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关系的数学表示。这样的方程以自然的方式出现在许多环境中，包括流体运动、化学反应、发育生物学和空间生态学。一般来说，这样的方程涉及对应于例如潜在的饱和效应和模型组件之间的相互作用的非线性效应。这样的效果虽然使模型更真实，但也使这样的方程的求解更具挑战性，并且实际上，除了在某些理想化的情况下，精确的封闭形式解通常是不可访问的。因此，替代方法的基础上近似和分析的定性特征，在某些限制的情况下，成为必不可少的理解模型的性质，他们的解决方案，反过来，基本的物理系统。在此背景下，该建议认为两类非线性偏微分方程和分析和计算方法来研究它们。首先，我打算继续进行非线性波动方程的共振解的研究，这是浅层流体（如地形上的大气流动）中波动的基础。主要目标涉及在一般情况下的解决方案的研究，以及分析方法的发展，这反过来又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普遍的方程，这些方程出现在涉及扩散和反应过程相结合的化学和生物应用的背景下。这方面的研究的重点将是研究一般的鲁棒性的某些解决方案的功能，如图案的形成，波和长时间的行为下的扰动效应的影响。此外，随着模型的日益复杂，解的数值近似正成为越来越重要的分析工具。因此，本研究还将研究和开发适用于这种类型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.