Linear stability of internal wave solitons in a deep stratified fluid

The linear stability problem of the internal wave solitons described by the Benjamin–Ono equation has been solved. The wave functions corresponding to the continuum spectrum are stable, showing no reflection and no phase shift. There are two bound states associated with the soliton symmetries. The one associated with the translational invariance shows no growth; the other one associated with the scale invariance shows a secular instability growing linearly with time. The closure is also proven and, therefore, the completeness of these wave functions.