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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Integrable Systems in Multidimensions - Maria Chri
stina van der Weele (University of Cambridge)
DTSTART;TZID=Europe/London:20191210T090000
DTEND;TZID=Europe/London:20191210T093000
UID:TALK135505AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/135505
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 [1]\, who deriv
ed equations of this type in four spatial dimensio
ns\, which however had the disadvantage of contain
ing two time dimensions. The associated initial va
lue problem for such equations\, where the depende
nt variables are specified for all space variables
at t1 = t2 = 0\, can be solved by means of a nonl
ocal d-bar problem. The next step in this program
is to formulate and solve nonlinear integrable sys
tems in 3+1 dimensions (i.e.\, with three space va
riables and a single time variable) in agreement w
ith physical reality. The method we employ is to f
irst construct a system in 4+2 dimensions\, with t
he aim to reduce this then to 3+1 dimensions.

<
br>In this talk we focus on the Davey-Stewartson s
ystem [2] and the 3-wave interaction equations [3]
. Both these integrable systems have their origins
in fluid dynamics where they describe the evoluti
on and interaction\, respectively\, of wave packet
s on e.g. a water surface. We start from these equ
ations in their usual form in 2+1 dimensions (two
space variables x\, y and one time variable t) and
we bring them to 4+2 dimensions by complexifying
each of these variables. We solve the initial valu
e problem of these equations in 4+2 dimensions. Su
bsequently\, in the linear limit we reduce this an
alysis to 3+1 dimensions to comply with the natura
l world. Finally\, we discuss the construction of
the 3+1 reduction of the full nonlinear problem\,
which is currently under investigation.

Thi
s is joint work together with my PhD supervisor Pr
of. A.S. Fokas.

References

[1] A.S. Fok
as\, Integrable Nonlinear Evolution Partial Differ
ential Equations in 4+2 and 3+1 Dimensions\, Phys.
Rev. Lett. 96 (2006)\, 190201.

[2] A.S. Fokas
and M.C. van der Weele\, Complexification and int
egrability in multidimensions\, J. Math. Phys. 59
(2018)\, 091413.

[3] M.C. van der Weele and A.
S. Fokas\, Solving the Initial Value Problem for t
he 3-Wave Interaction Equations in Multidimensions
(to be submitted\, 2019).
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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