Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions

Abstract

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- ε^2 Δψ+ V(x) ψ= |ψ|^{p-1} ψ$, on a manifold or in the 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 zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase is highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In this first part we provide the characterization of the limit set, with natural stationarity and non-degeneracy conditions. We then construct an approximate solution up to order $ε^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $ε$. Based on these, an existence result will be proved in the second part.