Asymptotics as $t\to\infty$ of the solution of the Cauchy problem for a two-dimensional generalization of the Toda lattice

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.
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