Special solutions of high order equation for waves in liquid with gas bubbles

TL;DR

This paper investigates special solutions of a fifth-order nonlinear PDE modeling waves in a bubbly liquid, constructing elliptic and periodic solutions and exploring their relation to Painleve transcendents.

Contribution

It introduces new special solutions for a high-order wave equation and links self-similar solutions to Painleve transcendents, advancing understanding of nonlinear wave behavior in bubbly liquids.

Findings

Constructed elliptic and simple periodic traveling wave solutions.

Connected self-similar solutions to Painleve transcendents.

Discussed high-order analogs of Painleve transcendents.

Abstract

A fifth--order nonlinear partial differential equation for the description of nonlinear waves in a liquid with gas bubbles is considered. Special solutions of this equation are studied. Some elliptic and simple periodic traveling waves solution are constructed. Connection of self--similar solutions with Painleve transcendents and their high--order analogous is discussed.