Formation and propagation dynamics of peakons and double-hump solitons of the generalized focusing/defocusing NLS equations with PT-symmetric δ(x)-sech optical potentials

In this paper, we investigate several important properties of the generalized nonlinear Schrodinger equation with PT-symmetric delta-sech optical potentials, such as the phase breaking of PT-symmetric delta-sech potentials, and existence, dynamics and excitations of new soliton solutions. Specifically, we identify the fully real spectral region of the non-Hermitian Hamiltonian and observe the phase-breaking phenomenon. Additionally, we discover a new soliton solution that represents a peakon solution, a smooth soliton solution, and a double-hump soliton solution under different potential parameters. We analyze the stability of these three types of solutions and determine their stability domains. Furthermore, we study the numerical peakon solution and its stability. In particular, we investigate the interaction of peakon solutions and observe semi-elastic interactions. Finally, we explore the stable adiabatic excitations of peakons. These results contribute to a deeper understanding of PT-symmetric optical fields and provide insights for related experimental works.