Riemann-Hilbert Approach to Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions: N Pairs of Higher-Order Poles Case

We investigate the inverse scattering transform for the focusing nonlinear Schrödinger (NLS) equation with a particular class of nonvanishing boundary conditions (NVBCs), especially in the case of reflectionless potentials that give rise to a transmission coefficient with an arbitrary finite number N pairs of higher-order poles. The inverse problem is characterized in terms of a \(2\times 2\) $2\times 2$ matrix Riemann-Hilbert (RH) problem equipped with several residue conditions at N pairs of higher-order poles. In the reflectionless case, we point out that the N-multipole soliton solutions including higher-order Kuznetsov-Ma breathers and Akhmediev breathers can be reconstructed by a linear algebraic system. Furthermore, we verify these special solutions by numerical simulations and display their density structures, also derive some Peregrine solitons by choosing appropriate parameters and taking limits.