For a long time it appeared that integrable (3+1)-dimensional systems (or, in terms used in physics, integrable 4D classical field theories, in general nonrelativistic and nonlagrangian) are quite exceptional, but in our recent article Multidimensional integrable systems from contact geometry it is shown that, surprisingly, this is not really the case and there is a large new class of (3+1)-dimensional integrable systems with nonisospectral Lax pairs. This new class exists alongside a few previously known important yet isolated examples like (anti-)self-dual Yang--Mills equations or (anti-)self-dual vacuum Einstein equations.
Just like (anti-)self-dual Yang--Mills equations or (anti-)self-dual vacuum Einstein equations with zero cosmological constant, integrable systems from the new class in question are dispersionless, i.e., can be written as first order quasilinear homogeneous systems; their Lax pairs are motivated by contact geometry, namely they involve contact vector fields.
The new class under study includes inter alia the first known example of an integrable (3+1)-dimensional system with a nonisospectral Lax pair which is algebraic, rather than just rational, in the variable spectral parameter, as well as integrable (3+1)-dimensional generalizations for the following (2+1)-dimensional integrable systems: dispersionless BKP, dispersionless asymmetric Nizhnik-Veselov-Novikov, dispersionless Gardner, dispersionless Davey-Stewartson, dispersionless (modified) KP equations, and the generalized Benney system.
You may wish to look at these slides for additional background and motivation before proceeding to the article itself.Â