Exact Solitary Wave Solutions in Shallow Water

Long's equation describes stationary flows to all orders of nonlinearity and dispersion. Dissipation is neglected. In this paper, Long's equation is used to attempt to model the propagation of a solibore -a train of internal waves in shallow water at the deepening phase of the internal tide. 1. The Solibore Phenomenon The internal tide in shallow water often has a sawtooth shape rather than a sinusoidal shape. The point of the tooth is not a simple jump as in a turbulent surface tidal bore, but consists of a train of waves. This wave train is called a solibore. The individual waves are often referred to as solitons, because they were originally modeled as solitons in the Korteweg-de Vries (KdV) equation. In many situations, the KdV modeling is highly inaccurate, as in Stanton and Ostrovsky, 1998; often an equation called the extended KdV, eKdV, is used . An example of a solibore is shown in Figure 1.