小展弦比波的Green-Naghdi渐进模型的行波解

本文研究了小展弦比波的Green-Naghdi渐进模型. 利用平面自治系统的稳定性分析方法, 在不同的参数条件下, 讨论了它的行波系统的分岔并且给出了对应的相图, 得到了光滑周期波解, 广义扭波解, 广义反扭波解, 广义紧波解, 周期尖波解, 孤波解和孤立尖波解的精确表达式. 进一步, 通过数学软件Maple模拟了这些解.     

小展弦比波的Green-Naghdi渐进模型的行波解（英文）

本文研究了小展弦比波的Green-Naghdi渐进模型.利用平面自治系统的稳定性分析方法,在不同的参数条件下,讨论了它的行波系统的分岔并且给出了对应的相图,得到了光滑周期波解,广义扭波解,广义反扭波解,广义紧波解,周期尖波解,孤波解和孤立尖波解的精确表达式.进一步,通过数学软件Maple模拟了这些解.     

非线性波方程的精确孤立波解

立了一种求解非线性波方程精确孤立波解的双曲函数方法,并在计算机代数系统上加以实现,推导出了一大批非线性波方程的精确孤立波解.方法的基本原理是利用非线性波方程孤立波解的局部性特点,将方程的孤立波解表示为双曲函数的多项式,从而将非线性波方程的求解问题转化为非线性代数方程组的求解问题.利用吴消元法或Gröbner基方法在计算机代数系统上求解非线性代数方程组, 最终获得非线性波方程的精确孤立波解,其中有很多新的精确孤立波解.     

Legendre小波函数求解非线性Volterra积分微分方程组的数值解

Legendre小波函数被用于逼近非线性Volterra积分微分方程组的解,方法是基于Legendre小波的性质构建相应的积分算子矩阵,进而将原问题转化为关于未知解系数的线性方程组,通过求解该方程组,即得原问题的数值解.数值结果表明所述方法对于求解此类问题是行之有效的.     

强非线性广义 Boussinesq 方程孤波解的波形分析及求解

上海理工大学理学院\quad 上海 200093该文建立了强非线性广义 Boussinesq 方程的耗散项、波速、渐进值与波形函数的导数之间的关系.利用适当变换和待定假设方法,作者求出了上述广义 Boussinesq 方程的扭状或钟状孤波解,还求出了以前文献中未曾提到过的余弦函数的周期波解.进一步给出了波速对波形影响的结论,即:``好'广义 Boussinesq 方程的行波当波速由小变大时,波形由钟状孤波变成余弦函数周期波解;``坏'广义 Boussinesq 方程的行波当波速由小变大时,波形由余弦函数周期波解变成钟状孤波.     

两层密度分层流体毛细重力波三阶Stokes波解

以小振幅波理论为基础,利用摄动方法研究了两层密度成层状态下的毛细重力波,求得了两层密度成层状态下各层流体速度势的三阶解及毛细重力波波面位移的三阶Stokes波解.结果表明:三阶方程的解均受到表面张力的影响.三阶Stokes波解描述了毛细重力波的三阶非线性修正,波速不仅取决于波数和各层流体的厚度,而且还与波幅及表面张力有关.     

广义带导数的非线性Schrdinger方程的动态分析和精确解

利用动力系统方法,针对广义带导数的非线性Schrdinger方程的精确解问题进行研究分析.采用行波变换,将其化为常微分方程动力系统;计算出该方程动力系统的首次积分,讨论了系统在不同参数条件下的奇点与相图,得到对应的精确解,包括孤立波解、周期波解、扭结波解和反扭结波解.运用数值模拟的方法,对方程的光滑孤立波解和周期波解等进行了数值模拟.分析计算获得的结果完善了相关文献已有的研究成果.     

有流存在下两层密度分层流体毛细重力波的三阶Stokes波解

以小振幅波理论为基础,利用奇异摄动方法研究了有背景流存在下两层密度成层状态下的毛细重力波,求得了两层密度成层状态下各层流体速度势的三阶解及毛细重力波波面位移的三阶Stokes波解,并讨论了毛细重力波的kelvin-Helmholtz的不稳定性.结果表明在有流存在的情况下,两层密度成层流体毛细重力波的一阶渐近解、频散关系,二阶渐近解及三阶渐近解不仅依赖于各层流体的厚度和密度,也依赖于表面张力和各层流体的背景流流场;毛细重力波的三阶解描述了背景流场与毛细重力波之问的三阶非线性相互作用.对于给定的波数k(实数)毛细重力波可能出现kelvin-Helmholtz不稳定性.     

一维可压缩Navier-Stokes方程组趋向于接触间断波的零耗散极限

本文研究了一维可压缩Navier-Stokes方程组趋向于接触间断波的零耗散极限问题.利用一个新的先验假设及一些精细的能量估计,证明了当可压缩Euler方程组的黎曼问题存在一个接触间断波解时,相应的可压缩Navier-Stokes方程组存在一个整体光滑解,并且当热传导系数κ趋于0时,此光滑解以κ~(7/8)的速率趋向于接触间断波,这里接触间断波的强度不需要小.本文改进了文献[1,2]中的主要结果.     

