A rigorous derivation of the extended KdV equation

Abstract

The interesting background and historical development of KdV equations were discussed widely. These equations describe the propagation of water waves in weakly non linear and weakly dispersive medium. Referring to physical derivation of KdV equations, scientists used to impose shallow water equations, thus the formal or physical derivation of KdV equations. However, these equations have rarely been derived rigorously. The aim of this paper is to giving insight into their rigorous mathematical derivation, instead of only referring to. Thereby, a rigorous derivation of two extended KdV equations: one on the velocity, other on the surface elevation. With this aim in mind, the primary research method for this paper will depend on the definition of consistency. Hence, a rigorous justification of new extended KdV equations will be provided thanks to this definition. This result provides a precise mathematical answer to a question raised by several authors in the last years, that is the verification of the extended KdV equations, derived previously, using formal methods.