@Article{Hwang2022,
author = "Hwang, Guenbo and Moon, Byungsoo",
title = "The d-bar formalism for the modified Veselov-Novikov equation on the half-plane",
journal = "Opuscula Math.",
year = "2022",
volume = "42",
number = "2",
pages = "179--217",
doi = "https://doi.org/10.7494/OpMath.2022.42.2.179",
url = "https://doi.org/10.7494/OpMath.2022.42.2.179",
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.",
keywords = "initial-boundary value problem, integrable nonlinear PDE, spectral analysis, \(d\)-bar",
issn = "1232-9274"
}