Phase flows generated by Cauchy problem for nonlinear Schr$\ddot{o}$dinger equation and dynamical mappings of quantum states

We consider the transformation of the initial data space for the Schr$\ddot{o}$dinger equation. The transformation is generated by nonlinear Schr$\ddot{o}$dinger operator on the segment $[-\pi,\,\pi]$ satisfying the homogeneous Dirichlet conditions on the boundary of the segment. The potential here has the type $\xi(u)=(1+|u|^2)^{p\over 2}u$, where $u$ is an unknown function, $p\geq 0$. The Schr$\ddot{o}$dinger operator defined on Sobolev space $H^2_0([-\pi,\,\pi])$ generates a vector field ${\bf v}: H^2_0([-\pi,\,\pi])\to H\equiv L_2(-\pi,\,\pi).$ First, we study the phenomenon of global existence of a solution of the Cauchy problem for $p\in [0,\,4)$ and, second, the phenomenon of rise of a solution gradient blow up during a finite time for $p\in [4,\,+\infty).$ In second case we study qualitative properties of a solution when it approaches to the boundary of its interval of existence. Moreover, we define a solution extension through the moment of a gradient blow up using both the one-parameter family of multifunctions and the one-parameter family of probability measures on the initial data space of the Cauchy problem. We show that this extension describes the destruction of a solution as the destruction of a pure quantum state and transition from the set of pure quantum states into the set of mixed quantum states