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We consider weak solutions to dispersive partial differential equations withperiodic boundary conditions and initial data with jump discontinuities. Theseare already known to be continuous at irrational times and piecewise constantat rational times; we show that as time approaches a rational value thesolution exhibits a ringing effect, with the characteristic overshoot of fixedamplitude near the discontinuities. Furthermore this effect is the same whetherthe sequence of times follows rational or irrational values.

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