Research of Artur Sergyeyev on Multidimensional Integrability
My key recent publication on the subject is
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
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.
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Last updated on December 20, 2022, 14:59 CET