变形的非线性Schrdinger方程的反散射解法的Zakhallov-Shabat方程

c_(a)(0)和r(ξ,0)为常量.对(2)中的ξ取ξ_m,ξ-i∈,-in+∈,其中ξ_m为第一象限的常数,ξ,η为正实数,∈为无限小正量,就得到决定ψ_1,ψ_2的完备方程组. MNLS方程(1)的解可以表为(7)式中(8)(9)这里R_1为(3)中的R的上分量,R_2为R的下分量,R_1(0,x)=R_1(ξ,x)|t=0,R_2(0,x)=[dR_2(ξ,x)/dξ]ξ=0。当r(ξ’)=r(iη’)=0时,即无反射的情况,方程(2)已由我们最近用亚纯变换矩阵方法首次导出,它的多孤子解的显式也得出了.     

障碍问题解的局部正则性和局部有界性

该文在算子A(x,ξ):Ω×R~n→R~n的强制性条件和控制增长条件下,考虑A-调和方程divA(x,▽u(x))=0的κ_(Ψ,θ)-障碍问题的解.A的原型是A(x,ξ)=(μ~2+|ξ|~2)~((p-2)/2)ξ,μ≥0.得到了局部正则性和局部有界性结果.     

(1+1)维Burgers方程新的行波解

通过采用新的exp(-ρ(ξ))展式法,得到了(1+1)维Burgers方程形如u(ξ)=αm(exp(-ρ(ξ)))m+αm-1(exp(-ρ(ξ)))m-1+…的新行波解.该方法也可以应用于求解其它许多的非线性演化方程.     

关于有限域上线性方程的原根解

设q是一个素数的方幂,F_q是q个元素的有限域.对F_q中任意确定的非零元素a_1,…,a_r及b,如果存在F_q中的原根ξ_1,…,ξ_r,使得a_1ξ_1+…+a_rξ_r=b,则称(ξ_1,…,ξ_r)是方程 a_1x_1+…+a_rx_r=b (1) 的一个原根解。令N(r,q)表示方程(1)的原根解的个数。1952年,文[1]证明     

一类双曲发展系统的爆破行为

考虑广义双曲型发展方程其中ξ是一元实函数,η是二元实函数,F是1+(n+1)+n(n+1)元实函数,Ω是R~n中有界域,u=u(t,x). 在方程(1)中,若取ξ=u,η=u,即得通常的非线性双曲方程,若取ξ=u,η=u_t,即得通常的非线性拟双曲方程,其它的取法可得多种形式的耗散方程.     

一类非线性高阶中立型方程的振动定理

本文对于一类具有连续分布滞量的高阶中立型微分方程dndtn[x( t) +c( t) x( t-τ) ]+∫baf ( t,ξ,x[g( t,ξ) ]) dσ(ξ) =0 ( 1 )进行讨论 ,得到了方程 ( 1 )的若干振动准则 .     

三阶非线性中立型方程的Philos型振动定理

研究三阶中立型分布时滞微分方程(r(t)[x(t)+p(t)x(r(t))]″)′+∫_a~b q(t,ξ)f(x[g(t,ξ)])dσ(ξ)=0的振动性.利用广义Riccati变换和积分平均技巧,建立了保证此方程一切解振动或者收敛到零的若干新的充分条件.     

非线性多时滞系统的二阶D型迭代学习控制

收敛性是迭代学习控制的重要研究内容之一.针对非线性多时滞系统,讨论了二阶D型迭代学习控制算法.通过引进新的(λ,ξ)-范数和新的分析方法,本文获得了算法收敛和目标跟踪精度较高的结果.仿真结果表明了该算法的有效性.     

与位势依赖于能量的特征值问题相联系的非线性发展方程

<正> 文[1]研究了特征值问题(?)得到了与此相联系的一类非线性发展方程.M.Jaulent,I.Miodek 又进一步研究了特征值问题(1.1)当位势 q 依赖于能量ξ的情形(r=1,q=u+ξv).在对 u,v 于无穷远处加一定条件下,当ξ_(?)=0 时,得到了相应的非线性发展方程(?)+Ω(L~*)(?)=0,(1.2)其中Ω(L~*)为 L~*的多项式或整函数,且     

在曲线边界区域上非线性拋物型方程的边界問題

<正> 在1954年与研究了非线性拋物型方程在长方形区域R{0≤x≤X,0≤t≤T}上的第一边界問題和在区域S{-∞相似文献   

一类非线性无穷多点边值问题正解的存在性

运用锥拉伸与锥压缩不动点理论,讨论了一类非线性二阶常微分方程无穷多点边值问题u″+a(t).f(u)=0,t∈(0,1),u(1)=∑a_iu(ζ_i),u′(0)=∑b_iu′(ζ_i)正解的存在性.其中a∈C([0,1],[0,∞)),ζ_i∈(0,1),a_i,b_i∈[0,∞),f∈C([0,∞),[0,∞))并且满足∑a_i<1,∑b_i<1.推广了已有文献中的一些结果.     

