Virtual solitons and the asymptotics of second harmonic generation

A model Goursat problem for second harmonic generation is treated by a form erly developed inverse scattering method on a finite or semi-infinite inter val. For small or moderate values of the scaled characteristic coordinates chi, tau, the approximate solution is given in terms of N-pole formulae, an d it is demonstrated that it becomes more accurate with increasing N. In an other limit, this same problem is studied in the asymptotic region for larg e tau. For our particular boundary values it is shown that the asymptotic s olution is a stationary wave solution (one for which there is no exchange o f energy) modified by a boundary layer.