一个2+1维可积方程的代数几何解

该文从1+1维的孤子方程出发,构造出一个2+1维在Lax意义下可积的方程.接着这个2+1维可积方程被分解为可解的常微分方程.随后引入超椭圆Riemann曲面和Abel-Jacobi坐标把流进行了拉直.再利用Riemannθ函数给出了这个2+1维方程的代数几何解.     

交通流流体力学模型与非线性波

介绍了交通流问题中的流体力学描述方法,分析了交通流在受压力和自驱动力等因素作用下所产生的非线性波动现象.这些描述包括LWR运动学模型,考虑动力学效应的高阶模型,考虑超车效应的多车种LWR(Lighthill-Whitham-Richards)模型,以及考虑流通量间断的模型方程.此外,还介绍了LWR网络推广模型在交叉口的Riemann问题求解;提出了描述二维行人流问题的Navier-Stokes-Eikon方程模型并描述了确定行人流运动期盼方向的基本思想.     

非线性退化波方程的Riemann问题

研究了弹性力学中一退化波方程的Riemann问题.其应力函数非凸非凹,从而使得激波条件退化.通过引入广义激波条件下的退化激波,构造性地得到了各种情形下Riemann问题的整体解.     

非稳态光传输的微分几何描述

引入四维Riemann流形描述光束在介质中的传输过程,并在Cartan结构方程的基础上研究了光束的传输方程并给出了其测地线方程解释和曲率解释,在此基础上研究了该系统的守恒流性质及其推论.     

一个(3+1)维非线性方程的B(a)cklund变换和精确解

利用双Bell多项式方法构造了一个(3+1)维非线性方程的双线性形式,得到了该方程的双线性Bcklund变换和相应的Lax对.同时利用Riemann theta函数,获得了该方程的周期波解.     

(3+1)维Klein-Gordon方程的新求解方法

给出第一种椭圆方程与函数变换相结合的方法,通过几个步骤,构造了(3+1)维Klein-Gordon方程的多种新解.步骤一、根据Jacobi椭圆函数的性质,获得了第一种椭圆方程的几种新解.步骤二、用第一种椭圆方程与函数变换相结合的方法,将(3+1)维Klein-Gordon方程的求解问题转化为非线性代数方程的求解问题.步骤三、借助符号计算系统Mathematica求出该方程组的解,并构造了由Riemannθ函数、Jacobi椭圆函数、双曲函数和三角函数两两组合的双周期解和双孤子解等多种复合型新解.     

一个(3+1)维非线性方程的B(a)cklund变换和精确解

利用双Bell多项式方法构造了一个(3+1)维非线性方程的双线性形式,得到了该方程的双线性B(a)cklund变换和相应的Lax对.同时利用Riemann theta函数,获得了该方程的周期波解.     

广义Broer-Kaup-Kupershmidt孤子方程的拟周期解

该文从新谱问题出发,得到一个新的(2+1)-维广义Broer-Kaup-Kupershmidt孤子方程在Lax对非线性化下被分解成可积的常微分方程.接着,给出了一个有限维Hamilton系统并且证明在Liouville意义下是完全可积的.通过引进Abel-Jacobi坐标把Hamilton流进行了拉直,借助Riemannθ函数得到了(2+1)-维Broer-Kaup-Kupershmidt孤子方程的拟周期解.     

Chaplygin气体带有三片常状态初值的二维Riemann问题

本文研究了具有三片常状态初值的Chaplygin气体的Euler方程组的二维Riemann问题.三片常状态初值由y轴的正半轴和x轴来划分.假设原点外的初始数据的每一次间断都恰好只产生一个平面激波、中心稀疏波或滑移面,我们利用广义特征分析的方法详细给出了解的结构.事实上,我们将其分为十种情形进行讨论,并说明其中只有四个子情形是合理的.     

二维双曲守恒律的大时间步长Godunov方法(英文)

考虑了关于二维守恒律的大时间步长Godunov方法.该方法是关于一维问题的自然推广.证明了文中给出的数值流函数下,该方法是守恒的.进一步还给出了近似Riemann解应满足的条件,并且证明了利用满足这些条件的近似Riemann解的大时间步长Godunov方法守恒.最后,补充证明了满足这些条件的近似Riemann解是满足熵条件的.     

A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUGH COMPLIANT AXISYMMETRIC VESSELS

We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisymmetric vessels.Early models derived are nonconservative and/or nonhomogeneous with measure source terms,which are endowed with infinitely many Riemann solutions for some Riemann data.In this paper,we derive a one-dimensional hyperbolic system that is conservative and homogeneous.Moreover,there exists a unique global Riemann solution for the Riemann problem for two vessels with arbitrarily large Riemann data,under a natural stability entropy criterion.The Riemann solutions may consist of four waves for some cases.The system can also be written as a 3×3 system for which strict hyperbolicity fails and the standing waves can be regarded as the contact discontinuities corresponding to the second family with zero eigenvalue.     

