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.