Particle orbits from the Lagrangian and the Eulerian Korteweg-de Vries equations

The particle orbits obtained by integrating the velocity field of the Eulerian Korteweg–deVries (KdV) equation and the trajectories given by the Lagrangian KdV equation are contrasted. It is shown that the two classes of orbits, while apparently equivalent, may be quite different. In particular, a spurious wave drift is generated by integrating the Eulerian velocities. It is shown that these differences are due to a mixing of perturbation orders inherent in the integration of the Eulerian velocity field. It is believed that these results may have some implications on the calculation of particle orbits in Eulerian flow models obtained by a perturbation expansion of the primitive equations.