Resonant Interactions of Poincare Waves in the Shallow Water Approximation

Abstract: The paper develops a weakly nonlinear theory of Poincare waves. The nondegeneracy of the Poincare wave dispersion law leads to the presence of resonant interactions in perturbation theory. A study of the dispersion relation of Poincare 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 Poincare waves. Equations of nonlinear interactions of four waves for the amplitudes of Poincare 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 Poincare waves, which is important for geophysical hydrodynamics, has been predicted. An analytical solution is obtained that describes the saturation effect of Poincare waves in time.