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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