Two-dimensional nonlinear Schrödinger equations with potential and dispersion given by arbitrary functions: Reductions and exact solutions

TL;DR

This work studies two-dimensional nonlinear Schrödinger equations with potentials and dispersions determined by arbitrary functions, yielding a broad class of models that encompass various physical contexts. By transforming w = r e^{iφ} into a real PDE system with h = r f(r), the authors derive numerous exact solutions and reductions in both Cartesian and polar coordinates, including traveling-wave, stationary, and radially symmetric forms, many expressible in quadratures or elementary functions. The results leverage generalized and functional separation techniques and identify invariances that generate new solutions, with particular attention to radially symmetric and time-periodic configurations. The obtained exact solutions serve as robust test problems for numerical and approximate analytical methods in nonlinear PDEs and have potential applications in nonlinear optics, superconductivity, and plasma physics. The findings extend the toolkit for solving variable-coefficient NLS equations and provide structured pathways for constructing multi-parameter families of solutions.

Abstract

The paper deals with nonlinear Schrödinger equations of the general form, depending on time and two spatial variables, the potential and dispersion of which are specified by one or two arbitrary functions. The equations under consideration naturally generalize a number of related nonlinear partial differential equations that occur in various areas of theoretical physics, including nonlinear optics, superconductivity, and plasma physics. Two- and one-dimensional reductions are described, which lead the studied nonlinear Schrödinger equations to simpler equations of lower dimension or ordinary differential equations (or systems of ordinary differential equations). Using methods of generalized and functional separation of variables, a number of new exact solutions of two-dimensional nonlinear Schrödinger equations of the general form are found, which are expressed in quadratures or elementary functions. To analyze the equations under consideration, both Cartesian and polar coordinate systems are utilized. Special attention is paid to finding solutions with radial symmetry. The exact solutions obtained in this work can be used as test problems intended to assess the accuracy of numerical and approximate analytical methods for solving complex nonlinear PDEs of mathematical physics.