Long-time behaviour of soliton ensembles. Part II––Periodical patterns of trajectories

The emergence and interaction of KdV solitons generated by harmonic initial conditions are studied in the long run (hundreds of recurrence time tR). After the initial train of solitons has emerged, the further process is characterised by propagation of soliton ensembles and in numerical calculations the local maxima of wave profiles can be traced (see Part I for the description of ensembles and the corresponding trajectories of single solitons vs those within an ensemble). The geometrical patterns of trajectories of those maxima are analysed. In the short run arc-like patterns are formed but in the long run regular rhombus-like patterns are detected. The backbones of those rhombi are balanced trajectories which correspond to interacting solitons whose phase-shifts to the right are balanced by phase-shifts to the left. Consequently, over long time and space intervals these trajectories can be approximated by straight lines. A rhombus in the x–t plane has the space periodicity 2π dictated by the initial excitation and time periodicity 2tP or tP, dictated by balanced trajectories and related to the recurrence time tR. Such a pattern is a steady feature of regularity in the soliton emergence process that usually is taken to be shadowed by the loss of the recurrence in the strict sense.