Far-Field Asymptotics for Multiple-Pole Solitons in the Large-Order Limit

The integrable focusing nonlinear Schrodinger equation admits soliton solutions whose associated spectral data consist of a single pair of conjugate poles of arbitrary order. We study families of such multiple-pole solitons generated by Darboux transformations as the pole order tends to infinity. We show that in an appropriate scaling, there are four regions in the space-time plane where solutions display qualitatively distinct behaviors: an exponential-decay region, an algebraic-decay region, a non-oscillatory region, and an oscillatory region. Using the nonlinear steepest-descent method for analyzing Riemann-Hilbert problems, we compute the leading-order asymptotic behavior in the algebraic-decay, non-oscillatory, and oscillatory regions.