Multidimensional Integrability via Geometry

2018-12-29T12:50:02Z (GMT) by Artur Sergyeyev
<div> The search for partial differential systems in four independent variables (<b>(3+1)D</b> or <b>4D</b> for short) that are <i>integrable</i> in the sense of <a href="" rel="nofollow" target="_blank">soliton theory</a> is a longstanding problem of mathematical physics as according to general relativity our spacetime is four-dimensional, and thus the (3+1)D case is particularly relevant for applications. <br><br>We address this problem and prove that integrable (3+1)D 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.<br></div><div><br></div><div>Explicit form of two infinite families of integrable (3+1)D systems from this class with polynomial and rational Lax pairs is given in the talk. For example, system (10) is a new (and the only known to date) <i>integrable</i> 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.<br></div><div><br></div><div>For further details please see the paper <br></div><div><br><a href="" rel="nofollow" target="_blank">A. Sergyeyev,<b> New integrable (3+1)-dimensional systems and contact geometry</b>, Lett. Math. Phys. 108 (2018), no. 2, 359-376 (arXiv:1401.2122)</a></div><div><br></div><div>and <a href=""></a><br></div><div><br> </div>