Mathematical Sciences: Differential Equations

将在涉及非线性微分方程组的问题上开展工作。这项工作将集中在三个方面。首先，研究了Painleve超越性与逆散射理论中的某些线性积分方程组之间的关系。复平面上的某些常微分方程只有奇点的极点。这些方程或超越方程与可用逆散射法求解的非线性偏微分方程组有关。目前，只有选定的例子说明了这种联系，但很明显，可以处理一整类方程。将对整个关系进行系统的研究。第二类研究是关于非线性扩散方程解的失效或爆破问题。解的爆破在很大程度上取决于初始条件的性质(以及方程中的非线性项)。由于人们经常要处理多个空间变量的函数，所以爆破可能在不同的时刻在不同的点发生。这就引出了该项目的主要猜想：爆炸必须局限于一组大小可以忽略的空间变量(零度)。第三，研究含有自由基的复杂化学燃烧理论模型的存在性、多解性和渐近性。该模型代表了预混燃烧中稳定的平面火焰前锋。这种分析超越了传统的简单反应。为了支持行波解，大多数模型不得不面对或假设不存在所谓的冷边界困难。如果中间物种或自由基被引入到反应体系中，主要研究者的方案将导致新的火焰层存在的证明。将继续开展工作，扩展所开发的方法，以涵盖其他复杂模型，验证其渐近展开，并考虑多重性或唯一性问题。

Work will be done on problems involving nonlinear differential equations. The work will focus on three areas. In the first, investigations will be carried out on the relation between Painleve transcendents and certain linear integral equations arising in inverse scattering theory. Certain ordinary differential equations in the complex plane have only poles for singularities. These equations, or transcendents, are related to nonlinear partial differential equations solvable by the method of inverse scattering. At this time only selected examples illustrate the connection but it is clear that an entire class of equations can be treated. A systematic study of the entire relationship will be undertaken. A second line of research deals with the failure or blow-up of solutions of nonlinear diffusion equations. Blow-up of solutions depends heavily on the nature of initial conditions (as well as the nonlinearity term in the equation). Since one is often dealing with functions of several space variables, blow-up can occur at different points at different times. This leads to the main conjecture of the project: blow-up must be confined to a set of space variables of negligible size (measure zero). Third, work will be done on existence, multiplicity and asymptotics for combustion theory models with complex chemistry including radicals. The models represent steady planar flame fronts in premixed combustion. This analysis goes beyond traditional simple reactions. Most models have to confront or hypothesize away what is known as the cold boundary difficulty to support traveling wave solutions. If intermediate species or radicals are introduced into a system of reactions, a scheme of the principal investigator leads to new existence proofs for flame layers. Work will proceed in expanding methods developed to cover other complex models, validate their asymptotic expansions and consider problems of multiplicity or uniqueness.