Spots and stripes in NLS-type equations with nearly one-dimensional potentials

We consider the existence of spots and stripes for a class of NLS-type equations in the presence of nearly one-dimensional localized potentials. Under suitable assumptions on the potential, we construct various types of waves which are localized in the direction of the potential and have singleor multihump, or periodic profile in the perpendicular direction. The analysis relies upon a spatial dynamics formulation of the existence problem, together with a center manifold reduction. This reduction procedure allows these waves to be realized as unior multipulse homoclinic orbits, or periodic orbits in a reduced system of ordinary differential equations.