Dirichlet energy and focusing NLS condensates of minimal 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 ${\mathcal{K}}^{\ast }$ consisting of critical trajectories of a quadratic differential. In many cases this quadratic differential coincides with the square of the real normalized quasimomentum differential $\mathrm{d}\mathbf{p}$ associated with the finite gap solutions of the focusing Nonlinear Schrödinger 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 ${\mathcal{K}}^{\ast }$ provides the fNLS soliton condensate of the least average intensity within a given ``connectivity class''.

37 pages, 6 (beautiful!) figures. Ver 2: 39 pages; improved introduction and summary of results, added references. Some re-rendered figures