00527nas a2200157 4500008004100000022001400041245010200055210006900157300001400226490000700240100001900247700001900266700002000285700002400305856004000329 2015 eng d a0010-364000aStrong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane0 aStrong asymptotics of the orthogonal polynomials with respect to a112–1720 v681 aBalogh, Ferenc1 aBertola, Marco1 aLee, Seung-Yeop1 aMcLaughlin, Kenneth uhttp://dx.doi.org/10.1002/cpa.2154101605nas a2200121 4500008004100000245008400041210006900125260002200194520117200216100002001388700002401408856005101432 2014 en d00aSemiclassical limit of focusing NLS for a family of square barrier initial data0 aSemiclassical limit of focusing NLS for a family of square barri bWiley Periodicals3 aThe 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.1 aJenkins, Robert1 aMcLaughlin, Kenneth uhttp://urania.sissa.it/xmlui/handle/1963/35066