Multidimensional beyond-all-orders asymptotics for problems in fluid mechanics

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In recent years, techniques in beyond-all-orders asymptotics or exponential asymptotics have led to significant breakthroughs in application areas ranging from water waves and interfacial flows, to pattern formation and quantum field theory. In such problems of interest, the objective is to understand the deep structure of asymptotic approximations beyond all orders (hence including algebraic and exponential corrections). In water-wave theories, for instance, such beyond-all-orders contributions have been shown to govern ripples on free-surface waves in singular limits, such as in the small surface-tension limit. However, while enormously successful, such techniques have been primarily limited to applications that are modeled via nonlinear ordinary differential equations (ODEs). Application to partial differential equations and similar multi-dimensional spatial and temporal systems remains very limited. This PhD will investigate a number of fluid mechanical problems where such tools are crucially in need of development. Such problems may include: (i) single and multi-dimensional reduced shallow-water (Korteweg-de Vries) models; (ii) models for free-surface flows with wave-structure interactions; (iii) time-dependent breathers in the Euler equations.