Solitons are interesting, I have always been curious if anyone has done any sort of genetic search for solitons for more general wave equations. There are only a couple of known analytic solutions to a few different wave equations, but finding approximate non-analytic solutions with something like a genetic algorithm might be interesting.
:-)
> Solitons are interesting, I have always been curious if anyone has done any sort of genetic search for solitons for more general wave equations. There are only a couple of known analytic solutions to a few different wave equations, but finding approximate non-analytic solutions with something like a genetic algorithm might be interesting.
My understanding of solitons is based on (classical/quantum) field theory. They correspond to "saddle points" of the underlying path integral (probability distribution, for practical purposes) which interpolate between different vacua (local minima, for practical purposes). For this reason, one cannot simply run an optimization algorithm on the log-likelihood distribution (i.e. action/Hamiltonian, etc) since we're not searching for minima!
I would expect this intuition to apply to the extent that the differential equations which we're trying to solve can be derived from an underlying action principle.
Framed the way solitons are usually framed, as a waveform that doesn't change shape when propagated, it seems one could frame a genetic algorithm with such a fitness function (how much the wavefunction changes).
In another sense, a saddle point is a minimum on one axis and a maximum on another... maybe optimization could be done in that way... though I'm not sure that is practical.