The \(t\to\infty\) asymptotic regime of the Cauchy problem solution for the Toda chain with threshold-type initial data
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 \(t\to\infty\).
scientific article; zbMATH DE number 1465980