The non-linear propagation of different kinds of ion-acoustic waves (IAWs) in multi-component plasma involving proton beam, positive ions and isothermal electrons has been studied. Using the derivative expansion method of basic equations, namely the hydrodynamic and Poisson equations, they are reduced to a single evolution equation of the non-linear Schrödinger (NLS) equation. By applying this model to plasma formed in Earth’s magnetosphere, different waves can be predicted that express the properties of the plasma. Using the separating variables method and the G /G-expansion method, we derived the exact analytical solutions to the evolution equation by using different solution regions in which non-linear waves are defined. A comparison has been made between the solutions describing the differential equation in each region in which the solution can appear using the data on the Earth’s magnetosphere in the study by Alotaibi et al. (2021).
