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.

and rational Lax pairs is given in the article. For example, system (40) is a new (and the only

known to date)

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.

Back to my main page.

**Last updated on **December 20, 2022, 14:59 CET