Classification of single-wave and bi-wave motion through fourth-order equations generated from the Ito model

Abstract

Recently, two fourth-order integrable equations were established by Wazwaz using the Boussinesq model. Tian and Zhang subsequently demonstrated that both equations are potential forms of the Ito model. This study investigates the dynamics of these equations using three effective schemes: the modified rational sine-cosine functions, Kudryashov-expansion, and the Hirota bilinear forms The study reports novel findings, including the observation that although these equations were derived from the same model, one propagates as a single-wave while the other propagates as a bi-wave. Additionally, some solutions of one equation can be obtained from the solutions of the other equation. These results are expected to be highly significant in the study of propagation solitary wave-solutions for nonlinear equations.