Inverse scattering and N-triple-pole soliton and breather solutions of the focusing nonlinear Schrödinger hierarchy with nonzero boundary conditions

Abstract

Based on the matrix Riemann-Hilbert problem we present the inverse scattering transform for the focusing nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBCs) at infinity and triple zeros of analytical scattering coefficients. As a consequence, the inverse scattering problem can be solved via the study of the matrix Riemann-Hilbert problem with triple poles. We find the general N-triple-pole solutions for the potential, and explicit N-triple-pole expression for the reflectionless potential in terms of two determinants. Besides, the trace formulae and theta conditions are also given. Finally, we discuss the one-triple-pole breather-breather-breather solution for the focusing NLS equation with NZBCs. Moreover, the general N-triple-pole breathers of NZBCs can reduce to general N-triple-pole solitons of the zero-boundary conditions. These results can also be extended to the focusing higher-order NLS equations (i.e., the NLS hierarchy such as the Hirota equation, complex mKdV equation, Lakshmanan-Porsezian-Daniel equation, complex fifth-order mKdV equation, fifth-, sixth-, and seventh-order NLS equations, etc.) with NZBCs to find their inverse scattering transforms (ISTs) and N-triple-pole solutions. These obtained triple-pole solutions are useful to explain the related nonlinear wave phenomena.