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 Izv. Akad. Nauk SSSR Ser. Mat., 1984, Volume 48, Issue 2, Pages 372–410 (Mi izv1450)

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

V. Yu. Novokshenov

Abstract: The leading term of the asymptotics of a solution of the nonlinear hyperbolic system
$$\square u_n=\exp(u_{n+1}-u_n)-\exp(u_n-u_{n-1}),\qquad n=1,2,…,N,$$
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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English version:
Mathematics of the USSR-Izvestiya, 1985, 24:2, 347–382

Bibliographic databases:

UDC: 517.946
MSC: Primary 35L70, 35B40; Secondary 35Q20

Citation: V. Yu. Novokshenov, “Asymptotics as $t\to\infty$ of the solution of the Cauchy problem for a two-dimensional generalization of the Toda lattice”, Izv. Akad. Nauk SSSR Ser. Mat., 48:2 (1984), 372–410; Math. USSR-Izv., 24:2 (1985), 347–382

Citation in format AMSBIB
\Bibitem{Nov84} \by V.~Yu.~Novokshenov \paper Asymptotics as $t\to\infty$ of the solution of the Cauchy problem for a~two-dimensional generalization of the Toda lattice \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1984 \vol 48 \issue 2 \pages 372--410 \mathnet{http://mi.mathnet.ru/izv1450} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=740796} \zmath{https://zbmath.org/?q=an:0566.35091} \transl \jour Math. USSR-Izv. \yr 1985 \vol 24 \issue 2 \pages 347--382 \crossref{https://doi.org/10.1070/IM1985v024n02ABEH001238} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. V. Kitaev, “The method of isomonodromy deformations and the asymptotics of solutions of the “complete” third Painlevé equation”, Math. USSR-Sb., 62:2 (1989), 421–444
2. V. V. Sukhanov, “An inverse problem for a selfadjoint differential operator on the line”, Math. USSR-Sb., 65:1 (1990), 249–266
3. O. M. Kiselev, “Asymptotics of solutions of higher-dimensional integrable equations and their perturbations”, Journal of Mathematical Sciences, 138:6 (2006), 6067–6230
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