Model equations for waves in stratified fluids

Model equations for gravity waves in horizontally stratified fluids are considered. The theories to be addressed focus on stratifications featuring either a single pycnocline or neighbouring pairs of pycnoclines. Particular models investigated include the general version of the intermediate long-wave equation derived by Kubota, Ko and Dobbs to simulate waves in a model system consisting of two homogeneous layers separated by a narrow region of variable density, and the related system of equations derived by Liu, Ko and Pereira for the transfer of energy between waves running along neighbouring pycnoclines. Issues given rigorous mathematical treatment herein include the well-posedness of the initial value problem for these models, the question of existence of solitary wave solutions, and theoretical results about the stability of these solitary waves.