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

Hwang Guenbo, Moon Byungsoo

Opuscula Mathematica

2022

Vol. 42, no. 2

179--217

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.