Invariance analysis, optimal system, and group invariant solutions of (3+1)-dimensional non-linear MA-FAN equation

The importance of research on non-linear evolution equations has increased over time. In the fields of plasma physics, fluid mechanics, optical fibers, and other sciences, such equations are not unrealistic. Discovering the exact answers to these equations needs to be a top priority. In this study, we will investigate a (3+1)$$ \left(3+1\right) $$-dimensional non-linear MA-FAN equation that is used to simulate weakly non-linear restoring forces and frequency dispersion in long wavelength water waves. The (3+1)$$ \left(3+1\right) $$-dimensional MA-FAN equation's solution is analyzed using the Lie symmetry analysis, and it is also used to pick the appropriate one-dimensional Lie subalgebra system. We discover invariant solutions of the (3+1)$$ \left(3+1\right) $$-dimensional MA-FAN equation, which may capture the dynamic behavior of non-linear waves, by solving the numerous ordinary differential equations that are reduced equations from the MA-FAN equation by using the Lie group of transformation approach. In conclusion, the method offered is robust and convenient tool for studying various classes of non-linear PDE solutions.