Abstract
We report a class of physically interesting PT-symmetric δ(x)-csch potentials containing three types of potentials: δ(x)-csch, δ(x), and csch-coth potentials. Firstly, we study the parameter regions for the PT phase transitions of the non-Hermitian Hamiltonians. Then we show that the both self-focusing and defocusing generalized nonlinear Schrödinger (NLS) equations with the PTδ(x)-csch potentials can support the physically intriguing csch-type peakon solitons. Moreover, we observe that they can stably propagate within certain parameter regions. And, we also study numerical peakon solutions and their stability for different propagation constants. In particular, the stable peakon solutions of the cubic and quintic NLS equations can be numerically found for the PT finite deep csch-coth potential, without involvement of the δ(x). Moreover, we find that the hyperbolic part of the potential can make the unstable states excite the stable ones in the cubic and quintic NLS equations. Finally, we investigate the interactions of nonlinear modes with exotic waves and stable adiabatic excitations of peakons. These results will have the implication for understanding the relevant physical phenomena.
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