# Exact solutions to three-dimensional generalized nonlinear Schrodinger equations with varying potential and nonlinearities

## TL;DR

This paper presents a method to derive exact localized and periodic solutions for three-dimensional generalized nonlinear Schrödinger equations with inhomogeneous coefficients by reducing them to one-dimensional stationary equations using similarity transformations.

## Contribution

The authors develop a systematic approach to obtain exact solutions of 3D nonlinear Schrödinger equations with varying potentials and nonlinearities through similarity transformations.

## Key findings

- Exact localized and periodic wave solutions are derived.
- The method applies to power potentials, including linear and nonlinear cases.
- Several case examples demonstrate the effectiveness of the approach.

## Abstract

It is shown that using the similarity transformations, a set of three-dimensional p-q nonlinear Schrodinger (NLS) equations with inhomogeneous coefficients can be reduced to one-dimensional stationary NLS equation with constant or varying coefficients, thus allowing for obtaining exact localized and periodic wave solutions. In the suggested reduction the original coordinates in the (1+3)-space are mapped into a set of one-parametric coordinate surfaces, whose parameter plays the role of the coordinate of the one-dimensional equation. We describe the algorithm of finding solutions and concentrate on power (linear and nonlinear) potentials presenting a number of case examples. Generalizations of the method are also discussed.