The first part reveals a phenomenon in fluid mechanics that can be viewed as paradoxical: horizontal momentum conservation is violated in the dynamics of a stratified ideal fluid filling an infinite horizontal channel between rigid bottom and lid boundaries, starting from localized initial conditions, even though external forces only act on the vertical direction. The paradox is shown to be a consequence of the rigid lid constraint coupling through incompressibility with the infinite inertia of the far ends of the channel, assumed to be at rest in hydrostatic equilibrium. By the perturbation theory based on small density variance, an analytical study quantifies the momentum development at the initial time. These results are compared with direct numerical simulations for variable density Euler fluids. The second part is a numerical study of strongly nonlinear models for internal waves. We emulate numerically the generation of solitary wave motivated by a laboratory experiment. The dam-break problem for internal waves can be solved by direct numerical simulations (DNS). By smoothing out the dam and symmetric extension of the wave tank, the strongly nonlinear model is ready for implementation. The Kelvin-Helmholtz instability associated with the model is treated by a time-dependent low-pass filter. The regularized strongly nonlinear model with less-restrictive stability criterion is also considered. The snapshots of the models and DNS show excellent agreements between models and DNS. The effect of numerical filters are considered to behave as reducing dissipation. The third part consists of the comparisons among weakly nonlinear models for internal waves by providing predictions for the two-layer dam-break problem. We regularized a completely integrable but ill-posed system, the two-layer Kaup equations. The new equations are numerically solvable and provide better agreement with the inverse scattering transform prediction for the Kaup equations than the Boussinesq equations, another weakly nonlinear model for bi-directional waves. A higher order uni-directional model is also considered to cope with moderate amplitude waves. These models are compared for their traveling wave solutions, phase speed and amplitude relations, and dispersion relations. The time evolutions for the two-layer dam-break problem from weakly nonlinear models are compared with DNS.