Exact solutions and reductions of nonlinear Schrödinger equations with delay

For the first time, Schrödinger equations with cubic and more complex nonlinearities containing the unknown function with constant delay are analyzed. The physical considerations that can lead to the appearance of a delay in such nonlinear equations and mathematical models are expressed. One-dimensional non-symmetry reductions are described, which lead the studied partial differential equations with delay to simpler ordinary differential equations and ordinary differential equations with delay. New exact solutions of the nonlinear Schrödinger equation of the general form with delay, which are expressed in quadratures, are found. To construct exact solutions, a combination of methods of generalized separation of variables and the method of functional constraints are used. Special attention is paid to three equations with cubic nonlinearity, which allow simple solutions in elementary functions, as well as more complex exact solutions with generalized separation of variables. Solutions representing a nonlinear superposition of two traveling waves, the amplitude of which varies periodically in time and space, are constructed. Some more complex nonlinear Schrödinger equations of a general form with variable delay are also studied. The results of this work can be useful for the development and improvement of mathematical models described by nonlinear Schrödinger equations with delay and related functional PDEs, and the obtained exact solutions can be used as test problems intended to assess the accuracy of numerical methods for integrating nonlinear equations of mathematical physics with delay.