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We present a brief overview of integrability of nonlinear ordinary andpartial differential equations with a focus on the Painleve property: an ODE ofsecond order has the Painleve property if the only movable singularitiesconnected to this equation are single poles. The importance of this propertycan be seen from the Ablowitz-Ramani-Segur conhecture that states that anonlinear PDE is solvable by inverse scattering transformation only if eachnonlinear ODE obtained by exact reduction of this PDE has the Painleveproperty. The Painleve property motivated motivated much research on obtainingexact solutions on nonlinear PDEs and leaded in particular to the method ofsimplest equation. A version of this method called modified method of simplestequation is discussed below.

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