Geodesic
Asymptotics as $t\to\infty$ of the solution of the Cauchy problem for a~two-dimensional generalization of the Toda lattice
Izvestiya. Mathematics , Tome 24 (1985) no. 2, pp. 347-382.

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The leading term of the asymptotics of a solution of the nonlinear hyperbolic system $$ \square u_n=\exp(u_{n+1}-u_n)-\exp(u_n-u_{n-1}),\qquad n=1,2,\dots,N, $$ for large times is constructed and justified. A version of the method of the inverse problem reducing to the solution of a matrix problem of linear conjugation on the complex plane of the spectral parameter is used to solve this system. The coefficients of the asymptotics of $u_n$ are expressed explicitly in terms of the elements of the Riemann matrix realizing the linear conjugation. A theorem is proved on the approximation of the exact solution by the asymptotics constructed. Bibliography: 14 titles. 
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V. Yu. Novokshenov. Asymptotics as $t\to\infty$ of the solution of the Cauchy problem for a~two-dimensional generalization of the Toda lattice. Izvestiya. Mathematics , Tome 24 (1985) no. 2, pp. 347-382. http://geodesic.mathdoc.fr/item/IM2_1985_24_2_a6/

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