The d-bar formalism for the modified Veselov-Novikov equation on the half-plane

Abstract

We study the modified Veselov-Novikov equation (mVN) posed on the half-plane via the Fokas method, considered as an extension of the inverse scattering transform for boundary value problems. The mVN equation is one of the most natural $(2+1)$-dimensional generalization of the $(1+1)$-dimensional modified Korteweg-de Vries equation in the sense as to how the Novikov-Veselov equation is related to the Korteweg-de Vries equation. In this paper, by means of the Fokas method, we present the so-called global relation for the mVN equation, which is an algebraic equation coupled with the spectral functions, and the $d$-bar formalism, also known as Pompieu's formula. In addition, we characterize the $d$-bar derivatives and the relevant jumps across certain domains of the complex plane in terms of the spectral functions.