Geodesic
Eigenfunctions of the Dirichlet and Neumann problems for the elliptic sinh-Gordon equation on a rectangle
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 19, Tome 179 (1989), pp. 32-36

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The Dirichlet and Neumann zero boundary problems on a rectangular for the equation $\Delta u+\mathrm{sh}\, u=0$ are considered. All solutions are constructed explicitely by the finite-gap integration technique. 
@article{ZNSL_1989_179_a3,
     author = {A. I. Bobenko},
     title = {Eigenfunctions of the {Dirichlet} and {Neumann} problems for the elliptic {sinh-Gordon} equation on a rectangle},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {32--36},
     publisher = {mathdoc},
     volume = {179},
     year = {1989},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1989_179_a3/}
}
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A. I. Bobenko. Eigenfunctions of the Dirichlet and Neumann problems for the elliptic sinh-Gordon equation on a rectangle. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 19, Tome 179 (1989), pp. 32-36. http://geodesic.mathdoc.fr/item/ZNSL_1989_179_a3/