On ringing effects near jump discontinuities for periodic solutions to dispersive partial differential equations

Details On ringing effects near jump discontinuities for periodic solutions to dispersive partial differential equations On ringing effects near jump discontinuities for periodic solutions to dispersive partial differential equations

2011-07-08

arxiv.org/abs/1107.1571v1

Mathematics

Analysis of PDEs

22 pages, 10 figures

Scientific paper

We consider weak solutions to dispersive partial differential equations with periodic boundary conditions and initial data with jump discontinuities. These are already known to be continuous at irrational times and piecewise constant at rational times; we show that as time approaches a rational value the solution exhibits a ringing effect, with the characteristic overshoot of fixed amplitude near the discontinuities. Furthermore this effect is the same whether the sequence of times follows rational or irrational values.

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