Holomorphic eigenfunctions of the vector field associated with the dispersionless Kadomtsev-Petviashvili equation

TL;DR

This paper proves the existence of holomorphic eigenfunctions in the spectral parameter for the vector field associated with the dispersionless Kadomtsev-Petviashvili equation, advancing spectral analysis in integrable PDEs.

Contribution

It establishes the existence of spectral eigenfunctions that are holomorphic in the spectral parameter for the vector field linked to the dispersionless KP equation, a novel result in the spectral theory of integrable PDEs.

Findings

Eigenfunctions are holomorphic in the spectral parameter in certain strips.

Existence proof for eigenfunctions associated with the dispersionless KP vector field.

Enhances understanding of spectral properties of dispersionless integrable equations.

Abstract

Vector fields naturally arise in many branches of mathematics and physics. Recently it was discovered that Lax pairs for many important multidimensional integrable partial differential equations (PDEs) of hydrodynamic type (also known as dispersionless PDEs) consist of vector field equations. These vector fields have complex coefficients and their analytic, in the spectral parameter, eigenfunctions play an important role in the formulations of the direct and inverse spectral transforms. In this paper we prove existence of eigenfunctions of the basic vector field associated with the celebrated dispersionless Kadomtsev-Petviashvili equation, which are holomorphic in the spectral parameter $\lambda$ in the strips $|\Im\lambda|> C_0$.