Derivation and simulation of the multiple lump solutions to an integrable generalized fifth-order Korteweg–de Vries equation

Abstract In this paper, we develop two methods for constructing lump solutions to the generalized fifth-order Korteweg–de Vries (KdV) equation, including the inverse scattering transform (IST) method and the ∂¯-dressing method. Making use of the IST method, we construct standard M-lump solutions originating from eigenfunctions containing simple poles, while also obtaining non-standard lump solutions corresponding to eigenfunctions with higher-order poles. Moreover, we present both standard three-lump solutions and non-standard four-lump solutions, which have not been reported in previous studies. By applying the Zakharov–Manakov ∂¯-dressing method, we construct two classes of lump solutions for the generalized fifth-order KdV equation with integrable boundary condition uy|y=0, corresponding to kernels with purely imaginary and real spectral points, respectively.