Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see e.g. references below). They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system.

The dispersionless Kadomtsev–Petviashvili equation (dKPE), also known (up to an inessential linear change of variables) as the Khokhlov–Zabolotskaya equation, has the form

It arises from the commutation

of the following pair of 1-parameter families of vector fields

where ${\displaystyle \lambda }$ is a spectral parameter. The dKPE is the ${\displaystyle x}$-dispersionless limit of the celebrated Kadomtsev–Petviashvili equation, arising when considering long waves of that system. The dKPE, like many other (2+1)-dimensional integrable dispersionless systems, admits a (3+1)-dimensional generalization.

The dispersionless KP system is closely related to the Benney moment hierarchy, each of which is a dispersionless integrable system:

These arise as the consistency condition between

and the simplest two evolutions in the hierarchy are: