Interfacial long traveling waves in a two-layer fluid with variable depth

Long wave propagation in a two-layer fluid with variable depth is studied for specific bottom configurations, which allow waves to propagate over large distances. Such configurations are found within the linear shallow-water theory and determined by a family of solutions of the second-order ordinary differential equation (ODE) with three arbitrary constants. These solutions can be used to approximate the true bottom bathymetry. All such solutions represent smooth bottom profiles between two different singular points. The first singular point corresponds to the point where the two-layer flow transforms into a uniform one. In the vicinity of this point nonlinear shallow-water theory is used and the wave breaking criterion, which corresponds to the gradient catastrophe is found. The second bifurcation point corresponds to an infinite increase in water depth, which contradicts the shallow-water assumption. This point is eliminated by matching the "nonreflecting" bottom profile with a flat bottom. The wave transformation at the matching point is described by the second-order Fredholm equation and its approximated solution is then obtained. The results extend the theory of internal waves in inhomogeneous stratified fluids actively developed by Prof. Roger Grimshaw, to the new solutions types.