Cauchy--Jost function and hierarchy of integrable equations

TL;DR

This paper explores the properties of the Cauchy--Jost function related to the KPII equation, providing explicit forms for the hierarchy of integrable equations and their evolutions using the $ar ext{-} ext{partial}$ problem.

Contribution

It introduces a compact, explicit formulation of the KPII hierarchy and its evolutions via the $ar ext{-} ext{partial}$ problem for the Cauchy--Jost function.

Findings

Explicit form of KPII hierarchy equations

Unified description of potential and solution evolutions

Application of $ar ext{-} ext{partial}$ problem to integrable systems

Abstract

Properties of the Cauchy--Jost (known also as Cauchy--Baker--Akhiezer) function of the KPII equation are described. By means of the $\bar\partial$-problem for this function it is shown that all equations of the KPII hierarchy are given in a compact and explicit form, including equations on the Cauchy--Jost function itself, time evolutions of the Jost solutions and evolutions of the potential of the heat equation.