Geodesic
The Dirac operator with elliptic potential
Sbornik. Mathematics, Tome 186 (1995) no. 8, pp. 1213-1221

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The Dirac operator with elliptic finite-gap potential $$ -\mathrm i\begin{pmatrix}10\\0-1\end{pmatrix}\Psi _x +\begin{pmatrix}0\\q0\end{pmatrix}\Psi =\lambda\Psi . $$ is considered. An Ansatz for the Krichever curves associated with elliptic (in $x$) finite-gap solutions of the 'decomposed' non-linear Schrödinger equation $$ \begin{cases} \mathrm ip_t+p_{xx}-2p^2q=0, \\iq_t-q_{xx}+2pq^2=0 \end{cases} $$ and of the modified $KdV$ ($mKdV$) equation $$ \begin{cases} p_t+p_{xxx}-6pqp_x=0, \\q_t+q_{xxx}-6pqq_x=0. \end{cases} $$ is presented. Examples of two- and three-sheeted coverings associated with the one- and twogap Dirac potential are discussed. 
@article{SM_1995_186_8_a6,
     author = {A. O. Smirnov},
     title = {The {Dirac} operator with elliptic potential},
     journal = {Sbornik. Mathematics},
     pages = {1213--1221},
     publisher = {mathdoc},
     volume = {186},
     number = {8},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_8_a6/}
}
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A. O. Smirnov. The Dirac operator with elliptic potential. Sbornik. Mathematics, Tome 186 (1995) no. 8, pp. 1213-1221. http://geodesic.mathdoc.fr/item/SM_1995_186_8_a6/