Papers › Dirichlet energy and focusing NLS condensates of minimal intensity
Dirichlet energy and focusing NLS condensates of minimal intensity
Marco Bertola, Alexander Tovbis
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We consider the family of (poly)continua $\K$ in the upper half-plane ℍ that contain a preassigned finite {\it anchor} set E∈ℍ. For a given harmonic external field we define a Dirichlet energy functional ℐ(𝒦) and show that within each ``connectivity class'' of the family, there exists a minimizing compact 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 d p associated with the finite gap solutions of the focusing Nonlinear Schr\"{o}dinger equation (fNLS) defined by a hyperelliptic Riemann surface ℜ branched at the points E∪E̅. 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 𝒦. An fNLS soliton condensate is defined by a compact 𝒦⊂ℍ (its spectral support) whereas the average intensity of the condensate is proportional to ℐ(𝒦). We prove that the spectral support 𝒦^* provides the fNLS soliton condensate of the least average intensity within a given ``connectivity class''.
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