Evolution equations for strongly nonlinear internal waves

This paper is concerned with shallow-water equations for strongly nonlinear internal waves in a two-layer fluid, and comparison of their solitary solutions with the results of fully nonlinear computations and with experimental data. This comparison is necessary due to a contradictory nature of these equations which combine strong nonlinearity and weak dispersion. First, the Lagrangian (Whitham’s) method for dispersive shallow-water waves is applied to derivation of equations equivalent to the Choi–Camassa (CC) equations. Then, using the Riemann invariants for strongly nonlinear, nondispersive waves, we obtain unidirectional, evolution equations with nonlinear dispersive terms. The latter are first derived from the CC equations and then introduced semiphenomenologically as quasistationary generalizations of weakly nonlinear Korteweg–de Vries and Benjamin–Ono models. Solitary solutions for these equations are obtained and verified against fully nonlinear computations. Comparisons are also made with available observational data for extremely strong solitons in coastal zones with well expressed pycnoclines.