Dirichlet energy and focusing NLS condensates of minimal intensity

TL;DR

We address the problem of selecting a minimal-energy spectral support for soliton condensates anchored by a finite set $E$ in the upper half-plane, by optimizing the Dirichlet energy $\,\\mathcal{I}(\\mathcal{K})$ over poly-continua $\\mathcal{K}$ containing $E$. The authors establish existence of minimizers within each connectivity class via Hausdorff-continuity and Jenkins’ interception framework, and prove that minimizers coincide with zero-level sets of a harmonic potential associated to Boutroux quadratic differentials of quasi-momentum type, i.e., $S$-curves, with energy matching a corresponding coefficient $I_Q$. They then connect these variational structures to the spectral theory of finite-gap focusing NLS, showing that the minimizer’s spectrum $\\mathfrak{S}$ yields the fNLS soliton condensate of least average intensity, through the relation $\mathbb I(\\psi)=2\\mathcal{I}(\\mathfrak{S})=-4\\mathfrak{J}(\\mathfrak{S})$ and the thermodynamic nonlinear dispersion relations. The approach unifies potential theory, complex analysis of quadratic differentials, and integrable-systems spectral theory, providing a principled framework for condensate optimization in nonlinear wave equations.

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 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''.

Figures (6)

• Figure 1: Various examples of minimal energy sets. These are also examples of solutions of the generalized Chebotarev problem discussed in Problem \ref{['chebo']}.
• Figure 2: Zero level curves (blue) $\mathfrak F_Q$ for all possible BMs $Q$ with the set of anchors $E = [-0.5 + 2i, 0.5+2i, 0.96i]$ are shown here. The Dirichlet energies of the left and right cases are the same, approximately $2.7299$, while for the central case the energy is approximately $2.7354$. This is an example of the fact that in $\mathbb K_E$ there might be not a unique minimizer (in this case there are two). Note that these two sets have "incommensurable" connectivity, namely, there is no connectivity $M$ that precedes both. If the anchor point on the imaginary axis is moved slightly down the connectivity on the left yields the absolute minimizer, while if we move it slightly up the one on the right is the unique minimizer.
• Figure 3: Examples of four Zakharov--Shabat spectra for the same configuration of anchor set $E$. Indicated also the stagnation points and the trajectories through them. The additional cuts $\Sigma$, used in Prop. \ref{['propximap']} (not depicted here), would be arcs of orthogonal trajectories extending from the stagnation points upwards toward the blue line representing $\mathcal{K}$, and downwards up to the real axis.
• Figure 4: Example of orthogonal flow-lines (level curves of $\mathrm {Re}\, \mathbf p(z)$) when $\Omega= {\mathrm {Out}}(\mathfrak F)$ is simply connected. The Zakharov-Shabat spectrum $\mathfrak F=\mathfrak F_{_{ZS}}\cap {{\mathbb H}}$ is shown by blue lines, which are zero level curves of $\mathrm {Im}\, \mathbf p(z)$. The red points at the end of blue lines form the set $E$. The level curves of $\mathrm {Re}\, \mathbf p(z)$ emanated from $\mathfrak F_{_{ZS}}$ and from ${\mathbb R}$ are shown in light blue and in black respectively. Note the absence of stagnation points in ${{\mathbb H}}$, so that the orthogonal flow from ${{\mathbb H}}$ to $\mathfrak F\cup {\mathbb R}$ is everywhere continuous.
• Figure 5: Example of orthogonal flow-lines (level curves of $\mathrm {Re}\, \mathbf p(z)$) when $\Omega= {\mathrm {Out}}(\mathfrak F)$ is not simply connected. The Zakharov-Shabat spectrum $\mathfrak F = \mathfrak F_{_{ZS}}\cap {{\mathbb H}}$ is shown by blue lines, which are zero level curves of $\mathrm {Im}\, \mathbf p(z)$ . The gradient lines of $\mathrm {Im}\, \mathbf p(z)$ emanating from $\mathfrak F$ and from ${\mathbb R}$ are shown in light blue and in black respectively. Note the stagnation point $z_0\in{{\mathbb H}}$ (the intersection of green curves) so that the orthogonal flow from ${{\mathbb H}}$ to $\mathfrak F\cup {\mathbb R}$ is discontinuous across the "caustic" (not depicted). The level curve $\mathrm {Im}\, \mathbf p(z)=\mathrm {Im}\, \mathbf p(z_0)$ is shown in green.

Theorems & Definitions (11)

• Remark 1.1
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 3.2
• Remark 3.3
• Remark 4.1
• Remark 4.2
• Remark A.1
• Corollary A.2