BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Integrable Systems in Multidimensions - Maria Christina van der We
 ele (University of Cambridge)
DTSTART:20191210T090000Z
DTEND:20191210T093000Z
UID:TALK135505@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:One of the main current topics in the field of integrable syst
 ems concerns the existence of nonlinear integrable evolution equations in 
 more than two spatial dimensions. The fact that such equations exist has b
 een proven by A.S. Fokas [1]\, 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 nonlocal d-bar problem. The next s
 tep in this program is to formulate and solve nonlinear integrable systems
  in 3+1 dimensions (i.e.\, with three space variables and a single time va
 riable) in agreement with physical reality. The method we employ is to fir
 st construct a system in 4+2 dimensions\, with the aim to reduce this then
  to 3+1 dimensions.<br><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 evolu
 tion and interaction\, respectively\, of wave packets on e.g. a water surf
 ace. We start from these equations 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 initia
 l value problem of these equations in 4+2 dimensions. Subsequently\, in th
 e linear limit we reduce this analysis to 3+1 dimensions to comply with th
 e natural world. Finally\, we discuss the construction of the 3+1 reductio
 n of the full nonlinear problem\, which is currently under investigation.<
 br><br>This is joint work together with my PhD supervisor Prof. A.S. Fokas
 .<br><br>References<br> [1] A.S. Fokas\, Integrable Nonlinear Evolution Pa
 rtial Differential Equations in 4+2 and 3+1 Dimensions\, Phys. Rev. Lett. 
 96 (2006)\, 190201.<br> [2] A.S. Fokas and M.C. van der Weele\, Complexifi
 cation and integrability in multidimensions\, J. Math. Phys. 59 (2018)\, 0
 91413.<br> [3] M.C. van der Weele and A.S. Fokas\, Solving the Initial Val
 ue Problem for the 3-Wave Interaction Equations in Multidimensions (to be 
 submitted\, 2019).
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR