Dirichlet energy and focusing NLS condensates of minimal intensity
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arXiv
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| Format: | Preprint |
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2024
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| author | Bertola, Marco Tovbis, Alexander |
| author_facet | Bertola, Marco Tovbis, Alexander |
| contents | 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''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19373 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dirichlet energy and focusing NLS condensates of minimal intensity Bertola, Marco Tovbis, Alexander Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems 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''. |
| title | Dirichlet energy and focusing NLS condensates of minimal intensity |
| topic | Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2412.19373 |