A Theoretical Study of an Extended KDV Equation

Discovered experimentally by Russell and described theoretically by Korteweg and de Vries, KdV equation has been a nonlinear evolution equation describing the propagation of weakly dispersive and weakly nonlinear waves. This equation received a lot of attention from mathematical and physical communities as an integrable equation. The objectives of this paper are: first, providing a rigorous mathematical derivation of an extended KdV equations, one on the velocity, other on the surface elevation, next, solving explicitly the one on the velocity. In order to derive rigorously these equations, we will refer to the definition of consistency, and to find an explicit solution for this equation, we will use the sine-cosine method. As a result of this work, a rigorous justification of the extended Kdv equation of fifth order will be done, and an explicit solution of this equation will be derived.