Internal waves in two-layer stratified flows

A problem of internal waves in a two-layer stratified fluid with a density that depends exponentially on the depth inside the layers and has a jump at the interface is considered. An equation for the second long-wavelength approximation is derived, which describes solitary waves. The spectral properties of the equations of small perturbations of a horizontal piecewise constant flow are studied and the possible mechanisms for the occurrence of shear instability of a stratified flow are characterized.