Singularity analysis and analytic solutions for the Benney-Gjevik equations

TL;DR

This paper applies the Painlevé Test to the Benney and Benney-Gjevik equations, demonstrating their integrability for traveling waves and providing algebraic Laurent solutions.

Contribution

It introduces the singularity analysis for these equations and derives explicit algebraic solutions, advancing understanding of wave behavior in falling liquids.

Findings

Benney and Benney-Gjevik equations pass the Painlevé Test for traveling waves

Explicit algebraic solutions are derived using Laurent expansions

The equations are shown to be integrable for certain solutions

Abstract

We apply the Painlev\'e Test for the Benney and the Benney-Gjevik equations which describe waves in falling liquids. We prove that these two nonlinear 1+1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.