On Madelung systems in nonlinear optics: A reciprocal invariance
The role of the de Broglie-Bohm potential, originally established as central to Bohmian quantum mechanics, is examined for two canonical Madelung systems in nonlinear optics. In a seminal case, a Madelung system derived by Wagner et al. via the paraxial approximation and in which the de Broglie-Bohm potential is present, is shown to admit a multi-parameter class of what are here introduced as "q-gaussons". In the limit as the Tsallis parameter q --> 1, the q-gaussons are shown to lead to standard gausson solitons as admitted by logarithmic nonlinear Schroedinger equation encapsulating the Madelung system. The q-gaussons are obtained for optical media with dual power-law refractive index. In the second case, a Madelung system originally derived via an eikonal approximation in the context of laser beam propagation and in which the de Broglie Bohm term is neglected, is shown to admit invariance under a novel class of two-parameter reciprocal transformations. Model optical laws analogous to the celebrated Karman-Tsien law of classical gasdynamics are introduced.