Multidimensional Integrability via Geometry

The search for partial differential systems in four independent variables ((3+1)D or 4D for short) that are integrable in the sense of soliton theory is a longstanding problem of mathematical physics as according to general relativity our spacetime is four-dimensional, and thus the (3+1)D case is particularly relevant for applications.

We address this problem and prove that integrable (3+1)D systems are significantly less exceptional than it appeared before: in addition to a handful of well-known important yet isolated examples like the (anti)self-dual Yang--Mills equations there is a large new class of integrable (3+1)D systems with Lax pairs of a novel kind related to contact geometry.

Explicit form of two infinite families of integrable (3+1)D systems from this class with polynomial and rational Lax pairs is given in the talk. For example, system (10) is a new (and the only known to date) integrable generalization from three to four independent variables for the Khokhlov--Zabolotskaya equation, also known as the dispersionless Kadomtsev--Petviashvili equation or the Lin--Reissner--Tsien equation and having many applications in nonlinear acoustics and fluid dynamics.

For further details please see the paper

and https://sites.google.com/site/artursergyeyev/research