The Bäcklund transformation and novel solutions for the Toda lattice (Q1776607)

The authors derive limit solutions to the Toda lattice \[ {d^2 x_n\over dt^2}= e^{x_{n-1}- x_n}- e^{x_n- x_{n+1}}. \] First, the authors modify the original bilinear Bäcklund transformation for the Toda lattice and give an alternative form of the Bäcklund transformation. Such changes lead to obtain the \(N\)-soliton solution which is slightly different from the one in Hirota form. However, this new form of the \(N\)-soliton solution can easily lead to limit solitons.