Research of Prof. Artur Sergyeyev on Multidimensional Integrability

The search for partial differential systems in four independent variables (4D or (3+1)D
for short) that are integrable in the sense of soliton theory is a longstanding problem of
mathematical physics as our spacetime is four-dimensional and thus the (3+1)D case is
especially relevant for applications. The above article addresses this problem and proves
that integrable 4D 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 article. For example, system (40) 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.

You may wish to look at the recent slides for additional background and motivation
before proceeding to the article itself.