Asymptotic and qualitive analysis of nonlinear partial differential equations

偏微分方程提供了未知函数或函数集的时间和空间变化之间的关系的数学表示。这样的方程在许多环境中以自然的方式产生，包括流体运动、化学反应、发育生物学和空间生态学。通常，这样的方程涉及与例如潜在的饱和效应和模型组件之间的相互作用相对应的非线性效应。这种效果在使模型更加逼真的同时，也使此类方程的解更具挑战性，实际上，除非在某些理想化的情况下，否则精确的闭合形式的解通常是无法获得的。因此，基于某些极限情况下的定性特征的近似和分析的替代方法对于理解模型的性质、其解决方案以及潜在的物理系统变得至关重要。在此背景下，本建议考虑了两类非线性偏微分方程组以及用于研究它们的分析和计算方法。首先，我打算继续正在进行的非线性波动方程的共振解的研究，该方程低于浅层流体中的波，例如地形上的大气流动。关键目标既包括在一般情况下研究解决办法，也包括发展分析方法，而分析方法反过来可能适用于更广泛的问题。第二是研究一类广泛适用且日益流行的方程，这些方程是在化学和生物应用的背景下产生的，涉及扩散和反应过程的组合。这方面研究的重点将是一般地研究某些解特征的稳健性，例如图案形成、波和在扰动效应影响下的长时间行为。此外，随着模型的日益复杂，解的数值逼近正成为越来越重要的分析工具。因此，这项研究还将调查和开发适用于这类模型的更有效的计算方法。

Partial differential equations provide a mathematical representation of the relationship between the temporal and spatial variation of an unknown function or set of functions. Such equations arise in a natural way in numerous settings including fluid motion, chemical reactions, developmental biology, and spatial ecology. In general such equations involve nonlinear effects corresponding to, for example, underlying saturation effects and interactions between model components. Such effects while making the models more realistic, also make the solution of such equations much more challenging and indeed exact closed form solutions are typically not accessible except in certain idealized circumstances. Therefore alternative methodologies based on approximation and analysis of qualitative features in certain limiting circumstances become essential in understanding the nature of the models, their solutions and in turn the underlying physical system. Within this context, this proposal considers two classes of nonlinear partial differential equations and the analytic and computational methodologies used to study them. First I intend to continue an ongoing study of resonant solutions of nonlinear wave equations which underly waves in shallow fluids such as atmospheric flows over topography. The key goals involve both the study of solutions in generalized circumstances, as well as the development of the analytic methodology which in turn may be applicable to a wider range of problems. Second is the study of a widely applicable and increasingly prevalent class of equations which arise in the context of chemical and biological applications involving the combination of diffusive and reaction processes. The focus of this aspect of the study will be to study in general the robustness of certain solution features such as pattern formation, waves and long time behaviour under the influence of perturbative effects. In addition with the growing complexity of the models, numerical approximation of solutions is becoming an increasingly crucial analytic tool. As such this study will also investigate and develop more efficient computational methodologies suited for models of this type.