The interaction of weakly nonlinear long internal gravity waves is studied. Weak interactions occur when the wave phase speeds are unequal; this case includes that of a head‐on collision. It is shown that each wave satisfies a Korteweg‐de Vries equation, and the main effect of the interaction is described by a phase shift. Strong interactions occur when the wave phase speeds are nearly equal although the waves belong to different modes. This case is described by a pair of coupled Korteweg‐de Vries equations, for which some preliminary numerical results are presented.

Solitary waves in density stratified fluids of shallow depth are described, to first order in wave amplitude, by the Korteweg–de Vries equation; the solution for a single solitary wave has the familiar ‘‘sech2’’ profile and a phase speed which varies linearly with the wave amplitude. This theory is here extended to second order in wave amplitude. The second-order correction to the wave profile and the phase speed and the first-order correction to the wavelength are all determined. Four special cases are discussed in detail. In certain special circumstances the first-order theory may fail due to the vanishing of the nonlinear coefficient in the Korteweg–de Vries equation. When this occurs a different theory is required which leads to an equation with both quadratic and cubic nonlinearities.

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Strong interactions between weakly nonlinear long waves are studied. Strong interactions occur when the linear long wave phase speeds are nearly equal although the waves belong to different modes. Specifically we study this situation in the context of internal wave modes propagating in a density stratified fluid. The interaction is described by two coupled Korteweg-deVries equations, which possess both dispersive and nonlinear coupling terms. It is shown that the coupled equations possess an exact analytical solution involving the characteristic" sech 2 "profile of the Korteweg-deVries equation. It is also shown that when the coefficients satisfy some special conditions, the coupled equations possess an n-soliton solution analogous to the Korteweg-deVries n-soliton solution. In general though the coupled equations are found not to be amenable to solution by the inverse scattering transform technique, and thus a numerical method has been employed in order to find solutions. This method is described in detail in Appendix A. Several numerical solutions of the coupled equations are presented.