Advection, Diffusion, Dispersion and All Things in Between

Abstract. As an example in their complex variables textbook, Mark Ablowitz and Athanassios Fokas apply the method of steepest descent to derive an asymptotic estimate for the solution of the linear Korteweg–de Vries (KdV) equation for large times. Here the analysis is extended to an equation with a fractional dispersion relation that contains this KdV equation (which models dispersion), the heat equation (diffusion), and the one-way wave equation (advection) as special cases.