Nonlinear effects in steady radiating waves: An exponential asymptotics approach

Exponential Asymptotics and Capillary Waves

Recently developed techniques in exponential asymptotics beyond all orders are employed on the problem of potential flows with a free surface and small surface tension, in the absence of gravity. Exponentially small capillary waves are found to be generated on the free surface where the equipotentials from singularities in the flow (for example, stagnation points and corners) meet it. The ampli...

Semiclassical Asymptotics for Weakly Nonlinear Bloch Waves

We study the simultaneous semi-classical and adiabatic asymptotics for a class of (weakly) nonlinear Schrödinger equations with a fast periodic potential and a slowly varying confinement potential. A rigorous two-scale WKB–analysis, locally in time, is performed. The main nonlinear phenomenon is a modification of the Berry phase.

Steady and unsteady nonlinear internal waves incident on an interface

Steady nonlinear internal waves are commonly described by the Dubreil-JacotinLong equation. This equation contains unknown functions of the stream function, representing the density and vorticity fields. Often these are determined by upstream conditions where the flow is assumed to be known. But for the case when the waves are periodic in the horizontal direction, these functions need to be det...

Exponential Asymptotics

Nonlinear Asymptotics for Hyperbolic Internal Waves of Small Width

In the limit ε → 0, the solutions converge to piecewise smooth functions which are discontinuous accross a characteristic surface. Such solutions have source terms which are also piecewise smooth. Discontinuous sources are idealizations of smooth sources with a thin (∼ ε) transition layer. The fundamental problem addressed here is to describe the dynamics of solutions with sources with such tra...