  We present here solutions of a non-linear Schroedinger equation in presence
of an arbitrary linear external potential. The non-linearity expresses a
self-focusing interaction. These solutions are the product of the pilot wave
with peaked solitons the velocity of which obeys the guidance equation derived
by Louis de Broglie in 1926. The degree of validity of our approximations
increases when the size of the soliton decreases and becomes negligible
compared to the typical size over which the pilot wave varies. We discuss the
possibility to reveal their existence by implementing a humpty-dumpty
Stern-Gerlach interferometer in the mesoscopic regime.
