Modulation Instability and Abundant Exact Solitons to the Fractional Mathematical Physics Model Through Two Distinct Methods

The paper consists of various types of wave solutions for the truncated M-fractional Bateman–Burgers equation, a significant mathematical physics equation. This model describes the nonlinear waves and solitons in different physical fields such as optical fibers, plasma physics, fluid dynamics, traffic flow, etc. Through the application of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo form="prefix">exp</mo><mi>a</mi></msub></semantics></math></inline-formula> function method and the modified simplest equation method, we are able to obtain exact series of soliton solutions. The results differ from the current solutions of the Bateman–Burgers model because of the fractional derivative. The achieved results could be helpful in various engineering and scientific domains. The Mathematica software is used to assist in obtaining and verifying the exact solutions and to obtain contour plots of the solutions in two and three dimensions. To ensure that the model in question is stable, a stability analysis is also carried out using the modulation instability method. Future research on the system in question and related systems will benefit from the findings. The methods used are simple and effective.