Soliton Perturbation Theory for the Compound KdV Equation

The compound Korteweg de-Vries (cKdV) equation that is going to be studied in this paper arises is in the study of nonlinear waves in fluids. In particular, it shows up in the context of internal gravity waves in a density-stratified ocean. A very common tool for describing these processes is the Korteweg-de Vries (KdV) equation and its modifications that are valid for small nonlinearity. But, in some cases, really strong nonlinear waves are observed. An example of this is the Coastal Ocean Probe Experiment (COPE) that was carried out in September 1995 in the Oergon Bay. The data of this experiment clearly indicated the presence of extremely strong trains of internal waves propagating to the shore. They had a form of isothermal depression consisting of solitary pulses with amplitudes of thermocline displacenments typically of order 10–20 m, and sometimes reaching 30 m. The character of these trains (internal bores) is rather typical of the tidally generated waves on shelves, but 30 m high solitons are usually encountered in deep ocean rather than on the shelf. More importantly, it is a fact that they propagate into the background of a very shallow, 5–7 m depth density jump (pynocline). This