Open AccessThe d-bar formalism for the modified Veselov-Novikov equation on the half-plane
Author(s) -
Guenbo Hwang,
Byungsoo Moon
Publication year - 2022
Publication title -
rocznik akademii górniczo-hutniczej im. stanisława staszica. opuscula mathematica/opuscula mathematica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 16
eISSN - 2300-6919
pISSN - 1232-9274
DOI - 10.7494/opmath.2022.42.2.179
Subject(s) - novikov self consistency principle , mathematics , mathematical analysis , inverse scattering transform , formalism (music) , korteweg–de vries equation , inverse , boundary value problem , mathematical physics , inverse scattering problem , pure mathematics , inverse problem , geometry , physics , quantum mechanics , nonlinear system , art , musical , visual arts
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.