The interaction of weakly nonlinear, long internal gravity waves in neighboring pycnoclines is studied. Two coupled equations which describe the evolution of the wave amplitudes are derived. These equations are shown to possess three conserved quantities. The numerical results demonstrate the existence of two types of periodic nonlinear wave solutions when mode-two waves propagate along each pycnocline with nearly equal speeds. The energy transfer between these resonant waves is discussed for two pycnocline separations.

Weak coupling between nonlinear internal solitary waves on neighbouring pycnoclines allows resonant energy exchange. The lagging wave increases its energy and speed at the expense of the front-running wave, so that the waves leapfrog about an average position. Analytical estimates for this process agree with the wave-tank experiments described in the companion paper by Weidman & Johnson (1982).

The propagation of an internal wave train in a stratified shear flow is investigated for a Boussinesq fluid in a horizontal channel. Linear effects are primarily reflected in the dispersion relation for the various modes. The phenomenon of Eckart resonance occurs for more realistic stratification profiles. The evolution of nonlinear internal wave packets is studied through a systematic perturbation analysis. A nonlinear Schrodinger equation for the envelope of the internal wave train is derived. Depending on the relative sign of the dispersive and nonlinear terms, a wave train may disperse or form an envelope soliton. The analysis demonstrates the existence of two types of critical layers: one the ordinary critical point where ū=c, while the other occurs where ū=cg. In order to calculate the coefficients of the nonlinear Schrodinger equation a numerical code has been developed which computes the second‐harmonic and induced mean motions. The existence of these envelope solitons and their dependence on environmental conditions are discussed.