Dirichlet Energy and Focusing NLS Condensates of Minimal Intensity

Abstract

We consider the family of (poly)continua \(\mathcal {K}\) in the upper half-plane \({{\mathbb {H}}} \) that contain a preassigned finite 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 \({\text {d}}{\textbf {p}}\) 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 \(\mathcal {K}^*\) provides the fNLS soliton condensate of the least average intensity within a given “connectivity class”.