Integrable Systems in Multidimensions

CATW03 - Computational complex analysis

One of the main current topics in the field of integrable systems concerns the existence of nonlinear integrable evolution equations in more than two spatial dimensions. The fact that such equations exist has been 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 step 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 variable) in agreement with physical reality. The method we employ is to first construct a system in 4+2 dimensions, with the aim to reduce this then to 3+1 dimensions.

In this talk we focus on the Davey-Stewartson system [2] and the 3-wave interaction equations [3]. Both these integrable systems have their origins in fluid dynamics where they describe the evolution and interaction, 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 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 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. Finally, we discuss the construction of the 3+1 reduction of the full nonlinear problem, which is currently under investigation.

This is joint work together with my PhD supervisor Prof. A.S. Fokas.

References
[1] A.S. Fokas, Integrable Nonlinear Evolution Partial Differential 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 integrability in multidimensions, J. Math. Phys. 59 (2018), 091413.
[3] M.C. van der Weele and A.S. Fokas, Solving the Initial Value Problem for the 3-Wave Interaction Equations in Multidimensions (to be submitted, 2019).

This talk is part of the Isaac Newton Institute Seminar Series series.