# Matrix logarithmic wave equation and multi-channel systems in fluid mechanics

## TL;DR

This paper develops a mathematical framework linking nonlinear wave equations with matrix logarithmic nonlinearity to multi-channel fluid flow systems, providing analytical solutions and insights into complex fluid behaviors.

## Contribution

It introduces a novel mapping between matrix logarithmic wave equations and multi-channel fluid flow equations, extending previous models to include matrix and multi-component systems.

## Key findings

- Derived Gaussian-type matrix solutions for specific cases
- Mapped nonlinear wave equations to multi-channel fluid flow models
- Provided analytical insights into complex fluid behaviors

## Abstract

We formulate the mapping between a large class of nonlinear wave equations and flow equations for barotropic fluid with internal surface tension and capillary effects. Motivated by statistical mechanics and multi-channel physics arguments, we focus on wave equations with logarithmic nonlinearity, and further generalize them to matrix equations. We map the resulting equation to flow equations of multi-channel or multi-component Korteweg-type materials. For some special cases, we analytically derive Gaussian-type matrix solutions and study them in the context of fluid mechanics.