Resonant Interactions of Poincaré Waves in the Shallow Water Approximation

The paper develops a weakly nonlinear theory of Poincaré waves. The nondegeneracy of the Poincaré wave dispersion law leads to the presence of resonant interactions in perturbation theory. A study of the dispersion relation of Poincaré waves showed that three-wave interactions are absent in the quadratic nonlinear approximation. In this paper, a linear equation of the envelope is derived. A qualitative study of the dispersion law showed the existence of four-wave interactions of Poincaré waves. Equations of nonlinear interactions of four waves for the amplitudes of Poincaré waves are derived. The Manley–Rowe equations are obtained, which determine the distribution of energy and its transfer between interacting waves. The nonlinear dynamics of interacting waves is investigated. The saturation effect of Poincaré waves, which is important for geophysical hydrodynamics, has been predicted. An analytical solution is obtained that describes the saturation effect of Poincaré waves in time.