On internal solitary waves. II

The asymptotic evolution of straight-crested internal solitary waves, from prescribed initial conditions, in that parametric domain in which quadratic nonlinearity, cubic nonlinearity, and dispersion are weak and of comparable significance is calculated through inverse-scattering theory. An initially rectangular displacement, which presumably serves as an example of any initial displacement of compact support, yields N solitary waves, where N ≥ 1. The fastest of these waves may resemble a bore (which, by definition, implies a net change in surface level between upstream and downstream limits) but is evanescent in both the upstream and downstream limits. An initial change of surface level of the right strength with a rise facing in the direction of propagation is found to yield (asymptotically) a true bore, but the asymptotic solution for an opposite facing rise does not comprise such a component. This last difficulty suggests a problem of uniqueness for initial displacements that are not of compact support.