Exact constant-intensity solutions and modulational instability in a higher-order nonlinear Schrödinger equation with real potential.

Constant-intensity (CI) waves are plane-wave-like solutions that are known to propagate in non-Hermitian inhomogeneous media with complex external potential (with the real part corresponding to the refractive index and the imaginary part representing a distribution of gain and loss) while keeping their field amplitudes space-time independent at the expense of acquiring a nonlinear phase. In this paper, it is shown that CI waves can exist in certain situations when the medium is Hermitian-in which case the external potential is purely real-provided that higher-order diffraction or dispersion effects are present. To this end, the one-dimensional nonlinear Schrödinger equation with fourth-order (quartic) diffraction is used to construct examples of localized and periodic real external potentials that support CI waves. Interestingly, the class of potentials presented is directly connected to solutions of the modified Korteweg-de Vries equation and the more general Gardner equation, both of which are well-studied integrable partial differential equations. Throughout a combination of linear stability analysis and direct numerical simulations, regions in parameter space are identified for which CI solutions are predicted to be stable.