Uniform Asymptotics Near Standard And Nonstandard Caustics For Linear Dispersive Water Waves Generated By Localized Source

Duration: 36 mins 19 secs

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Description:	Sergey Dobrokhotov Institute for Problems in Mechanics of Russian Academy of Sciences 30 June 2021 – 14:30 to 15:00


Created:	2021-07-01 11:52
Collection:	New connections in number theory and physics
Publisher:	Isaac Newton Institute for Mathematical Sciences
Copyright:	Dobrokhotov, S
Language:	eng (English)


Abstract:	From a geometric point of view, caustics are singularities of Lagrangian manifolds in phase space that define the wave field described by the corresponding (pseudo)differential equation. In many problems, Lagrangian manifolds are smooth, then we have standard caustics, which are the singularities of projecting a manifold into a configuration space. On the other hand, in many interesting situations, Lagrangian non-smooth manifolds appear, and the singularities of such manifolds we call "non-standard" caustics. In problems about water waves, these include, for example, the coastline and the leading edge front of the wave generated by localized sources. We discuss a recently proposed effective method for constructing uniform asymptotics in the form of special functions in large caustic neighborhoods for problems of dispersed water waves. In some cases, (for example, in the case of a constant bottom) the resulting asymptotics are global. One of our main observations is that the asymptotic solutions are represented in a form of parametrically given functions, and the appropriate parameters are the coordinates on a Lagrangian manifold defining this solution. We also discuss the passage from a strong dispersion situation to a weak dispersion one and the role of the source type in obtaining explicit asymptotic formulas. The talk is based on joint work with V.E.Nazaikinskii and A.A.Tolchennikov supported by Russian Science Foundation.

Abstract:

From a geometric point of view, caustics are singularities of Lagrangian manifolds in phase space that define the wave field described by the corresponding (pseudo)differential equation. In many problems, Lagrangian manifolds are smooth, then we have standard caustics, which are the singularities of projecting a manifold into a configuration space. On the other hand, in many interesting situations, Lagrangian non-smooth manifolds appear, and the singularities of such manifolds we call "non-standard" caustics. In problems about water waves, these include, for example, the coastline and the leading edge front of the wave generated by localized sources. We discuss a recently proposed effective method for constructing uniform asymptotics in the form of special functions in large caustic neighborhoods for problems of dispersed water waves. In some cases, (for example, in the case of a constant bottom) the resulting asymptotics are global. One of our main observations is that the asymptotic solutions are represented in a form of parametrically given functions, and the appropriate parameters are the coordinates on a Lagrangian manifold defining this solution. We also discuss the passage from a strong dispersion situation to a weak dispersion one and the role of the source type in obtaining explicit asymptotic formulas.

The talk is based on joint work with V.E.Nazaikinskii and A.A.Tolchennikov supported by Russian Science Foundation.

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