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The Caudrey–Dodd–Gibbon (CDG) model, a variation of the fifth–order KdV equation (fKdV) with significant practical consequences, is solved in this study using a precise and numerical technique. This model shows how gravity-capillary waves, shallow–water waves driven by surface tension, and magneto-acoustic waves move through a plasma medium. With a focus on accuracy, new computational and approximation methods have been made possible by recent improvements in analytical and numerical methods. Numeric information is represented visually in the tables. All simulation results are shown in two and three dimensions to show both the numerical and fundamental behavior of the single soliton. Recent research shows that this method is the best way to solve nonlinear equations that are common in mathematical physics.