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arXiv · 0907.5571

Existence and Regularity for an Energy Maximization Problem in Two Dimensions

Spyridon Kamvissis·Evguenii A. Rakhmanov
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Abstract

We consider the variational problem of maximizing the weighted equilibrium Green's energy of a distribution of charges free to move in a subset of the upper half-plane, under a particular external field. We show that this problem admits a solution and that, under some conditions, this solution is an S-curve (in the sense of Gonchar-Rakhmanov). The above problem appears in the theory of the semiclassical limit of the integrable focusing nonlinear Schr\"odinger equation. In particular, its solution provides a justification of a crucial step in the asymptotic theory of nonlinear steepest descent for the inverse scattering problem of the associated linear non-self-adjoint Zakharov-Shabat operator and the equivalent Riemann-Hilbert factorization problem.

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BibTeXRIS

Spyridon Kamvissis, Evguenii A. Rakhmanov. 2009-07-31. Existence and Regularity for an Energy Maximization Problem in Two Dimensions. https://arxiv.org/abs/0907.5571

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