On the Integrability of an Extended Coupled Kuramoto–Sivashinsky System: Exact Solution and Conserved Vectors

Nonlinear waves described by an extended coupled Kuramoto–Sivashinsky system (KS) is analytically studied. The addition of the last second-order dissipative terms, which will radically change the characteristics of the equation, is investigated. We implement the classical symmetry method to construct exact solutions of the adopted model, in the presence of the last second-order dissipative terms. In order to achieve a variety of exact solutions of distinct physical structures for the model, we will use simplest equation methods and an ansatz method. The derived results indicate that the extended coupled Kuramoto–Sivashinsky system with second-order dissipative terms shows the richness of analytical solutions. Moreover, the conserved vectors will be constructed using the multiplier approach. In order to have a better comprehensive of the results, profile structures of the derived solutions will be analysed in detail. The findings can well exhibit complex waves and their underlying properties in a variety of physical features.