Nonlinear Schrödinger Equation with Delay and Its Regularization

The properties of an initial-boundary value problem for a nonlinear Schödinger equation including terms with a delay of the time argument in the nonlinear potential or in whole part of the nonlinear term are studied. Conditions of local and global existence or arising of a gradient blow up phenomenon are obtained. The relation of a gradient blow up phenomenon with a self-focusing and a destruction of a pure quantum state are described. We consider an evolution of a quantum state generated by a solution of a nonlinear Schrödinger equation as a curve in the set of pure quantum states. We introduce the regularization procedure, which defines a continuation of a solution through the moment of blow up by a curve in the set of quantum states with the transition into the set of mixed states.