Opuscula Math. 42, no. 2 (2022), 179-217
https://doi.org/10.7494/OpMath.2022.42.2.179

 
Opuscula Mathematica

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

Guenbo Hwang
Byungsoo Moon

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.

Mathematics Subject Classification: 35G31, 35Q53, 37K15.

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  • Guenbo Hwang
  • Department of Mathematics and Institute of Natural Sciences, Daegu University, Gyeongsan Gyeongbuk 38453, Korea
  • Byungsoo Moon (corresponding author)
  • Department of Mathematics, Incheon National University, Incheon 22012, Korea
  • Communicated by Runzhang Xu.
  • Received: 2021-06-11.
  • Revised: 2021-07-28.
  • Accepted: 2021-08-09.
  • Published online: 2022-02-25.
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Cite this article as:
Guenbo Hwang, Byungsoo Moon, The d-bar formalism for the modified Veselov-Novikov equation on the half-plane, Opuscula Math. 42, no. 2 (2022), 179-217, https://doi.org/10.7494/OpMath.2022.42.2.179

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