Integrable Systems Seminar
Wednesday, February 27, 2002, 4:00pm, 120 Physics
Peter Miller (University of Michigan)
The Degree of Approximation of Analytic Functions by Solitons
Abstract:- The problem of approximating a given real analytic
bell-shaped function A(x), x real, by semiclassical soliton
ensembles is described. A semiclassical soliton ensemble is in this
context a reflectionless potential of the nonselfadjoint Zakharov-Shabat
operator with derivatives proportional to a small parameter hbar
having approximately 1/hbar purely imaginary eigenvalues accumulating
according to a Bohr-Sommerfeld quantization rule. In other words, it is
a nonlinear superposition of stationary solitons. By means of a novel
and rigorous asymptotic analysis of a matrix Riemann-Hilbert problem in
the spirit of Deift and Zhou, it is shown that the approximation error
is of the order of hbar^{1/7} pointwise in x.
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