Inverse scattering transform for the focusing PT-symmetric nonlinear Schrödinger equation with nonzero boundary conditions: Higher-order poles and multi-soliton solutions

Abstract

In this paper, we extend the theory of inverse scattering transform for the focusing PT-symmetric nonlinear Schrödinger equation with nonzero boundary conditions by considering the reciprocals of scattering coefficients have multiple higher-order poles. For the inverse problem with the presence of simple, double and triple poles, we study the pole contributions, trace formulas and reconstruction formulas. On the other hand, we present the general N-soliton solutions in the determinant form for the reflectionless case, and particularly analyze the dynamics of heteroclinic multi-soliton solutions which admit the asymptotic phase difference $\pi$ as $x\to \pm \infty$. It turns out that the solutions are nonsingular with a wide range of parameters and can display abundant multi-soliton interactions. The discrete eigenvalues correspond to two different localized waves: one is the conventional soliton exhibiting the dark/antidark profile, the other is the heteroclinic breather-like wave. In addition, the asymptotic solitons associated to the double- or triple-pole eigenvalues are localized in some logarithmical curves, and thus they have the variable velocities with the time dependence of attenuation.