Multidimensional integrable systems from contact geometry

Upon having presented a bird’s eye view of history of integrable systems, we give a brief review of certain earlier advances (arXiv:1401.2122 & arXiv:1812.02263) in the longstanding problem of search for partial differential systems in four independent variables, often referred to as (3+1)-dimensional or 4D systems, that are integrable in the sense of soliton theory. Namely, we review a recent construction for a large new class of (3+1)-dimensional integrable systems with Lax pairs involving contact vector fields. This class contains inter alia two infinite families of such systems, thus establishing that there is significantly more integrable (3+1)-dimensional systems than it was believed for a long time. In fact, the construction under study yields (3+1)-dimensional integrable generalizations of many well-known dispersionless integrable (2+1)-dimensional systems like the dispersionless KP equation, as well as a first example of a (3+1)-dimensional integrable system with an algebraic, rather than rational, nonisospectral Lax pair. To demonstrate the versatility of the construction in question, we employ it here to produce novel integrable (3+1)-dimensional generalizations for the following (2+1)-dimensional integrable systems: dispersionless BKP, dispersionless asymmetric Nizhnik–Veselov–Novikov, dispersionless Gardner, and dispersionless modified KP equations, and the generalized Benney system.

💡 Research Summary

The paper surveys recent advances in the construction of integrable partial differential systems in four independent variables—commonly referred to as (3+1)-dimensional or 4‑D systems—by exploiting contact geometry. After a brief historical overview of integrability, the author focuses on a systematic method introduced in earlier works (arXiv:1401.2122, arXiv:1812.02263) that produces a large new class of dispersionless (hydrodynamic‑type) equations possessing Lax pairs built from contact vector fields.

The central object is a contact Hamiltonian (h(p,u)) defined on a three‑dimensional contact manifold with local coordinates ((x,z,p)) and contact one‑form (dz+p,dx). From (h) one constructs the contact vector field \