Spots and stripes in nonlinear Schrödinger-type equations with nearly one-dimensional potentials. (English) Zbl 1282.35346
Summary: We consider the existence of spots and stripes for a class of nonlinear Schrödinger-type equations in the presence of nearly one-dimensional localized potentials. Under suitable assumptions on the potential, we construct various types of waves that are localized in the direction of the potential and have single- or 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 unipulse or multipulse homoclinic orbits, or periodic orbits in a reduced system of ordinary differential equations.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
| 35B32 | Bifurcations in context of PDEs |
Keywords:
NLS-like equations (nonlinear Schrödinger); normal forms; center manifold theory; bifurcation theory; bifurcationReferences:
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