Wave packets in boundary problems of quantum mechanics

System of two independent linear quantum equations with symbols representing polynoms of the n-th order is considered. Boundary conditions are non-linear. They functionally connect amplitudes of the direct and inverse wave functions by mapping $\Phi :I \mapsto I$. It is demonstrated that 1) if mapping $ \Phi $ is linear, the amplitude of the falling wave at $ t\rightarrow\infty $ tends to zero or infinity; 2) if $ \Phi $ is nonlinear but single-valued, at $ t\rightarrow\infty $, the amplitude of the falling wave tends to a double-periodic — constant function with one singular point per a period; 3) if $ \Phi $ is multi-valued, asymptotically periodic — constant distributions of square amplitude of the wavefunction with finite or infinite number of singularities per a period are possible. The limiting solutions of this type we shall call distributions of pre-turbulent or turbulent type. Applications to the study of the emergence of spatial-temporal bright and dark asymptotic solitons in a limited resonator with non-linear feedback between the amplitudes of two optical beams on the resonator surface are presented.