Frobenius manifold for the dispersionless Kadomtsev-Petviashvili equation

TL;DR

This paper constructs a Frobenius manifold structure related to the dispersionless KP equation, extending the classical hierarchy by developing a set of infinitely many commuting flows using a continuous analogue of finite-dimensional theory.

Contribution

It introduces a novel Frobenius structure for the dispersionless KP equation using Schwartz functions and extends the classical hierarchy with new commuting flows.

Findings

Frobenius potential is a logarithmic function with quadratic external field.

Constructed an extended hierarchy with infinitely many commuting flows.

Established a continuous analogue of finite-dimensional Frobenius theory.

Abstract

We consider a Frobenius structure associated with the dispersionless Kadomtsev-Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic potential with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.