Research of Artur
Sergyeyev on Multidimensional Integrability
The
search for partial differential systems in four independent
variables (known as (3+1)-dimensional, or (3+1)D, for short) that are integrable in the sense
of soliton theory (in
terms used in physics, these are classical 4D integrable
field theories, in general non-relativistic and
non-Lagrangian) is a longstanding problem of
mathematical physics, naturally motivated by our spacetime
being four-dimensional according to Einstein's general
relativity.
The recent article Multidimensional
integrable systems from contact geometry reviews our
results addressing this problem in a positive fashion and
shows, in particular, that integrable (3+1)-dimensional
systems are significantly less exceptional than it appeared
before: it turns out that in addition to a handful of
well-known important yet isolated examples like the
(anti)self-dual Yang--Mills equations or (anti)self-dual
vacuum Einstein equations there is a large new class
containing infinitely many integrable (3+1)-dimensional
systems with Lax pairs of a novel kind related to contact
geometry.
You may wish to look also e.g. at this video
and the accompanying slides for
additional background and motivation before proceeding to the
article itself.
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