LAUSR

Wave motions play an important role in fluid dynamics and engineering issues. In this paper, a systematic study to the generalized KdV–Burgers–Kuramoto equation is presented resort to symmetry method. First, based on the Lie point symmetries of the generalized KdV–Burgers–Kuramoto equation, we get invariants and invariant solutions. In particular, we obtain series solutions of generalized KdV–Burgers–Kuramoto equation. Meanwhile, we find that this equation just exists in Lie point symmetries. Then, we present a conservation law, and derive reciprocal Bäcklund transformations of conservation law for the first time. Furthermore, Bäcklund transformation of KdV–Burgers–Kuramoto equation associated with truncated Painlevé expansion is studied for the first time. Subsequently, mCK method is employed to study KdV–Burgers–Kuramoto equation. Lastly, we investigate symmetries of generalized time fractional KdV–Burgers–Kuramoto equation, and derive a conservation law. The obtained results illustrate that symmetry method is a very powerful method to deal with nonlinear partial differential equations. The results provide theoretical support for explaining complex nonlinear phenomena.