Layman Summary
The goal of this paper is to study how waves move over a fluid in with an underlying current. Oceans displace debris and heat. Ocean models can be used to predict the movement of debris and even predict the effect of the movements of oceans on the climate. Modeling water can be improved by introducing more information from observations into the models. The goal of this paper is to combine the effect of varying density and an underlying current in a model for ocean water near the equator. Our model focuses on long, shallow waves. This allows us to simplify the model and solve it more easily. We combine methods from different sources to create a model for our problem. The model we create shows there are two types of solution waves: Periodic travelling waves and Solitons. Solitons are single waves that don’t change shape through time. We find that density influences the amplitude of waves and the perturbation of the current. We find that the underlying current influences the wave profile’s width as well as the amplitude.

Summary
Models of oceans can be used to predict the displacement of debris and even trace its path back to its origin. Oceans are a large influence on the weather and climate all over the world. Improving these models is therefore very useful.
There isn’t a general equation that describes all water dynamics. Even if there would be, we would not have a computer good enough to make all the necessary calculations for the model. Though there will not be a perfect model, there is still a lot of room to improve the current models. The accuracy of water models, specifically for oceans, can be improved in different ways. We can increase the resolution to model smaller, more intricate behavior. The model can also be improved by coupling more different phenomena.
The goal of this paper is to combine existing methods for modeling the propagation of waves in a model that describes the propagation of waves over a current in a stratified fluid. We assume the fluid to be inviscid and incompressible. We also assume there is no thermal conductivity. We choose to focus the model on long and shallow waves and changes in the current. These
assumptions mean we can choose to base the governing equations on the Euler equations for inviscid fluids. Through non-dimensionalisation and scaling transformations, we transform our model to unitless equations. We examine only the behavior of the leading order solutions by expanding the unknown variables as asymptotic series. The final equations that describe the wave propagation belong to the family of Korteweg-deVries equations. One solution to these equations is a sech2(θ) function. They represent soliton waves. These are
solitary waves that hold their shape through the combination of dispersion and the non-linear character of the waves.
We find that the density influences the amplitude of waves and the perturbation of the current because it shows up as a multiplicative factor. We find that the underlying current influences the wave profile’s width as well as the amplitude because it also shows up inside the θ term of the sech^2(θ) function.