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A nonlinear wave mechanical equation is proposed by inserting an imaginaryquantum potential into the Schr\"{o}dinger equation. An explicit expression forits solution is given under certain assumptions and it is shown that it entailsattenuation related effects as non-unitary evolution, non-exponential quantumdecay and entropy production. In the quantum hydrodynamical formulation theexistence of circulation effects for the osmotic velocity field is established.Finally, a time-invariant equation for the probability density is derived,analogous to the tensor Lighthill equation in aeroacoustics, which admits bothretarded and advanced solutions.

Induced osmotic vorticity in the quantum hydrodynamical picture