Here on this page is an interesting proposition that crossed my mind, and another method of utilizing the device overall. In general the structure of the device doesn't change at all. Although, the number of windings on what would now be considered the primary and no longer the field coil would require less windings for reasons concerning frequency response. Then this device could be successfully driven by an AC current and it would respond as a standard transformer. It's apparent there may be several advantages significant to the electrode-less electrolysis process. The main difference is that the system would not be effected by the degradation of a metal cathode or build up of byproducts produced by the electrodes on the anode and in the solution because, the voltage and current are induced in the same conductive solution containing the catalyst. Winding the coil may seem difficult until you consider machining glass as you would metals. Then it's possible to leave a gap between male, and female threaded fittings where the water and catalyst can co-exist as the conductive secondary winding.
The main difference between this coil and it's next of kin would be the core material. I would have to exchange the neodymium for some other ferro-magnetic material based upon the frequency responce to something like ferrite, or possibly soft iron, and reduce the number of windings to accomodate a higher frequency. Then it would operate in the same fashion as a standard power transformer within the selected frequency range. If you would like to take a look at what kind of circuit I have in mind as a driver for this go to my old page on the topic of chaotic resonance, at this link.
Now, analyzing the conductivity of most catalysts based upon conductivity. The conductivity is controlled by the quantity of a catalyst present in the solution. The more catalyst is added to the water, the more conductive it typically becomes until the solution is saturated. Therefore, once the fluid has filled the wound glass tube, there is no requirement for metal, wire, or electrodes involved in the process of electrolysis.