Nonlinear Schrödinger Equations with Delay: Closed-Form and Generalized Separable Solutions

Nonlinear Schr & ouml;dinger equations with constant delay are considered for the first time. These equations are generalizations of the classical Schr & ouml;dinger equation with cubic nonlinearity and the more complex nonlinear Schr & ouml;dinger equation containing functional arbitrariness. From a physical point of view, considerations are formulated about the possible causes of the appearance of a delay in nonlinear equations of mathematical physics. To construct exact solutions, the principle of structural analogy of solutions of related equations was used. New exact solutions of nonlinear Schr & ouml;dinger equations with delay are obtained, which are expressed in elementary functions or in quadratures. Some more complex solutions with generalized separation of variables are also found, which are described by mixed systems of ordinary differential equations without delay or ordinary differential equations with delay. The results of this work can be useful for the development of new mathematical models described by nonlinear Schr & ouml;dinger equations with delay, and the given exact solutions can serve as the basis for the formulation of test problems designed to evaluate the accuracy of numerical methods for integrating nonlinear partial differential equations with delay.