Soliton and rogue wave excitations in the Chen–Lee–Liu derivative nonlinear Schrödinger equation with two complex $P T$-symmetric potentials

We demonstrate that fundamental nonlinear localized modes can exist in the Chen–Lee–Liu equation modified by several parity-time (⁠ $P T$⁠) symmetric complex potentials. The explicit formula of analytical solitons is derived from the physically interesting Scarf-II potential, and families of spatial solitons in internal modes are numerically captured under the optical lattice potential. By the spectral analysis of linear stability, we observe that these bright solitons can remain stable across a broad scope of potential parameters, despite the breaking of the corresponding linear $P T$-symmetric phases. When these bright spatial solitons interact with external incident waves, they can always maintain their original shape, while the external incident wave may remain unchanged or may generate a reflected wave after the interaction. Then, the adiabatic switching of potential parameters is carried out in a way that allows these bright solitons to be excited from one unstable bound state to another alternative stable bound state. Many other intriguing properties associated with these nonlinear localized modes including the lateral power flow are further analyzed meticulously. Various high-order rogue waves induced by modulation instability in these $P T$-symmetric systems are generated too. These results may be useful to construct novel optical soliton communication schemes or design related optical materials.