On integrability of a third-order complex nonlinear wave equation

TL;DR

This paper investigates the integrability of a recently introduced third-order complex nonlinear wave equation, demonstrating it does not pass the Painlevé test and analyzing its reductions to understand solution behaviors.

Contribution

It provides the first analysis of the integrability of Müller-Hoissen's third-order wave equation, identifying both integrable and non-integrable reductions.

Findings

The equation does not pass the Painlevé test for integrability.

Two reductions are identified: one integrable, one non-integrable.

Solutions of the reductions cover all solutions of the original equation.

Abstract

We show that the new third-order complex nonlinear wave equation, introduced recently by M\"{u}ller-Hoissen [arXiv:2202.04512], does not pass the Painlev\'{e} test for integrability. We find two reductions of this equation, one integrable and one non-integrable, whose solutions jointly cover all solutions of the original equation.