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

TL;DR

This paper proves the existence of complex solutions to the nonlinear Schrödinger equation that concentrate along curves and carry momentum, analyzing their limit set and constructing approximate solutions in the semiclassical regime.

Contribution

It characterizes the limit set of solutions and constructs approximate solutions up to order epsilon squared, laying groundwork for proving existence in the second part.

Findings

Solutions concentrate along closed curves as epsilon tends to zero

Phase of solutions is highly oscillatory, carrying momentum

Approximate solutions constructed up to order epsilon squared

Abstract

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger 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 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 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 $\epsilon^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $\epsilon$. Based on these, an existence result will be proved in the second part.