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SUMMARY:The Maxwell-Bloch System in the Sharp-Line Limit - Peter Miller (U
 niversity of Michigan)
DTSTART:20220905T133000Z
DTEND:20220905T143000Z
UID:TALK177926@talks.cam.ac.uk
DESCRIPTION:We study the characteristic Cauchy problem for the Maxwell-Blo
 ch system that describes the interaction of an optical pulse with an activ
 e quantum medium. It is well known that in the sharp-line limit that the a
 toms in the medium are not Doppler-shifted in frequency\, this system can 
 be embedded in the integrable hierarchy of the nonselfadjoint Zakharov-Sha
 bat spectral problem. However\, it is also known that there are certain di
 fficulties with formulating and using the inverse-scattering transform bas
 ed on this spectral problem in the usual way. We construct a Riemann-Hilbe
 rt problem that returns the unique causal solution of the Cauchy problem a
 nd use it to explain features of solutions such as the stimulated decay by
  a suitable optical pulse of an unstable medium to its stable state and th
 e spontaneous generation of a dispersive tail of the optical pulse with po
 sitive time that ruins absolute integrability that would be needed for the
  standard inverse-scattering transform to make sense. This tail is related
  to a specific self-similar solution of the Maxwell-Bloch system that in t
 urn is connected with a concrete special solution of the Painlev&eacute\;-
 III equation that has become important in several recent application probl
 ems for the focusing nonlinear Schr&ouml\;dinger equation.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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