Rouge Wave, W-Shaped, Bright, and Dark Soliton Solutions for a Generalized Quasi-1D Bose–Einstein Condensate System with Local M-Derivative

In this paper, a generalized quasi-1D Bose–Einstein condensate system with contact repulsion and dipole–dipole attraction (QBECS) and with the local M-derivative of order α is introduced. Using similarity transformation, the generalized QBECS is transformed into the same system but with constant coefficients under certain conditions. Finally, the travelling wave transformation is used for getting rogue waves and soliton solutions for the original equation. The effect of the fractional order $$\alpha$$ on the wave profile is discussed using some figures.