Abstract
•The QC nonlinear media with perturbation terms for NLSE are considered.•The bright, dark and singular soliton solutions are constructed.•The modified simple method and the trial equation method are implemented.•In addition, periodic singular wave solutions are deduced by the integration schemes.
In this work, we derive bright, dark and singular soliton solutions to quadratic-cubic nonlinear media with perturbation terms being present. We perform the modified simple and the trial equation algorithms to the considered model. In addition, periodic singular wave solutions will be constructed by the integration schemes.