Lax pair and Darboux transformation for the generalized Fornberg–Whitham equation: Lump, breather, periodic, and rogue wave solutions

In this article, we investigate a generalized Fornberg–Whitham (gFW) equation. Based on the Lax pair and Darboux transformation, one-soliton solution, breather solutions, lump solution, lump–1-strip, lump–periodic, manifold periodic, and rogue wave solutions for the gFW equation are obtained by choosing appropriate transformations and symbolic computation. In addition, we evaluate Ma breather, Kuznetsov–Ma breather and their corresponding rogue waves, generalized breather, and Akhmediev breathers. The obtained solutions reveal rich nonlinear wave phenomena such as localized structures, recurrence patterns, and energy concentration effects. These findings have direct applications in fluid dynamics, optical fiber communications, and plasma physics, where such nonlinear behaviors are prevalent. Moreover, the explicit construction of rogue waves and breather interactions provides insight into the prediction and control of extreme wave events in oceanography and nonlinear optics. The results further demonstrate the utility of analytical methods in exploring complex wave structures in integrable and near-integrable systems.