带梯度算子二阶方程的渐近概周期解

利用渐近概周期函数的性质得到带梯度算子二阶方程的渐近概周期解在C（R^-）中的存在性．同时利用迭代法和线性常微分方程的概周期解的存在性和唯一性,得到R上此方程渐近概周期解的存在和唯一性．     

积分中值ξ=ξ(x)的渐近性

对一类不满足g(n)≠0的函数g讨论了第一积分中值定理中ξ=ξ(x)在x→+∞时的渐近性质,并对第二积分中值定理的中值ξ=ξ(x)的渐近性进行了探讨,给出一些相关的结果.     

Linear integral equations with asymptotically almost periodic solutions

Conditions are given that ensure that bounded solutions of $\text{∫}{}_{\mathrm{?\xi }}{}^{\mathrm{\xi }}\text{x(t ? ξ)dA(ξ) = ?(t)}$ are asymptotically almost periodic. The result strengthens and extends a recent theorem of Levin and Shea. Generalizations to systems of integral equations as well as to integrodifferential systems are included.     

On convergence of the inexact Rayleigh quotient iteration with the Lanczos method used for solving linear systems

For the Hermitian inexact Rayleigh quotient iteration(RQI),we consider the local convergence of the in exact RQI with the Lanczos method for the linear systems involved.Some attractive properties are derived for the residual,whose norm is ξk,of the linear system obtained by the Lanczos method at outer iteration k+1.Based on them,we make a refned analysis and establish new local convergence results.It is proved that(i) the inexact RQI with Lanczos converges quadratically provided that ξk≤ξ with a constant ξ1 and (ii) the method converges linearly provided that ξk is bounded by some multiple of1/||rk|| with rkthe residual norm of the approximate eigenpair at outer iteration k.The results are fundamentally diferent from the existing ones that always require ξk<1,and they have implications on efective implementations of the method.Based on the new theory,we can design practical criteria to control ξkto achieve quadratic convergence and implement the method more efectively than ever before.Numerical experiments confrm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.     

Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations

Hermitian and skew-Hermitian splitting(HSS) method has been proved quite successfully in solving large sparse non-Hermitian positive definite systems of linear equations. Recently, by making use of HSS method as inner iteration, Newton-HSS method for solving the systems of nonlinear equations with non-Hermitian positive definite Jacobian matrices has been proposed by Bai and Guo. It has shown that the Newton-HSS method outperforms the Newton-USOR and the Newton-GMRES iteration methods. In this paper, a class of modified Newton-HSS methods for solving large systems of nonlinear equations is discussed. In our method, the modified Newton method with R-order of convergence three at least is used to solve the nonlinear equations, and the HSS method is applied to approximately solve the Newton equations. For this class of inexact Newton methods, local and semilocal convergence theorems are proved under suitable conditions. Moreover, a globally convergent modified Newton-HSS method is introduced and a basic global convergence theorem is proved. Numerical results are given to confirm the effectiveness of our method.     

A Super-Halley Type Approximation in Banach Spaces

The Super-Halley method is one of the best known third-order iteration for solving nonlnear equations. A Newton-like method which is an approximation of this method is studied. Our approach yields a fourth R-order iterative process which is more efficient than its classical predecessor. We establish a Newton-Kantorovich-type convergence theorem using a new system of recurrence relations, and give an explicit expression for the a priori error bound of the iteration.     

A Super-Halley Type Approximation in Banach Spaces

The Super-Halley method is one of the best known third-order iteration for solving nonlnear equations. A Newton-like method which is an approximation of this method is studied. Our approach yields a fourth R-order iterative process which is more efficient than its classical predecessor. We establish a Newton-Kantorovich-type convergence theorem using a new system of recurrence relations, and give an explicit expression for the a priori error bound of the iteration.     

Convergence rates for Kaczmarz-type algorithms

By making use of the normal and skew-Hermitian splitting (NSS) method as the inner solver for the modified Newton method, we establish a class of modified Newton-NSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Under proper conditions, the local convergence theorem is proved. Furthermore, the successive-overrelaxation (SOR) technique has been proved quite successfully in accelerating the convergence rate of the NSS or the Hermitian and skew-Hermitian splitting (HSS) iteration method, so we employ the SOR method in the NSS iteration, and we get a new method, which is called modified Newton SNSS method. Numerical results are given to examine its feasibility and effectiveness.     

一种新的迭代收敛阶数的证明与推广

迭代方法是求解非线性方程近似根的重要方法.本文基于隐函数存在定理,提出了一种新的迭代方法收敛性和收敛阶数的证明方法,并分别对牛顿(Newton)和柯西(Cauchy)迭代方法迭代收敛性和收敛阶数进行了证明.最后,利用本文提出的证明方法,证明了基于三次泰勒(Taylor)展式构成的迭代格式是收敛的,收敛阶数至少为4,并提出猜想,基于n次泰勒展式构成的迭代格式是收敛的,收敛阶数至少为(n+1).