Analysis Seminar

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Gibbs-Like Behavior of Dispersive PDEs

Speaker: Tom Trogdon, Courant

Location: Warren Weaver Hall 1302

Date: Thursday, October 16, 2014, 11 a.m.

Synopsis:

The classical Gibbs phenomenon is an artifact of non-uniform convergence. More precisely, it arises from the approximation of a discontinuous function with an analytic partial sum of the Fourier series. It is known from the work of DiFranco and McLaughlin (2005) that a similar phenomenon occurs when a box initial condition is taken for the free Schrödinger equation in the short-time limit. This talk is focused extending the linear theory of this work in two ways. First, we establish sufficient conditions for the classical smoothness of the solutions of linear dispersive equations for positive times. Second, we derive a highly-oscillatory and computable short-time asymptotic expansion of the solution of general linear dispersive PDEs with a large class of discontinuous initial data. Boundary-value problems can also be treated.