Integrability of differential equations with fluid mechanics application: from Painleve property to the method of simplest equation

TL;DR

This paper reviews the integrability of nonlinear differential equations in fluid mechanics, emphasizing the Painleve property and introducing the method of simplest equations for finding exact solutions.

Contribution

It provides an overview of the Painleve property and discusses the modified method of simplest equations as a novel approach for solving nonlinear PDEs.

Findings

The Painleve property is crucial for integrability of nonlinear differential equations.

The method of simplest equations aids in obtaining exact solutions of nonlinear PDEs.

The modified method of simplest equations extends the original approach for broader applicability.

Abstract

We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order has the Painleve property if the only movable singularities connected to this equation are single poles. The importance of this property can be seen from the Ablowitz-Ramani-Segur conhecture that states that a nonlinear PDE is solvable by inverse scattering transformation only if each nonlinear ODE obtained by exact reduction of this PDE has the Painleve property. The Painleve property motivated motivated much research on obtaining exact solutions on nonlinear PDEs and leaded in particular to the method of simplest equation. A version of this method called modified method of simplest equation is discussed below.