## 6 Discussion  
The choice to construct the analytical solutions presented in Sections 4 and 5 as combinations (sums) of distinct special functions, such as hyperbolic, trigonometric, or elliptic functions with different independent variables, is motivated by the need to capture richer and more flexible dynamical behavior in the time-fractional Boussinesq-type systems. In contrast to classical (integer-order) models, where solutions typically describe coherent solitary or periodic waves with a single traveling wave velocity, the presence of fractional-order time derivatives introduces memory effects and nonlocal temporal dynamics, which naturally lead to the coexistence of multiple propagation modes with different spatialand temporal scales. Different types of special functions used in the construction of solutions correspond to distinct classes of wave behavior. For example, the hyperbolic functions typically present localized structures such as solitons or traveling fronts, while the trigonometric functions describe regular harmonic oscillations characteristic of linear or weakly nonlinear wave regimes. In addition, the elliptic functions, in turn, capture more complex, anharmonic periodicity and modulated patterns observed in structured or heterogeneous media.  
In the same context, the superposition of different functions associated with different wave speeds presented in the analytical solutions derived in Sections 4 and 5 enables the modeling of interactions between qualitatively different wave structures, such as slowly decaying dispersive waves, rapidly oscillating wave trains, and modulated periodic patterns within a unified analytical framework. The use of hyperbolic and trigonometric functions in combination reflects the interplay between localization and regular periodicity, while combinations involving elliptic functions introduce additional modulation effects, allowing for the description of anharmonic or double-periodic regimes. Notably, the pairing of hyperbolic and elliptic components can model solitary wave fronts modulated by periodic structures, whereas combinations of trigonometric and elliptic functions may represent interference between regular and anharmonic oscillations. This interpretation is supported by the representative solutions visualized in Figures 2-5.  
Figure 2 shows a solution composed of a hyperbolic and a trigonometric function with different traveling wave variables. The resulting profile exhibits a localized structure superimposed on a regular oscillatory background, illustrating the interaction between solitary and harmonic wave modes. Figure 3 presents a solution constructed from a hyperbolic and an elliptic function with distinct wave variables. The profile shows modulated anharomonic periodicity with a localized deformation, characteristic of complex wave interactions in fractional dispersive media. Figure 4 illustrates a multi-wave solution involving a fractional trigonometric and a fractional hyperbolic function. The surface plots reveal harmonic waves modulated by localized structures, representing multi-scale propagation behavior driven by memory effects. Figure 5 displays another combination of fractional trigonometric and hyperbolic functions, with amplitude-modulated oscillations exhibiting asymmetry and localized energy concentration again indicative of fractional dynamics where distinct wave types coexist and interact.  
Therefore, the multi-wave solutions obtained in Sections 4 and 5 can describe more complex wave processes that may arise from the competition and interaction of different spatial and temporal scales. Although the full physical interpretation of such solutions may be nontrivial, especially for arbitrary fractional orders, we hope that they offer some qualitative insights into the complex behavior of fractional dispersion systems beyond the scope of classical theory.