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SUMMARY:Plane wave decompositions: general notes - Andrey Shanin (Moscow S
 tate University)
DTSTART:20240701T133000Z
DTEND:20240701T140000Z
UID:TALK214828@talks.cam.ac.uk
DESCRIPTION:A starting point of solving a diffraction problem is choosing 
 an Ansatz for the field\, and such an Ansatz is usually a plane wave decom
 position. Some of plane waves decompositions follow directly from the Four
 ier analysis (this is the case of reprepresentation of Green's functions)\
 , and some are the matter of guess (the Sommerfeld integrals). The talk de
 scribes common properties of the plane wave decompositions in various phys
 ical situations (2D and 3D Helmholtz equations\, 2D discrete lattice equat
 ion\, WFEM equation for a waveguide\, &nbsp\;Laplace-Beltrami equation on 
 a sphere).\nA starting point of the consideration is introducing of a disp
 ersion manifold of the space\, that is a set of all plane waves possibly a
 dmitted by the equation. For planar geometries\, the plane waves are usual
  complex plane waves\, and for the the sphere the sitiation is a bit more 
 complicated.&nbsp\;\nIn all cases studied in the talk\, we assume that the
  dispersion manifold posseses a structure or complex manifold. Thus\, one 
 can study a "wave field" that is an integral over some contour (or\, more 
 generally\, a cycle) on the diffraction manifold. The integrand is a produ
 ct of the plave wave\, a transformant and a holomorphic differental form. 
 The transformant is assumed to be a meromorphic function of the dispersion
  manifold. The complex structure on the dispersion manifold enables one to
  use the Cauchy's theorem and deform the integration contour if necessary.
 \nThe next step is introducing of "sliding" contours of integration on the
  dispersion manifold. Usually\, it is impossible to describe the field in 
 the whole domain of interest by a single integral. Thus\, the plane wave d
 ecomposition comprises a family of contours\, by means of which the field 
 is described in overlapping domains covering the whole domain of interest.
  For consistency\, the contours should be deformed into each other for the
  areas of overlapping.&nbsp\;\nThe representations of Green's function and
  of solutions of plane wave diffraction problems are different by the choi
 ce of the families of contours. The latter can be described using the Somm
 erfeld contours that can be localized near infinity\, while the former nec
 essarily pass through the finite parts of the dispersion manifold.&nbsp\;\
 nThe talk is based on common works with O.I.Makarov and K.S.Kniazeva from 
 Moscow State University\, and with R.C.Assier\, A.I.Korolkov\, and V.Kunz 
 from the University of Manchester.
LOCATION:Seminar Room 1\, Newton Institute
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