Numerous Multi–Wave Solutions of the Time–Fractional Boussinesq–Like System via a Variant of the Extended Simple Equations Method (SEsM)

In this study, we extend the Simple Equations Method (SEsM) and adapt it for finding exact solutions of systems of fractional nonlinear partial differential equations (FNPDEs). In accordance with the extended SEsM, the analytical solutions to the studied FNPDEs are constructed as complex composite functions which combine two single composite functions, comprising power series of solutions of two simple equations or two special functions. The novel and innovative aspect of the proposed extended methodology is that the simple equations or the special functions used have different independent variables (different wave coordinates). In this context, the proposed approach is mostly suitable for application to systems of FNPDEs (or NPDEs) that describe the complex wave dynamics of real processes in different scientific fields such as fluid mechanics, plasma physics, optics, etc. The extended SEsM is applied to the time–fractional Boussinesq-like system, assuming that each system variable supports multi-wave dynamics, which involves the combined propagation of two distinct waves traveling at different wave speeds. As a result, numerous complex multi-wave solutions including combinations of different hyperbolic, elliptic and trigonometric functions are derived. In order to visualize the complex wave dynamics obtained through the applied analytical approach, some analytical solutions have been numerically simulated.