Far-field asymptotics for multiple-pole solitons in the large-order limit

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