Research

The search for partial differential systems that can be solved exactly rather than just numerically, that is, are integrable in the sense of soliton theory, is a longstanding problem in mathematical physics. According to general relativity our spacetime is four-dimensional, so the search for integrable partial differential systems in four independent variables, also known as integrable (3+1)-dimensional systems, is particularly relevant inter alia for applications in physics, where such systems are referred to as classical integrable 4D field theories (in general nonrelativistic and non-Lagrangian).

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 was believed for a long time: 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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