Abstract: The algebraic solitary wave and its associated eigenvalue problem in a deep stratified fluid with a free surface and a shallow upper layer were studied.And its vertical structure was examined. An exact solution for the derived 2D Benjamin-Ono equation was obtained, and physical explanation was given with the corresponding dispersion relation. As a special case, the vertical structure of the weakly nonlinear internal wave for the Holmboe density distribution was numerically investigated, and the propagating mechanism of the internal wave was studied by using the ray theory.

Key words: two-dimensional Benjamin-Ono equation, eigenvalue, vertical structure, stratified fluid