A discrete analogue of the modified Novikov-Veselov hierarchy

We construct a discrete analogue of the integrable two-dimensional Dirac operator and describe the spectral properties of its eigenfunctions. We construct an integrable discrete analogue of the modified Novikov-Veselov hierarchy. We derive the first two equations of the hierarchy and give explicit formulas for the eigenfunctions in terms of the theta-functions of the associated spectral curve.

💡 Research Summary

The paper presents a systematic construction of a discrete analogue of the two‑dimensional Dirac operator and uses it to formulate an integrable discrete version of the modified Novikov‑Veselov (mNV) hierarchy. Starting from the continuous Dirac operator (D=\begin{pmatrix}\partial & u\ -\bar\partial & v\end{pmatrix}) that underlies the mNV hierarchy, the authors replace the continuous variables by two integer lattice directions (n,m\in\mathbb Z) and introduce shift operators (T_1) and (T_2). The discrete Dirac operator is defined as \