Formally second-order correct, mathematical descriptions of long-crestedwater waves propagating mainly in one direction are derived. These equationsare analogous to the first-order approximations of KdV- or BBM-type. Theadvantage of these more complex equations is that their solutions correspondingto physically relevant initial perturbations of the rest state may be accurateon a much longer time scale. The initial-value problem for the class ofequations that emerges from our derivation is then considered. A localwell-posedness theory is straightforwardly established by way of a contractionmapping argument. A subclass of these equations possess a special Hamiltonianstructure that implies the local theory can be continued indefinitely.
展开▼
