The Maxwell class exact solutions to the Schr\"odinger equation and continuum mechanics models

TL;DR

This paper derives exact solutions to the Schrödinger and continuum mechanics equations using a nonlinear Legendre transform and a generalized Maxwell distribution, providing explicit expressions for various physical quantities.

Contribution

It introduces a novel method applying the nonlinear Legendre transform to obtain exact solutions for quantum and continuum models, with explicit formulas and analysis.

Findings

Explicit solutions for vector fields and potentials

Use of generalized Maxwell distribution in modeling

Mathematical and physical analysis of solutions

Abstract

By applying the nonlinear Legendre transform to the continuity equation, this paper derives exact solutions to the Schr\"odinger equation and the equations of continuum mechanics. A generalized Maxwell distribution has been used as the momentum density function. Explicit expressions for the vector fields of time independent flows, density distributions, quantum and classical potentials have been found, and a detailed mathematical and physical analysis of the results obtained has been carried out.