相变演化Suliciu模型中波的相互作用

描述相变演化的Suliciu模型，其基本波可由行波分析得到．对于任何给定分两段常值的初始状态，相应的Riemann解是某些基本波的组合．对分三段常值的初始状态，解由上述Piemann解构成，其中相邻两状态间以基本波连接．当基本波发生碰撞时，新的Riemann问题形成．通过研究这些Riemann。问题，可以在适当的参数空间中对基本波之间复杂的相互作用加以分类．     

Rarefaction Wave Interaction and Shock-Rarefaction Composite Wave Interaction for a Two-Dimensional Nonlinear Wave System

In order to construct global solutions to two-dimensional(2 D for short)Riemann problems for nonlinear hyperbolic systems of conservation laws,it is important to study various types of wave interactions.This paper deals with two types of wave interactions for a 2 D nonlinear wave system with a nonconvex equation of state:Rarefaction wave interaction and shock-rarefaction composite wave interaction.In order to construct solutions to these wave interactions,the authors consider two types of Goursat problems,including standard Goursat problem and discontinuous Goursat problem,for a 2 D selfsimilar nonlinear wave system.Global classical solutions to these Goursat problems are obtained by the method of characteristics.The solutions constructed in the paper may be used as building blocks of solutions of 2 D Riemann problems.     

The asymptotic limits of Riemann solutions for the isentropic drift-flux model of compressible two-phase flows

The formation of vacuum state and delta shock wave are observed and studied in the limits of Riemann solutions for the one-dimensional isentropic drift-flux model of compressible two-phase flows by letting the pressure in the mixture momentum equation tend to zero. It is shown that the Riemann solution containing two rarefaction waves and one contact discontinuity turns out to be the solution containing two contact discontinuities with the vacuum state between them in the limiting situation. By comparison, it is also proved rigorously in the sense of distributions that the Riemann solution containing two shock waves and one contact discontinuity converges to a delta shock wave solution under this vanishing pressure limit.     

Conservation Laws Possessing Contact Characteristic Fields with Singularities

In many applications, there arise systems of two nonlinear conservation laws with a single linearly degenerate characteristic field, or contact field, the speed of which may coincide with that of the genuinely nonlinear characteristic field along a curve. Along this coincidence curve, the contact field may have isolated singular points. We prove that under generic assumptions the singular points can be centers or saddles for the contact field. We construct the local Riemann solution for each of the two generic cases. This work sheds light on the classification of local Riemann solutions of systems of two conservation laws with a linearly degenerate characteristic field.     

The singular solutions to a nonsymmetric system of Keyfitz-Kranzer type with initial data of Riemann type

The solutions to the Riemann problem for a nonsymmetric system of Keyfitz-Kranzer type are constructed explicitly when the initial data are located in the quarter phase plane. In particular, some singular hyperbolic waves are discovered when one of the Riemann initial data is located on the boundary of the quarter phase plane, such as the delta shock wave and some composite waves in which the contact discontinuity coincides with the shock wave or the wave back of rarefaction wave. The double Riemann problem for this system with three piecewise constant states is also considered when the delta shock wave is involved. Furthermore, the global solutions to the double Riemann problem are constructed through studying the interaction between the delta shock wave and the other elementary waves by using the method of characteristics. Some interesting nonlinear phenomena are discovered during the process of constructing solutions; for example, a delta shock wave is decomposed into a delta contact discontinuity and a shock wave.     

气体动力学燃烧模型的广义Riemann问题

考察了在(x,t)平面上原点(t>0)的邻域内气体动力学燃烧模型的广义Riemann问题.在改进的熵条件下构造了此问题的唯一解.它们是自相似ZND燃烧模型的极限.发现对某些情形,广义Riemann问题的解与相应的Riemann问题的解有本质的不同.特别地,扰动会使得相应Rie-mann问题的强爆轰波转化为由预压激波点燃的弱爆燃波.在一些情形,尽管相应的Riemann解中不含燃烧波,扰动后燃烧波会出现.这反映了未燃气体的不稳定性.     

非等熵Chaplygin气体极限黎曼解 关于扰动的依赖性

主要研究了非等熵Chaplygin气体黎曼问题初值扰动后解的结构,分析了经典的黎曼问题和扰动问题解的结构及极限结构,发现后者的极限解在δ质量权趋于零时不同于前者解的结构.该结果表明对非等熵Chaplygin气体而言,经典的黎曼问题与带δ初值的黎曼问题有着本质的区别.     

Periodic wave solutions and asymptotic analysis of the Hirota–Satsuma shallow water wave equation

In this paper, we devise a simple way to explicitly construct the Riemann theta function periodic wave solution of the nonlinear partial differential equation. The resulting theory is applied to the Hirota–Satsuma shallow water wave equation. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function. We obtain the one‐periodic and two‐periodic wave solutions of the equation. The relations between the periodic wave solutions and soliton solutions are rigorously established. Copyright © 2014 John Wiley & Sons, Ltd.     

Wave front sets of the Riemann function of elastic interface problems

We obtain an inner and an outer estimates of wave front sets and analytic wave front sets of the Riemann function of elastic interface problems by using the localization method due to Wakabayashi. In our problem the outer estimate of wave front sets and analytic wave front sets of the Riemann function coincides with the inner estimate of those. The strong point of our results is to catch the lateral wave as well as the incident, the reflected, and the refracted waves. Copyright © 2004 John Wiley & Sons, Ltd.