深水表面波可积发展方程的行波解与分支

本文研究了small-aspect-ratio波方程和深水表面波可积发展方程的行波解问题.利用微分方程定性理论的方法,分析了行波系统的相图分支,获得了孤立波解的精确表达式.     

Simultaneous Subgradient Algorithms for the Generalized Split Variational Inclusion Problem in Hilbert Spaces

In this article, we study the generalized split variational inclusion problem. For this purpose, motivated by the projected Landweber algorithm for the split equality problem, we first present a simultaneous subgradient extragradient algorithm and give related convergence theorems for the proposed algorithm. Next, motivated by the alternating CQ-algorithm for the split equality problem, we propose another simultaneous subgradient extragradient algorithm to study the general split variational inclusion problem. As applications, we consider the split equality problem, split feasibility problem, split variational inclusion problem, and variational inclusion problem in Hilbert spaces.     

First kind symmetric periodic solutions and their stability for the Kepler problem and anisotropic Kepler problem plus generalized anisotropic perturbation

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As an application of this general situation, we discuss the existence of symmetric periodic solutions for the anisotropic Kepler problem plus a generalized anisotropic perturbation, (shortly, p-AKPQ problem) and for the Kepler problem plus a generalized anisotropic perturbation (shortly, p-KPQ problem), as continuation of circular orbits of the two-dimensional Kepler problem. To get this objective, we consider different types of perturbations and then we apply our main result.     

A study of mixed discrete bilevel programs using semidefinite and semi-infinite programming

In this paper, we study the bilevel programming problem with discrete polynomial lower level problem. We start by transforming the problem into a bilevel problem comprising a semidefinite program (SDP for short) in the lower level problem. Then, we are able to deduce some conditions of existence of solutions for the original problem. After that, we again change the bilevel problem with SDP in the lower level problem into a semi-infinite program. With the aid of the exchange technique, for simple bilevel programs, an algorithm for computing a global optimal solution is suggested, the convergence is shown, and a numerical example is given.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.     

Solving semi-infinite programs by smoothing projected gradient method

In this paper, we study a semi-infinite programming (SIP) problem with a convex set constraint. Using the value function of the lower level problem, we reformulate SIP problem as a nonsmooth optimization problem. Using the theory of nonsmooth Lagrange multiplier rules and Danskin’s theorem, we present constraint qualifications and necessary optimality conditions. We propose a new numerical method for solving the problem. The novelty of our numerical method is to use the integral entropy function to approximate the value function and then solve SIP by the smoothing projected gradient method. Moreover we study the relationships between the approximating problems and the original SIP problem. We derive error bounds between the integral entropy function and the value function, and between locally optimal solutions of the smoothing problem and those for the original problem. Using certain second order sufficient conditions, we derive some estimates for locally optimal solutions of problem. Numerical experiments show that the algorithm is efficient for solving SIP.     

Optimal control of a two-dimensional contact problem

We consider a frictionless contact problem with unilateral constraints for a 2D bar. We describe the problem, then we derive its weak formulation, which is in the form of an elliptic variational inequality of the first kind. Next, we establish the existence of a unique weak solution to the problem and prove its continuous dependence with respect to the applied tractions and constraints. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.     

Hyperbolic Yamabe problem

In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the(1 + n)-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n = 2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem;while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.     

On the Main Aspects of the Inverse Conductivity Problem

We consider a nonlinear inverse problem for an elliptic partial differential equation known as the Calder{\''o}n problem or the inverse conductivity problem. Based on several results, we briefly summarize them to motivate this research field. We give a general view of the problem by reviewing the available results for $C^2$ conductivities. After reducing the original problem to the inverse problem for a Schr\"odinger equation, we apply complex geometrical optics solutions to show its uniqueness. After extending the ideas of the uniqueness proof result, we establish a stable dependence between the conductivity and the boundary measurements. By using the Carleman estimate, we discuss the partial data problem, which deals with measurements that are taken only in a part of the boundary.     

Heterogeneous time-harmonic Maxwell equations in bidimensional domains

This note deals with the low-frequency time-harmonic Maxwell equations for a heterogeneous media in bidimensional bounded domains. We propose a three step method to solve this problem. First, we construct an extension of the boundary data solving a scalar Neumann problem for the Laplace operator. Second, we solve a problem in the conductor with an unusual boundary condition of nonlocal type. Third, we solve a boundary value problem in the insulator using the solution calculated in the conductor. Also, this third problem can be reduced to a Neumann problem for the Laplace operator.     

Optimal harvesting for nonlinear age-dependent population dynamics

In this paper, we investigate an optimal harvesting problem for nonlinear age-dependent population dynamics. Using the concept of the normal cone, we also obtain the necessary conditions for optimality for the optimal control problem. Using Ekeland’s variational principle, we demonstrate the existence and uniqueness of solutions for the optimal control problem. Finally, we give the synthesis of the optimal feedback law.