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Semi-analytical solutions for dispersive shock waves in colloidal media

Journal Article


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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 hard-sphere gas with the

    Carnahan–Starling formula used for the gas compressibility. The governing equation is a

    focusing nonlinear Schr¨odinger-type 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 semi-analytical solution is developed in the

    (1 + 1) dimensional case by approximating the dispersive shock wave as a train of uniform

    solitary waves. A semi-analytical 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 S-shaped 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 steady-state

    response diagrams, also typically found in combustion applications, are unusual in

    applications involving solitary waves. The predictions of the semi-analytical 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 semi-analytical results for a dispersive shock wave in a focusing medium.

Publication Date


  • 2012

Citation


  • Marchant, T. R. & Smyth, N. F. (2012). Semi-analytical solutions for dispersive shock waves in colloidal media. Journal of Physics B: Atomic, Molecular and Optical Physics, 45 (14), 1-9.

Scopus Eid


  • 2-s2.0-84863713458

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2297&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/1276

Number Of Pages


  • 8

Start Page


  • 1

End Page


  • 9

Volume


  • 45

Issue


  • 14

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 hard-sphere gas with the

    Carnahan–Starling formula used for the gas compressibility. The governing equation is a

    focusing nonlinear Schr¨odinger-type 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 semi-analytical solution is developed in the

    (1 + 1) dimensional case by approximating the dispersive shock wave as a train of uniform

    solitary waves. A semi-analytical 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 S-shaped 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 steady-state

    response diagrams, also typically found in combustion applications, are unusual in

    applications involving solitary waves. The predictions of the semi-analytical 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 semi-analytical results for a dispersive shock wave in a focusing medium.

Publication Date


  • 2012

Citation


  • Marchant, T. R. & Smyth, N. F. (2012). Semi-analytical solutions for dispersive shock waves in colloidal media. Journal of Physics B: Atomic, Molecular and Optical Physics, 45 (14), 1-9.

Scopus Eid


  • 2-s2.0-84863713458

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2297&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/1276

Number Of Pages


  • 8

Start Page


  • 1

End Page


  • 9

Volume


  • 45

Issue


  • 14