Envelope soliton solution for finite amplitude equatorial waves

Using shallow water equations on an equatorial beta plane, the nonlinear dynamics of the equatorial waves is investigated. A general mathematical procedure to study the nonlinear dynamics of these waves is developed using the asymptotic method of multiple scales. On faster temporal and spatial scales the equations describe the equatorial wavesviz, the Rossby waves, Rossby gravity waves, the inertia gravity waves and the Kelvin waves. Assuming that the amplitude of these waves are functions of slower time and space scales, it is shown that the evolution of the amplitude of these waves is governed by the nonlinear Schrodinger equation. It is then shown that for the dispersive waves like Rossby waves and Rossby-gravity waves, the envelope of the amplitude of the waves has a ‘soliton’ structure.