Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions

The generalized derivative higher order non-linear Schrödinger (DNLS) equation describes pluses propagation in optical fibers and can be regarded as a special case of the generalized higher order non-linear Schrödinger equation. We derive a Lagrangian and the invariant variational principle for DNLS equation. Using the amplitude ansatz method, we obtain the different cases of the exact bright, dark and bright–dark solitary wave soliton solutions of the generalized higher order DNLS equation. By implementing the modulation instability analysis and stability analysis solutions, the stability analysis of the obtained solutions and the movement role of the waves are analyzed. All solutions are analytic and stable.