## Multidimensional Integrability via Geometry

2018-12-29T12:50:02Z (GMT)
by

The search for partial differential systems in four
independent variables (

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.

**(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