Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

An Eight Component Integrable Hamiltonian Hierarchy from a Reduced Seventh-Order Matrix Spectral Problem

We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.

EFFICIENT SPECTRAL METHODS FOR EIGENVALUE PROBLEMS OF THE INTEGRAL FRACTIONAL LAPLACIAN ONABALLOFANYDIMENSION

An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper.The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis.And a sharp estimate on the algebraic system's condition number is established which behaves as N4s with respect to the polynomial degree N,where 2s is the fractional derivative order.The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces.Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived.Meanwhile,rigorous error estimates of the eigenvalues and eigenvectors are ob-tained.Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.

A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problem

This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation.A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out,which exhibits the Liouville integrability of each model in the resulting hierarchy.Two specific examples,consisting of novel generalized combined nonlinear Schrodinger equations and modified Korteweg-de Vries equations,are given.

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

A Finite-Dimensional Completely Integrable System Associated with Boussinesq Hierarchy

In this paper, a new completely integrable system related to the complex spectral problem -φ xx＋（i/4）wpx＋（i/4）（wp）x＋（1/4）vφ=iλφxand the constrained flows of the Boussinesq equations axe generated. According to the viewpoint of Hamiltonian mechanics, the Euler-Lagrange equations and the Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the constrained conditions between the potentiaJ u, v and the eigenfunction φ, the involutive representations of the solutions for the Boussinesq equation hieraxchy axe given.

一族新Lax和Liouville可积广义Hamilton方程

考虑了一个新的具有 4个位势的等谱问题 ,利用屠格式获得一族新的含有任意函数的Lax可积演化方程 .进一步由迹恒等式得到其广义 Hamilton结构并且证明是 Liouville可积的 .Burgers方程是所得方程族的特例 .

Integral Equation Method for a Non-Selfadjoint Steklov Eigenvalue Problem

We propose a numerical method for a non-selfadjoint Steklov eigenvalue problem of the Helmholtz equation.The problem is formulated using boundary integrals.The Nyström method is employed to discretize the integral operators,which leads to a non-Hermitian generalized matrix eigenvalue problems.The spectral indicator method(SIM)is then applied to calculate the(complex)eigenvalues.The convergence is proved using the spectral approximation theory for(non-selfadjoint)compact operators.Numerical examples are presented for validation.

RECURSIVE INTEGRAL METHOD FOR THE NONLINEAR NON-SELFADJOINT TRANSMISSION EIGENVALUE PROBLEM

The transmission eigenvalue problem is an eigenvalue problem that arises in the scatter- ing of time-harmonic waves by an inhomogeneous medium of compact support. Based on a fourth order formulation, the transmission eigenvalue problem is discretized by the Mor- ley element. For the resulting quadratic eigenvalue problem, a recursive integral method is used to compute real and complex eigenvalues in prescribed regions in the complex plane. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

SOLUTION OF ARBITRARY CROSS-SECTION WAVEGUIDE USING METHOD OF EIGEN WEIGHTED BOUNDARY INTEGRAL EQUATION

Based on the second kind of Green’s identity,a boundary integral equation forarbitrary cross-section waveguide is transformed to a system of linear homogeneous algebraicequations by means of expansion of boundary bases and by using the eigenfunctions of a fictitiousregular boundary as weighting functions,which corresponds to less algebraic equations than BEMand simpler coefficients than the modified BEM.The numerical results for some typical metallicwaveguides are given by using the method of eigen-weighted boundary integral equation,and theyare accurate enough with fast convergence.

A NEW HIERARCHY OF LAX AND LIOUVILLE INTEGRABLE EVOLUTION EQUATIONS ASSOCIATED WITH AN ISOSPECTRAL PROBLEM IN THE LOOP ALGEBRA■

In this paper, an isospectral problem with five potentials is investigated in loop algebra A2 such that a new hierarchy of evolution equations with five arbitrary functions is obtained. And then by fixing the five arbitrary functions to be certain flmctions and using the trace identity, the generalized Hamiltonian structure of the hierarchy of evolution equations is given, it is shown that this hierarchy of equations is Liouville integrable. Finally some special cases of the isospectral problem are also given.

