Abstract

The diffractive resolution of a discontinuity at the edge of an optical beam in a colloidal
suspension of spherical dielectric nanoparticles by a collisionless shock, or dispersive shock
wave, is studied. The interaction of the nanoparticles is modelled as a hardsphere gas with the
Carnahan–Starling formula used for the gas compressibility. The governing equation is a
focusing nonlinear Schr¨odingertype equation with an implicit nonlinearity. It is found that the
discontinuity is resolved through the formation of a dispersive shock wave which forms before
the eventual onset of modulational instability. A semianalytical solution is developed in the
(1 + 1) dimensional case by approximating the dispersive shock wave as a train of uniform
solitary waves. A semianalytical solution is also developed for a (2 + 1) dimensional circular
dispersive shock wave for the case in which the radius of the bore is large. Depending on the
value of the background packing fraction, three qualitatively different solitary wave amplitude
versus jump height diagrams are possible. For large background packing fractions a single
stable solution branch occurs. At moderate values an Sshaped response curve results, with
multiple solution branches, while for small values the upper solution branch separates from the
middle unstable branch. Hence, for low to moderate values of the background packing fraction
the dispersive shock bifurcates from the low to the high power branch as the jump height, the
height of the input beam’s edge discontinuity, is increased. These multiple steadystate
response diagrams, also typically found in combustion applications, are unusual in
applications involving solitary waves. The predictions of the semianalytical theory are found
to be in excellent agreement with numerical solutions of the governing equations for both line
and circular dispersive shock waves. The method used represents a new technique for
obtaining semianalytical results for a dispersive shock wave in a focusing medium.