Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part II: proof of the existence result

TL;DR

This paper proves the existence of complex solutions to the nonlinear Schrödinger equation that concentrate along curves and carry momentum, especially in the semiclassical limit, extending previous approximate constructions to a rigorous proof.

Contribution

It provides a rigorous proof of the existence of solutions with prescribed concentration and momentum properties along curves, advancing the understanding of semiclassical states.

Findings

Solutions concentrate along closed curves as epsilon tends to zero

Solutions exhibit highly oscillatory phase behavior

Existence is established rigorously for a special class of solutions

Abstract

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroedinger Equation $- \epsilon^2 \Delta \psi + V(x) \psi = |\psi|^{p-1} \psi$ on a manifold or in the Euclidean space. Here V represents the potential, p is an exponent greater than 1 and $\epsilon$ a small parameter corresponding to the Planck constant. As $\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase in highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In the first part of this work we identified the limit set and constructed approximate solutions, while here we give the complete proof of our main existence result.