Solutions to the nonlinear Schrödinger equation carrying momentum along a curve

Abstract

Abstract We prove existence of a special class of solutions to the (elliptic) nonlinear Schrödinger equation −ε 2 Δψ + V ( x )ψ = |ψ| p − 1 ψ on a manifold or in Euclidean space. Here V represents the potential, p an exponent greater than 1, and ε a small parameter corresponding to the Planck constant. As ε tends to 0 (in the semiclassical limit) we exhibit complex‐valued solutions that concentrate along closed curves and whose phases are highly oscillatory. Physically these solutions carry quantum‐mechanical momentum along the limit curves. © 2008 Wiley Periodicals, Inc.