PT symmetric cubic-quintic nonlinear Schr?dinger equation with dual power nonlinearities and its solitonic solutions

We propose a more general nonlocal integrable nonlinear Schrödinger equation with nonlocal cubic and quintic nonlinearities in addition to external potential. We employ similarity transformation and Lax-pair method to achieve integrability of the model and then suitably exploit the Darboux transformation approach to generate more general solitonic solutions which enable us to exhibit all kinds of soliton pairs and breather solutions by fine-tuning spectral parameters. We also witness how one can manoeuver the parameters to suppress or retain the amplitude of the solitons desirably. We also show how the interaction parameter can be manipulated to transfer energy from unstable mode to stable mode in a two soliton solution. © 2020