Nonhomogeneous Dispersive Water Waves and Painlev\'{e} equations

TL;DR

This paper demonstrates that certain nonhomogeneous deformations of the Dispersive Water Wave (DWW) soliton equation have stationary flows equivalent to the well-known Painlevé equations, revealing deep connections between water wave models and integrable systems.

Contribution

It establishes a novel link between nonhomogeneous DWW deformations and Painlevé equations, expanding understanding of integrable structures in water wave models.

Findings

Stationary flows of three nonhomogeneous DWW deformations are equivalent to Painlevé II, III, and IV.

The work reveals new integrable structures within water wave equations.

Provides a mathematical bridge between soliton equations and Painlevé transcendents.

Abstract

In this letter we consider three nonhomogeneous deformations of Dispersive Water Wave (DWW) soliton equation and prove that their stationary flows are equivalent to three famous Painlev\'{e} equations, i.e. $P_{II}$, $P_{III}$ and $P_{IV},$ respectively.