The Toda lattice with step-like initial data. Soliton asymptotics

We study the asymptotic behaviour of solutions of the Toda lattice Cauchy problem with step-like initial data near the wavefront. The initial data are supposed to tend to the discrete Laplacian on one half-axis and to the periodic Jacobi matrix on the other. The solution is shown to split into an infinite series of solitons when rime tends to infinity. We obtain an exact asymptotic formula for these solutions.