Mathematics – Complex Variables
Scientific paper
2009-07-31
J.Math.Phys., v.46, n.8, 1 August 2005
Mathematics
Complex Variables
43 pages
Scientific paper
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.
Kamvissis Spyridon
Rakhmanov Evguenii A.
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