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We consider the transformation of the initial data space of the Cauchy problem for the focusing nonlinear Schrodinger equation with power nonlinearity in potential ([1]- [3]). Also we study properties of the initial-boundary value problem for a nonlinear Schrodinger equation including terms with delay of time argument. First, we prove the local existence for the Cauchy problem and for the initial problem. After that we study the phenomenon of global existence as a solution of the Cauchy problem for small powers of nonlinearity. In the case of large power of nonlinearity the phenomenon of rise of a solution gradient blow up during a finite time is proven. In the last case we study qualitative properties of a solution when it approaches the boundary of its interval of existence. Same properties are studied for the initial problem. The relation of the gradient blow up phenomenon with the self-focusing and the destruction of the pure quantum state are described [2]. Moreover, we define a solution extension through the moment of a gradient blow by means of the random process with values in the set of pure quantum states. We show that this extension describes the destruction of a solution as the destruction of a pure quantum state and the transition from the set of pure quantum states into the set of mixed quantum states [3].

References:

[1] A.D. Grekhneva & V.Zh. Sakbaev, Dynamics of a Set of Quantum States Generated by a Nonlinear Liouville–von Neumann Equation. Computational Mathematics & Mathematical Physics, 60:8 (2020), 1337-1347.

[2] L.S. Efremova, A.D. Grekhneva, V.Zh. Sakbaev, Phase flow generated by Cauchy problem for nonlinear Schrodinger equation and dynamical mappings of quantum states. Lobachevskii Journal of Mathematics, 40:10 (2019), 1455-1469.

[3] V.Zh. Sakbaev & A.D. Shiryaeva, Blow-Up of States in the Dynamics Given by the Schrödinger Equation with a Power-Law Nonlinearity in the Potential. Differential Equation, 58:4 (2022), 498-508.