Exact Elliptic and Solitary Waves in a Fourth-Order Generalized Burgers–Huxley-Type Equation with Truncated M-Fractional Derivatives

Most studies of generalized Burgers–Huxley equations model spatial transport with a second-order diffusion term. Fourth-order spatial terms have received less attention, although they occur in long-range dispersal and regularized reaction–diffusion models and change the dominant balance of traveling-wave reductions. We consider a fourth-order generalized Burgers–Huxley-type equation with local truncated M-fractional derivatives and reaction polynomials of degrees two, three, or five. For the differentiable solutions considered here, the truncated M-operator is local, and its parameters enter through an exact nonlinear reparameterization of the independent variables rather than through a memory kernel. A derivative-enriched elliptic expansion reduces the resulting fourth-order differential equation to algebraic coefficient systems. The bounded auxiliary profiles are Jacobi sn, cn, and dn functions, with tanh and sech obtained when roots of the auxiliary quartic coalesce. Within this ansatz, the coefficient systems yield two convective quadratic, two stationary quadratic, three stationary cubic, and one stationary quintic class in centered and shifted forms; only the convective quadratic families admit nonzero speed. Reality conditions and equivalences under reflection, phase and sign changes, and singular limits identify distinct bounded profiles. These profiles may provide closed-form reference solutions for numerical studies and illustrate the effect of the local coordinate transformation.