Dispersive deformations of hydrodynamic reductions of 2D dispersionless integrable systems

TL;DR

This paper shows how hydrodynamic reductions of 2D dispersionless integrable systems can be uniquely deformed into their dispersive versions, revealing constraints on dispersive terms and offering new insights into 2+1 dimensional integrability.

Contribution

It introduces a method to deform hydrodynamic reductions of dispersionless systems into dispersive ones, establishing uniqueness modulo the Miura group and providing a new perspective on integrability.

Findings

Deformations are unique modulo the Miura group.

Strong constraints on dispersive terms are derived.

An alternative approach to 2+1 dimensional integrability is proposed.

Abstract

We demonstrate that hydrodynamic reductions of dispersionless integrable systems in 2+1 dimensions, such as the dispersionless Kadomtsev-Petviashvili (dKP) and dispersionless Toda lattice (dTl) equations, can be deformed into reductions of the corresponding dispersive counterparts. Modulo the Miura group, such deformations are unique. The requirement that any hydrodynamic reduction possesses a deformation of this kind imposes strong constraints on the structure of dispersive terms, suggesting an alternative approach to the integrability in 2+1 dimensions.