The nonlinear Schrödinger equation is a second-order nonlinear, integrable partial differential equation describing the propagation of nonlinear waves in a variety of media, including light propagation in optical fibres. Inspired by recently reported experiments, here we consider its generalization to higher, even orders, of derivatives corresponding in optics to higher orders of dispersion. We show that none of these equations are integrable and investigate the nature of singularities that cause the equations to fail the Painlevé test.
