  In this paper, we investigate the dichotomous behavior of solutions to the
Kawahara equation with bounded variation initial data, analogous to the Talbot
effect. Specifically, we observe that the solution is quantized at rational
times, whereas at irrational times, it is a nowhere continuous differentiable
function with a fractal profile. This phenomenon, however, has not been
explored for the Kawahara equation, which is a fifth-order KdV type equation.
To achieve this, we derive smoothing estimates for the nonlinear Duhamel
solution, which, when combined with the known results on the linear solution,
provides a mathematical description of the Talbot effect.
