On Madelung systems in nonlinear optics: A reciprocal invariance

Abstract

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 Schrödinger 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 class of reciprocal transformations. Model optical laws analogous to the celebrated Kármán-Tsien law of classical gasdynamics are introduced.

Moving Boundary Problems for Heterogeneous Media. Integrability via Conjugation of Reciprocal and Integral Transformations

“…Reciprocal-type transformations have previously had diverse physical applications in such areas as gasdynamics, magnetogasdynamics, nonlinear heat conduction, the theory of discontinuity wave propagation and invariance properties of classical capillairity and nonlinear optics systems (see e.g. [7,[24][25][26][27][28] and literature cited therein). In terms of practical moving boundary problems, reciprocal transformations have, in particular, been applied in the analysis of methacrylate distribution in wood saturation processes [8].…”

On relativistic gasdynamics: invariance under a class of reciprocal-type transformations and integrable Heisenberg spin connections

“…Such model constitutive laws also arise in Korteweg capillarity theory [39] and in the context of reciprocal invariance of Madelung systems in nonlinear optics [40]. In most recent work in [38], isentropic, 1+1-dimensional relativistic gasdynamics systems have been investigated with regard to the formation of shock waves and vacuum states for media with an extended model Chaplygin-type gas law p = Ae − B e − α , 0 < α ≤ 1.…”

“…In particular, attention has been commonly directed to relativistic gasdynamics motions with model pressureenergy density laws of Chaplygin-Kármán-Tsien type p ∼ e −1 and its variants, with origin in the approximation theory of plane, irrotational subsonic gasdynamics. Such model constitutive laws also arise in Korteweg capillarity theory [39] and in the context of reciprocal invariance of Madelung systems in nonlinear optics [40]. In most recent work in [38], isentropic, 1+1-dimensional relativistic gasdynamics systems have been investigated with regard to the formation of shock waves and vacuum states for media with an extended model Chaplygin-type gas law p = Ae − B e −α , 0 < α ≤ 1.…”

Recursive formulation of Madelung continuity equation leads to propagation equation

“…To rewrite the probability density G[v] δ[r − ξ(t 0 )] we perform the steps from equation (12) to equation (10) in reverse which leads us to…”

“…This concept revealed to be very fruitful in a number of applications and is now established in several branches of quantum mechanics 3 , such as Bose-Einstein-condensation 4,5 , condensed matter physics 6,7 and quantum cosmology 8,9 . It has been also applied as a useful tool to solve linear and non-linear partial differential equations [10][11][12][13] . We extend the Madelung fluid description by applying a recursive approach to its continuity equation which leads to a propagation equation.…”