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SUMMARY:Integrable Systems in 4+2 Dimensions and their Reduction to 3+1 Di
 mensions - Maria Christina van der Weele\, University of Cambridge
DTSTART:20190306T160000Z
DTEND:20190306T170000Z
UID:TALK121084@talks.cam.ac.uk
CONTACT:Angeliki Menegaki
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\, who derived equations of this type in four spa
 tial dimensions\, which however had the disadvantage of containing two tim
 e dimensions. The associated initial value problem for such equations\, wh
 ere the dependent variables are specified for all space variables at t1 = 
 t2 = 0\, can be solved by means of a nonlocal d-bar problem.\n\nThe next s
 tep in this program is to formulate and solve nonlinear integrable systems
  in 3+1\ndimensions (i.e.\, with three space variables and a single time v
 ariable) in agreement with physical reality. The method we employ is to fi
 rst construct a system in 4+2 dimensions\,\nwhich we then aim to reduce to
  3+1 dimensions.\n\nIn this talk I will focus on the Davey-Stewartson syst
 em and the 3-wave interaction equations. Both these integrable systems hav
 e their origins in fluid dynamics where they\ndescribe the evolution and i
 nteraction\, respectively\, of wave packets on e.g. a water surface. We st
 art 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 dimensio
 ns by complexifying each of these variables. We solve the initial value pr
 oblem of these equations in 4+2 dimensions. Subsequently\, in the linear l
 imit we reduce this analysis to 3+1 dimensions to comply with the natural 
 world. Finally\, we discuss the construction of the 3+1 reduction of the f
 ull nonlinear problem\, which is currently under investigation.\n\nThis is
  joint work together with my PhD supervisor Prof. A. S. Fokas.\n
LOCATION:MR14\, Centre for Mathematical Sciences
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