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 Schrödinger equation and the ${\varphi }^4$ 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.

如果在给定输出的情况下始终可以以适当方式恢复输入，则称动态系统是可逆的。然而，我们认为具有时间反转对称性的可逆系统，例如非线性Schrödinger方程和${\varphi }^4$等式可能会变成“物理上不可逆的”。这样，我们的意思是，在测量输出时实际较小的实验误差会导致恢复的输入与原始输入之间的巨大差异。可逆性的丧失揭示了自然的“时间箭”，让人联想到热力学，即热辐射向外发射的方向。我们的结果与非线性光学中的成像和反转应用有关。