  In this paper, the Talbot effect for the multi-component linear and nonlinear
systems of the dispersive evolution equations on a bounded interval subject to
periodic boundary conditions and discontinuous initial profiles is
investigated. Firstly, for a class of two-component linear systems satisfying
the dispersive quantization conditions, we discuss the fractal solutions at
irrational times. Next, the investigation to nonlinear regime is extended, we
prove that, for the concrete example of the Manakov system, the solutions of
the corresponding periodic initial-boundary value problem subject to initial
data of bounded variation are continuous but nowhere differentiable
fractal-like curve with Minkowski dimension 3/2 at irrational times. Finally,
numerical experiments for the periodic initial-boundary value problem of the
Manakov system, are used to justify how such effects persist into the
multi-component nonlinear regime. Furthermore, it is shown in the nonlinear
multi-component regime that the interplay of different components may induce
subtle different qualitative profile between the jump discontinuities,
especially in the case that two nonlinearly coupled components start with
different initial profile.
