In this paper, by using a transformation and an application of Fan subequation, we study a class of generalized Korteweg–de Vries (KdV) equation with generalized evolution. As a result, more types of exact solutions to the generalized KdV equation with generalized evolution are obtained, which include more general single-hump solitons, multihump solitons, kink solutions and Jacobian elliptic function solutions with double periods.

By constructing auxiliary equations and combining the expansion method of $(G'/G, 1/G)$, we study a class of nonlinear Schrödinger equation with perturbation terms which describes the propagation of the waves in optical metamaterials. More types of exact solutions, particularly solitary wave solutions, are obtained for the first time.