Dispersive shock waves (DSWs) in nonlocal nonlinear optics are useful tools to simulate the quantization of decay rates predicted by time asymmetric quantum mechanics (TA-QM). 
Foundations of QM are designed by its physical interpretation, hence each observable must be a Hermitian operator to guarantee the reality of its spectrum on a Hilbert space. TA-QM shows that the reality of the Hamiltonian spectrum is not necessary. On the contrary, the imaginary part of a complex energy takes into account the presence of an irreversible evolution. In order to get this, TA-QM considers Hamiltonians that are still Hermitian, but act on a rigged Hilbert space, where wavefunctions are superpositions of non normalizable functions, called Gamow vectors (GVs). These GVs exponentially decay with quantized decay rates, a peculiar signature of the theory.  
We show that classical DSWs in a nonlocal nonlinear optical medium sustain this TA-QM peculiarity, exhibiting exponential decays with quantized decay rates.