Utilizing a Coupled Nonlinear Schr\"{o}dinger Model to Solve the Linear Modal Problem for Stratified Flows*

The spatial structure of small disturbances in stratified flows without background shear, usually named the `Taylor-Goldstein equation', is studied by employing the Boussinesq approximation (variation in density ignored except in the buoyancy). Analytical solutions are derived for special wavenumbers when the Brunt-V\"{a}is\"{a}l\"{a} frequency is quadratic in hyperbolic secant, by comparison with coupled systems of nonlinear Schr\"{o}dinger equations intensively studied in the literature. Cases of coupled Schr\"{o}dinger equations with four, five and six components are utilized as concrete examples. Dispersion curves for arbitrary wavenumbers are obtained numerically. The computations of the group velocity, second harmonic, induced mean flow, and the second derivative of the angular frequency can all be facilitated by these exact linear eigenfunctions of the Taylor-Goldstein equation in terms of hyperbolic function, leading to a cubic Schr\"{o}dinger equation for the evolution of a wavepacket. The occurrence of internal rogue waves can be predicted if the dispersion and cubic nonlinearity terms of the Schr\"{o}dinger equations are of the same sign.