Exact solitary wave solutions and linear stability of the β-time-fractional Gardner equation in shallow water...

This study investigates the nonlinear dynamics of soliton structures governed by the fractional Gardner equation incorporating the $\beta$-time derivative. The model is of significant importance because it generalizes classical nonlinear evolution equations to capture memory effects that arise in dispersive media, plasma physics, and shallow-water waves. Using the improved modified extended tanh function method, a variety of exact analytical solutions are derived, including bright, dark, and singular solitons, as well as periodic wave profiles. The influence of the fractional-order parameter $\beta$ on the amplitude and stability of the resulting waveforms is analyzed in detail. Linear stability analysis is also performed to identify the stable regions of soliton propagation. Numerical simulations and three-dimensional plots confirm the validity of the obtained analytical results and illustrate the impact of $\beta$ on the soliton structure. The findings reveal that decreasing $\beta$ enhances the amplitude and steepness of the soliton, demonstrating the strong memory-dependent behavior of the system. Compared with previous works on the classical and time-fractional Gardner equations, the present results extend analytical solution classes by employing the $\beta$-fractional derivative, providing a deeper physical interpretation of fractional-order wave interactions. This novel approach contributes to a better understanding of nonlinear wave phenomena in fractional dispersive systems and bridges the gap between analytical and numerical frameworks.