic distribution the recurrent formulas are obtained. Further, using obtained recurrent formulas the necessary and sufficient conditions for almost regularity of spectral problem for the equation of the second order is proved.

Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

开始从一新分离光谱问题，非线性的格子方程的相应层次被建议。格子 soliton 层次拥有 bi-Hamiltonian 结构，这被显示出，无穷地，许多普通变换保存功能。进一步，层次的无限的能量守恒定律被介绍。

Decomposition of Soliton Hierarchy Associated with a Schrodinger Type Spectral Problem

soliton 层次与一种 Schrodinger 类型联系了光谱有四个潜力的问题被分解成新有限维的 Hamiltonian 系统的一个类由用 nonlinearized， approach.It 是价值指 soliton 层次的解决方案被归结为解决平常的微分方程的兼容 Hamiltonian 系统。

A Finite-Dimensional Integrable System Related to the Complex 3 ×3 Spectral Problem and the Coupled Nonlinear Schrödinger Equation

The relation between the 3 × 3 complex spectral problem and the associated completely integrable system is generated. From the spectral problem, we derived the Lax pairs and the evolution equation hierarchy in which the coupled nonlinear Schr?dinger equation is included. Then, with the constraints between the potential function and the eigenvalue function, using the nonlineared Lax pairs, a finite-dimensional complex Hamiltonian system is obtained. Furthermore, the representation of the solution to the evolution equations is generated by the commutable flows of the finite-dimensional completely integrable system.

On the Local Solvability and Stability for the Inverse Spectral Problem of the Generalized Dirichlet–Regge Problem

For the generalized Dirichlet–Regge problem with complex coefficients,we prove the local solvability and stability for the inverse spectral problem,which indicates an improved result of the previous work([Journal of Geometry and Physics,159,103936(2021)]).

A New 4×4 AKNS Spectral Problem and Its Associated Integrable Decomposition of the AKNS Equation

A new approach to construct a new 4×4 matrix spectral problem from a normal 2×2 matrix spectral problem is presented.AKNS spectral problem is discussed as an example.The isospectral evolution equation of the new 4×4 matrix spectral problem is nothing but the famous AKNS equation hierarchy.With the aid of the binary nonlino earization method,the authors get new integrable decompositions of the AKNS equation. In this process,the r-matrix is used to get the result.

A New m × m Matrix Kaup-Newell Spectral Problem and Its Associated Integrable Decomposition

A new m×m matrix Kaup-Newell spectral problem is constructed from a normal 2×2 matrix Kaup-Newell spectral problem,a new integrable decomposition of the Kaup-NeweU equation is presented.Through this process,we find the structure of the r-matrix is interesting.

A Spectral Element/Laguerre Coupled Method to the Elliptic Helmholtz Problem on the Half Line

A Legendre spectral element/Laguerre coupled method is proposed to numerically solve the elliptic Helmholtz problem on the half line. Rigorous analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results. The advantage of this method is demonstrated by a numerical comparison with the pure Laguerre method.

Mixed spectral method for exterior problems of Navier-Stokes equations by using generalized Laguerre functions

In this paper,we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations.A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc.Numerical results demonstrate the spectral accuracy in space.

A Spectral Method in Time for Initial-Value Problems

A time-spectral method for solution of initial value partial differential equations is outlined. Multivariate Chebyshev series are used to represent all temporal, spatial and physical parameter domains in this generalized weighted residual method (GWRM). The approximate solutions obtained are thus analytical, finite order multivariate polynomials. The method avoids time step limitations. To determine the spectral coefficients, a system of algebraic equations is solved iteratively. A root solver, with excellent global convergence properties, has been developed. Accuracy and efficiency are controlled by the number of included Chebyshev modes and by use of temporal and spatial subdomains. As examples of advanced application, stability problems within ideal and resistive magnetohydrodynamics (MHD) are solved. To introduce the method, solutions to a stiff ordinary differential equation are demonstrated and discussed. Subsequently, the GWRM is applied to the Burger and forced wave equations. Comparisons with the explicit Lax-Wendroff and implicit Crank-Nicolson finite difference methods show that the method is accurate and efficient. Thus the method shows potential for advanced initial value problems in fluid mechanics and MHD.

On a Boundary Value Problem for a Polynomial Pencil of the Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions

The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.

Time-Spectral Solution of Initial-Value Problems—Subdomain Approach

Temporal and spatial subdomain techniques are proposed for a time-spectral method for solution of initial-value problems. The spectral method, called the generalised weighted residual method (GWRM), is a generalisation of weighted residual methods to the time and parameter domains [1]. A semi-analytical Chebyshev polynomial ansatz is employed, and the problem reduces to determine the coefficients of the ansatz from linear or nonlinear algebraic systems of equations. In order to avoid large memory storage and computational cost, it is preferable to subdivide the temporal and spatial domains into subdomains. Methods and examples of this article demonstrate how this can be achieved.

对称五对角矩阵加箭型矩阵的广义逆谱问题

研究了一类对称五对角矩阵加箭型矩阵的广义逆谱问题,将该矩阵所有主子阵的极端特征值作为其特征数据,利用几何学上的圆锥曲线、五对角矩阵及箭型矩阵的相关性质,重构此类特殊箭状矩阵。最后给出了该问题解的充分条件以及问题构造的算法与实例,验证了结果的准确性。

耦合MRT方程的三哈密顿对偶系统

本文将三哈密顿方法应用于耦合Merola-Ragnisco-Tu方程,获得了新的离散可积系统,并求出了它的线性谱问题、双哈密顿结构以及Darboux-Backlund变换.进一步,利用DarbouxBacklund变换求出了对偶系统的精确解.

RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations.A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix,corresponding to the reflectionless inverse scattering transforms,is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.

椭圆域上二阶/四阶变系数问题有效的谱Galerkin逼近

本文提出了椭圆域上二阶/四阶变系数问题的一种有效的谱Galerkin逼近.首先,我们将原问题化为极坐标下的等价形式,并建立其弱形式及相应的离散格式.其次,针对二阶情形,我们证明了弱解和逼近解的存在唯一性及它们之间的误差估计.另外,根据极条件和勒让得多项式的正交性,我们构造了一组有效的径向基函数,并在θ方向作截断的傅立叶展开,推导了离散格式等价的矩阵形式.最后,我们给出了大量的数值算例,数值结果表明了我们算法的收敛性和谱精度.

周期边界条件下四阶特征值问题的一种有效的Fourier谱逼近

文章提出了周期边界条件下四阶特征值问题的一种有效的Fourier谱逼近方法.首先,根据周期边界条件引入了适当的Sobolev空间和相应的逼近空间,建立了原问题的一种弱形式及其离散格式,并推导了等价的算子形式.其次,定义了正交投影算子,并证明了其逼近性质,结合紧算子的谱理论证明了逼近特征值的误差估计.另外,构造了逼近空间中的一组基函数,推导了离散格式基于张量积的矩阵形式.最后,文章给出了一些数值算例,数值结果表明其算法是有效的和谱精度的.

BEST APPROXIMATION BY NORMAL MATRICES WITH SPECTRAL CONSTRAINTS

考虑在给定谱约束和Ｆｒｏｂｅｎｉｕｓ范数意义下用正规矩阵最佳逼近一个给定复方阵的问题。给出了这个问题可解的充分必要条件，提出了求解这个问题的一个数值算法，并给出了一个数值例子。