Dirichlet energy and focusing NLS condensates of minimal intensity

TL;DR

This paper studies minimal intensity soliton condensates of the focusing NLS equation via Dirichlet energy minimization in a family of continua, linking spectral support to quadratic differentials and hyperelliptic Riemann surfaces.

Contribution

It introduces a variational approach to identify minimal intensity soliton condensates using Dirichlet energy and quadratic differentials related to finite gap solutions of fNLS.

Findings

Existence of a minimizing continuum in each connectivity class.

The minimizer corresponds to a quadratic differential linked to finite gap solutions.

The spectral support of the minimal condensate minimizes average intensity.

Abstract

We consider the family of (poly)continua $\K$ in the upper half-plane ${\mathbb H} $ that contain a preassigned finite {\it anchor} set $E\in\mathbb H$. For a given harmonic external field we define a Dirichlet energy functional $\mathcal I(\mathcal K)$ and show that within each ``connectivity class'' of the family, there exists a minimizing compact $\mathcal K^*$ consisting of critical trajectories of a quadratic differential. In many cases this quadratic differential coincides with the square of the real normalized quasimomentum differential ${\rm d} {\bf p}$ associated with the finite gap solutions of the focusing Nonlinear Schr\"{o}dinger equation (fNLS) defined by a hyperelliptic Riemann surface $\mathfrak R$ branched at the points $E\cup\bar E$. The motivation for this work lies in the problem of soliton condensate of least average intensity such that a given anchor set $E$ belongs to the poly-continuum $\mathcal K$. An fNLS soliton condensate is defined by a compact $\mathcal K\subset{\mathbb H} $ (its spectral support) whereas the average intensity of the condensate is proportional to $\mathcal I(\mathcal K)$. We prove that the spectral support $\mathcal K^*$ provides the fNLS soliton condensate of the least average intensity within a given ``connectivity class''.