Abstract:
We study solutions of the Volterra lattice satisfying the stationary equation for its nonautonomous symmetry. We show that the dynamics in t and n are governed by the respective continuous and discrete Painlevé equations and describe the class of initial data leading to regular solutions. For the lattice on the half-axis, we express these solutions in terms of the confluent hypergeometric function. We compute the Hankel transform of the coefficients of the corresponding Taylor series based on the Wronskian representation of the solution.
This research was performed under the State
Assignment 0033-2019-0006 (Integrable systems of mathematical physics) of
the Ministry of Science and Higher Education of the Russian Federation.
Citation:
V. E. Adler, A. B. Shabat, “Some exact solutions of the Volterra lattice”, TMF, 201:1 (2019), 37–53; Theoret. and Math. Phys., 201:1 (2019), 1442–1456
This publication is cited in the following 8 articles:
Mariusz Białecki, “Simple Rules of a Discrete Stochastic Process Leading to Catalan-like Recurrences”, Algorithms, 18:3 (2025), 149
V.E. Adler, “Bogoyavlensky Lattices and Generalized Catalan Numbers”, Russ. J. Math. Phys., 31:1 (2024), 1
V. E. Adler, “Negative flows and non-autonomous reductions of the Volterra lattice”, Open Communications in Nonlinear Mathematical Physics, Special Issue in Memory of... (2024)
Jonathan Colen, Alexis Poncet, Denis Bartolo, Vincenzo Vitelli, “Interpreting Neural Operators: How Nonlinear Waves Propagate in Nonreciprocal Solids”, Phys. Rev. Lett., 133:10 (2024)
M. N. Kuznetsova, I. T. Habibullin, A. R. Khakimova, “On the problem of classifying integrable chains with three independent variables”, Theoret. and Math. Phys., 215:2 (2023), 667–690
V. E. Adler, “Painleve type reductions for the non-abelian Volterra lattices”, J. Phys. A-Math. Theor., 54:3 (2021), 035204
I. T. Habibullin, A. R. Khakimova, “Invariant manifolds and separation of the variables for integrable chains”, J. Phys. A-Math. Theor., 53:38 (2020), 385202
A. Zemlyanukhin, A. Bochkarev, “Exact solutions and numerical simulation of the discrete Sawada-Kotera equation”, Symmetry-Basel, 12:1 (2020), 131
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