TY - JOUR
T1 - Strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane
JF - Comm. Pure Appl. Math.
Y1 - 2015
A1 - Ferenc Balogh
A1 - Marco Bertola
A1 - Lee, Seung-Yeop
A1 - Kenneth McLaughlin
VL - 68
UR - http://dx.doi.org/10.1002/cpa.21541
ER -
TY - JOUR
T1 - Semiclassical limit of focusing NLS for a family of square barrier initial data
Y1 - 2014
A1 - Robert Jenkins
A1 - Kenneth McLaughlin
AB - The small dispersion limit of the focusing nonlinear Schrödinger equation (NLS) exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales. The non-self-adjoint scattering problem and ill-posed limiting Whitham equations associated to focusing NLS make rigorous asymptotic results difficult. Previous studies have focused on special classes of analytic initial data for which the limiting elliptic Whitham equations are wellposed. In this paper we consider another exactly solvable family of initial data,the family of square barriers,ψ 0(x) = qχ[-L,L] for real amplitudes q. Using Riemann-Hilbert techniques, we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing NLS globally in space and up to an O(1) maximal time. In particular, we show that the discontinuities in our initial data regularize by the immediate generation of genus-one oscillations emitted into the support of the initial data. To the best of our knowledge, this is the first case in which the genus structure of the semiclassical asymptotics for focusing NLS have been calculated for nonanalytic initial data.
PB - Wiley Periodicals
UR - http://urania.sissa.it/xmlui/handle/1963/35066
U1 - 35301
U2 - Mathematics
U4 - 1
ER -