  The concept of spectrum for a class of non-linear wave equations is studied.
Instead of looking for stability, the key to the spectral structure is found in
the instability phenomena (bifurcations). This aspect is best seen in the
`classical model' of the non-linear wave mechanics. The solitons
(macro-localizations) are a part of the non-linear spectral problem; their
bifurcations reflect the dynamical symmetry breaking. The computer simulations
suggest that the bifurcations of the asymptotic behaviour occur also for the
general, non-stationary states. A~phenomenon of the soliton splitting is
observed.

Quantum Physics