Based on the inverse scattering method, the formulae of one higher‐order pole solitons and multiple higher‐order poles solitons of the nonlinear Schrödinger equation (NLS) equation are obtained. Their denominators are expressed as , where is a matrix frequently constructed for solving the Riemann‐Hilbert problem, and the asterisk denotes complex conjugate. We take two methods for proving is invertible. The first one shows matrix is equivalent to a self‐adjoint Hankel matrix , proving . The second one considers the block‐matrix form of , proving . In addition, we prove that the dimension of is equivalent to the sum of the orders of pole points of the transmission coefficient and its diagonal entries compose a set of basis.

中文翻译：

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NLS方程的多个高阶极孤子的正则性

基于逆散射法，得到了非线性薛定ding方程（NLS）方程的一个高阶极子孤子和多个高阶极子孤子的公式。它们的分母表示为，其中是经常用来解决Riemann-Hilbert问题的矩阵，星号表示复共轭。我们采取两种方法证明是可逆的。第一个证明矩阵等于自伴汉克尔矩阵，证明。第二个问题考虑了证明的块矩阵形式。此外，我们证明了 等于传输系数极点的阶数之和，其对角线项构成一组基础。