TENSOR NEURAL NETWORK AND ITS NUMERICAL INTEGRATION

In this paper,we introduce a type of tensor neural network.For the first time,we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension.Based on the tensor product structure,we develop an efficient numerical integration method by using fixed quadrature points for the functions of the tensor neural network.The corresponding machine learning method is also introduced for solving high-dimensional problems.Some numerical examples are also provided to validate the theoretical results and the numerical algorithm.

一个新的含非线性位势的谱问题及可积系

对一个新的带非线性位势u2 的特征值问题及相应的辅谱问题进行研究 ,得到一族新的孤子方程 ,并由此导出一个新的 ( 2 + 1)维可积模型 .通过将特征值问题非线性化 ,得到一个完全可积的有限维Hamilton系统 。

带有积分边值条件的三阶边值问题正解的存在性

应用特征值准则研究了一类三阶带有积分边值条件边值问题正解的存在性,其中非线性项f:[0,1]×[0,∞)→[0,∞)满足Caratheodory条件。在赋予非线性项一定条件下,得到该边值问题至少存在3个正解的充分条件。

2n维空间中的广义自对偶Yang-Mills方程的达布变换

从带负幂次谱参数的谱问题出发,构造了一类广义自对偶Yang-Mills方程.这类方程包括若干著名的Lax可积方程,如Takasaki情形、Belavin-Zakharov情形、AblowitzChakravarty-Takhtajall情形和Ma情形.进而建立了这类方程的达布变换的精确表达式.

基于高阶辛算法的纳米器件本征问题仿真

研究一种准确、有效的数值方法是现代纳米器件建模和优化的重要目标之一,而分析大多数纳米器件特性的起始点是确定器件的本征值和本征态。提出了一种新算法—高阶辛时域有限差分法(Symplectic finitedifference time-domain,SFDTD(3,4)),求解含时薛定谔方程。在时间上采用三阶辛积分,空间上采用四阶差分格式,建立了针对含时薛定谔方程数值求解的高阶辛时域有限差分算法。将高阶辛算法SFDTD(3,4)用于一维量子阱中盒中粒子和一维谐振子的仿真中,实验结果表明SFDTD(3,4)法比传统的时域有限差分算法以及高阶时域有限差分算法更加准确,适用于对纳米器件本征问题的长时间仿真。

基于小波理论光子晶体能带计算中小波积分的解析求解

详细推导了基于双尺度关系的小波积分的矩阵本征值求解方法.并选择常用的Harr小波和CDF小波进行了实际计算,与数值积分的方法进行了比较,结果表明该方法具有良好的精度.同时,还针对光子晶体的特点,对该方法进行了改进,这将有利于进一步提高基于小波理论的光子晶体计算方法的性能.

二阶积分边值问题的恒号解

研究积分边界条件下,证明二阶微分方程边值问题恒号解的存在性,应用单侧全局分歧定理,得到该边值问题至少存在一个正解和一个负解.

小波Galerkin方法在Sturm-Liouville边值问题紧积分算子中的应用

考虑一类由Sturm-Liouville边值问题引出的核对称紧积分算子。给出该类边值问题解的紧积分算子表达形式,构造小波Galerkin方法求解该类积分算子的特征值问题,利用周期构造法构造平方可积空间中的紧支集小波基,讨论特征值在该算法下的收敛性。通过实例验证了小波Galerkin方法在Sturm-Liouville边值问题紧积分算子中应用的可行性。

和推广的Harry Dym方程族相联系的一族完全可积的Hamilton系统(英文)

将和谱问题φ_(zz)+sum from i=1 to v u_iλ~iφ=αφ相联系的推广的Harry Dym方程族限制到它们递推算子的不变子空间,我们得到一族Hamilton系统。利用和谱问题有关的递推关系式,可以构造这族系统的守恒积分和Hamilton函数,从而证明,这些Hamilton系统在Liouville异义下是完全可积的且两两可交换的,同时它们的解满足推广的Harry Dym方程。