  A dynamical system is said to be reversible if, given an output, the input
can always be recovered in a well-posed manner. Nevertheless, we argue that
reversible systems that have a time-reversal symmetry, such as the Nonlinear
Schr{\"o}dinger equation and the $\phi ^4$ equation can become "physically
irreversible". By this, we mean that realistically-small experimental errors in
measuring the output can lead to dramatic differences between the recovered
input and the original one. The loss of reversibility reveals a natural "arrow
of time", reminiscent of the thermodynamic one, which is the direction in which
the radiation is emitted outward. Our results are relevant to imaging and
reversal applications in nonlinear optics.
