Exactly solvable models on 4D spacetime are not exceptional

What is it about?

The search for exactly solvable (a.k.a. integrable in the sense of soliton theory, i.e. roughly speaking as nice as the celebrated Korteweg -- de Vries equation -- having plenty of exact solutions, infinitely many conservation laws, etc.) partial differential systems in four independent variables ((3+1)D or 4D) is a longstanding problem in mathematical physics. In the present paper we address this problem by introducing a new systematic and effective construction for integrable 4D systems using a novel kind of Lax pairs inspired by contact geometry. In particular, we show that there is significantly more integrable (3+1)D systems than it appeared before.

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Why is it important?

Searching for new integrable partial differential systems in four independent variables is among the most important challenges in the modern mathematical physics, because the spacetime we live in is four-dimensional. For a long time it appeared that such systems are very exceptional and cannot be effectively constructed in a systematic fashion. The present article dispels this illusion by presenting an efficient systematic construction of a large new class of systems in question. It turns out that the construction under study is closely related to contact geometry on 3-manifolds.