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CATEGORIES:Cambridge Analysts' Knowledge Exchange
SUMMARY:Integrable Systems in 4+2 Dimensions and their Red
uction to 3+1 Dimensions - Maria Christina van der
Weele\, University of Cambridge
DTSTART;TZID=Europe/London:20190306T160000
DTEND;TZID=Europe/London:20190306T170000
UID:TALK121084AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/121084
DESCRIPTION:One of the main current topics in the field of int
egrable systems concerns the existence of nonlinea
r integrable evolution equations in more than two
spatial dimensions. The fact that such equations e
xist has been proven by A. S. Fokas\, who derived
equations of this type in four spatial dimensions\
, which however had the disadvantage of containing
two time dimensions. The associated initial value
problem for such equations\, where the dependent
variables are specified for all space variables at
t1 = t2 = 0\, can be solved by means of a nonloca
l d-bar problem.\n\nThe next step in this program
is to formulate and solve nonlinear integrable sys
tems in 3+1\ndimensions (i.e.\, with three space v
ariables and a single time variable) in agreement
with physical reality. The method we employ is to
first construct a system in 4+2 dimensions\,\nwhic
h we then aim to reduce to 3+1 dimensions.\n\nIn t
his talk I will focus on the Davey-Stewartson syst
em and the 3-wave interaction equations. Both thes
e integrable systems have their origins in fluid d
ynamics where they\ndescribe the evolution and int
eraction\, respectively\, of wave packets on e.g.
a water surface. We start from these equations in
their usual form in 2+1 dimensions (two space vari
ables x\, y and one time variable t) and we bring
them to 4+2 dimensions by complexifying each of th
ese variables. We solve the initial value problem
of these equations in 4+2 dimensions. Subsequently
\, in the linear limit we reduce this analysis to
3+1 dimensions to comply with the natural world. F
inally\, we discuss the construction of the 3+1 re
duction of the full nonlinear problem\, which is c
urrently under investigation.\n\nThis is joint wor
k together with my PhD supervisor Prof. A. S. Foka
s.\n
LOCATION:MR14\, Centre for Mathematical Sciences
CONTACT:Angeliki Menegaki
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