Nonlinear Schrödinger Equation of General Form: Multifunctional Model, Reductions, and Exact Solutions

A new mathematical model based on the nonlinear Schrödinger equation with six arbitrary functions is presented to take into consideration different factors. This multifunctional model is a generalization of simpler related nonlinear models, which are frequently encountered in different areas of theoretical physics, including nonlinear optics, superconductivity, and plasma physics. The nonlinear equation under consideration is analyzed by using the method of functional constraints in combination with methods of generalized separation of variables. One-dimensional non-symmetry reductions that lead the considered complex partial differential equation (PDE) to simpler ordinary differential equations (ODEs) or systems of such equations are described. For the nonlinear Schrödinger equation, a number of exact solutions are found in the form of quadrature or elementary functions. Both periodic solutions in time and in spatial variable are obtained. Particular attention is paid to some narrower PDE classes with a smaller number of arbitrary functions. The described general multifunctional model makes it possible to efficiently analyze numerous simpler models and find their exact solutions by specifying the form of arbitrary functions. The exact solutions found in this paper can be used as test problems to verify the adequacy and assess the accuracy of numerical and approximate analytical methods of integrating the nonlinear equations of mathematical physics.