Abstract:
Formally second-order correct approximations of long-crested water waves propagating mainly in one direction are derived. These equations are analogous to the first-order approximations of KdV- or BBM-type. The advantage of these more complex equations is that their solutions corresponding to physically relevant initial perturbations of the rest state are expected to be accurate on a much longer time scale. The initial-value problem for the class of equations that emerges from our derivation is then considered. A local well-posedness theory is straightforwardly established by way of a contraction mapping argument. A subclass of these equations possess a special Hamiltonian structure that may be used to show that the local theory can be continued indefinitely. |