The \(t\to\infty\) asymptotic regime of the Cauchy problem solution for the Toda chain with threshold-type initial data
type: Item
instance of: scholarly article
title: The \(t\to\infty\) asymptotic regime of the Cauchy problem solution for the Toda chain with threshold-type initial data
zbMATH Open document ID: 0985.37080
DOI: 10.1007/BF02557384
published in: Theoretical and Mathematical Physics
publication date: 2000-07-19T00:00:00Z
review text: Consider the Toda chain \[ \begin{aligned} &D_t a_n= \textstyle{{1\over 2}} a_n(b_n- b_{n+1}),\quad a_n= a_n(t)> 0,\\ & D_t b_n= a^2_{n-1}- a^2_n,\quad b_n= b_n(t),\quad n\in\mathbb{Z}\end{aligned}\tag{1} \] with the asymptotics \[ \begin{aligned} &a_n(0)\to c_1,\quad b_n(0)\to m_1,\quad n\to\infty,\\ & a_n(0)\to c_2,\quad b_n(0)\to m_2,\quad n\to-\infty,\end{aligned}\tag{2} \] where \(c_1> 0\) and \(c_2> 0\). The main goal of this paper is to construct the \(t\to\infty\) asymptotic expansion of the solutions to (1) and (2). To this end the authors analyze the scattering problem for second-order difference equations whose coefficients tend to different constant limits at \(\pm\infty\). As a result they obtain formulas which allow to find solutions of (1) and study the asymptotic behaviour of solutions as \(t\to\infty\).
Mathematics Subject Classification ID: 37K15
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K40
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35Q51
zbMATH DE Number: 1465980
zbMATH Keywords: Toda chain
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: scattering problem
author: Hidayat M. Huseynov
author: Agil Kh. Khanmamedov
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
cites work: On the Toda Lattice. II: Inverse-Scattering Solution
cites work: Theory of nonlinear lattices
cites work: Asymptotic expansion of the solution to the Cauchy problem for the Volterra chain with a step-like initial condition
cites work: Characterization of the scattering data of the Schrödinger and Dirac operators
description: scientific article; zbMATH DE number 1465980