The regularity of the multiple higher‐order poles solitons of the NLS equation
Yongshuai Zhang, Xiangxing Tao, Teng‐Teng Yao, Jingsong He
Abstract
Abstract 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.