Advanced fractional soliton solutions of the Joseph–Egri equation via Tanh–Coth and Jacobi function methods

This study introduces new exact soliton solutions of the time-fractional Joseph-Egri equation by employing the Tanh-Coth and Jacobi Elliptic Function methods. Using Jumarie's modified Riemann-Liouville derivative, a wide variety of soliton structures-such as periodic, bell-shaped, W-shaped, kink, and anti-bell-shaped waves-are obtained and expressed through hyperbolic, trigonometric, and Jacobi functions. The analysis reveals the significant impact of fractional-order derivatives on soliton dynamics, with graphical illustrations highlighting their physical relevance. This work expands the known solution space of the fractional Joseph-Egri equation, demonstrates the effectiveness of advanced analytical techniques, and provides fresh insights into the behavior of fractional nonlinear waves, with potential applications in physics and engineering.