The integrable focusing nonlinear Schrödinger 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.

可积聚焦非线性薛定谔方程允许孤子解，其相关光谱数据由一对任意阶的共轭极组成。我们研究了由 Darboux 变换产生的这种多极孤子族，因为极序趋于无穷大。我们表明，在适当的缩放中，时空平面中有四个区域，其中解显示出性质不同的行为：指数衰减区域、代数衰减区域、非振荡区域和振荡区域。使用非线性最速下降法分析 Riemann-Hilbert 问题，我们计算代数衰减、非振荡和振荡区域中的领先阶渐近行为。