# Analytic solutions of the Madelung equation

## TL;DR

This paper derives analytic self-similar solutions for the Madelung equation in multiple dimensions, revealing a link between fluid density zeros and quantum potential magnitude.

## Contribution

It provides the first explicit analytic solutions to the multidimensional Madelung equation, enhancing understanding of quantum hydrodynamics.

## Key findings

- Analytic solutions for 1D, 2D, and 3D Madelung equations.
- Identified relationship between density zeros and quantum potential.
- Clarified the structure of quantum fluid dynamics.

## Abstract

We present analytic self-similar solutions for the one, two and three dimensional Madelung hydrodynamical equation for a free particle. There is a direct connection between the zeros of the Madelung fluid density and the magnitude of the quantum potential.