EXACT SOLUTIONS AND REDUCTIONS OF THE NONLINEAR SCHRO¨ DINGER EQUATION OF THE GENERAL FORM

The nonlinear Schro¨dinger equation of a general form is investigated, in which the chromatic dispersion and the potential are given by two arbitrary functions. The equation under consideration is a natural generalization of a wide class of related nonlinear equations that are often encountered in various sections of theoretical physics, including nonlinear optics, superconductivity, and plasma physics. Exact solutions of the nonlinear Schro¨dinger equation of general form are found, which are expressed in quadratures. One-dimensional reductions are described, which reduce the studied partial differential equation to simpler ordinary differential equations or systems of such equations. The exact solutions obtained in this work can be used as test problems intended to assess the accuracy of numerical methods for integrating nonlinear equations of mathematical physics.