Approximate Techniques for Dispersive Shock Waves in Nonlinear Media

Marchant, Timothy R.; Smyth, Noel F.

doi:10.1142/s021886351250035x

Cited by 21 publications

(22 citation statements)

References 39 publications

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“…(69) These will now be used to find an approximate solution for the nematic CDSW by assuming that it consists of a uniform series of solitary waves [69], as discussed above.…”

Section: Nematic Crossover Dswmentioning

confidence: 99%

Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals

Baqer

Smyth

2020

Physica D: Nonlinear Phenomena

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A full analysis of all regimes for optical dispersive shock wave (DSW) propagation in nematic liquid crystals is undertaken. These dispersive shock waves are generated from step initial conditions for the optical field and are resonant in that linear diffractive waves are in resonance with the DSW, resulting in a resonant linear wavetrain propagating ahead of it. It is found that there are six regimes, which are distinct and require different solution methods. In previous studies, the same solution method was used for all regimes, which does not yield solutions in full agreement with numerical solutions. Indeed, the standard DSW structure disappears for sufficiently large initial jumps. Asymptotic theory, approximate methods or Whitham modulation theory are used to find solutions for these resonant dispersive shock waves in a given regime. The solutions are found to be in excellent agreement with numerical solutions of the nematic equations in all regimes. It is found that for small initial jumps, the resonant wavetrain is unstable, but that it stabilises above a critical jump height. It is additionally found that the DSW is unstable, except for small jump heights for which there is no resonance and large jump heights for which there is no standard DSW structure.

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“…(69) These will now be used to find an approximate solution for the nematic CDSW by assuming that it consists of a uniform series of solitary waves [69], as discussed above.…”

Section: Nematic Crossover Dswmentioning

confidence: 99%

Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals

Baqer

Smyth

2020

Physica D: Nonlinear Phenomena

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“…In order to obtain an approximation for the amplitude of the solitary waves, we use uniform soliton theory [11,12]. The mass conservation equation of (1) is…”

Section: Uniform Soliton Theory and The Dispersive Shock Wavementioning

confidence: 99%

“…We assume that solitary waves generated by the bore have this same mass to energy ratio. However, for NLStype equations, the solitary waves amplitude is independent of wavenumber, so (12) applies for all wavenumbers [3]. We now use the semi-analytical expression for a single colloidal solitary wave (6) which gives < M >= 2a 2 w, < H >= P, where…”

Section: Uniform Soliton Theory and The Dispersive Shock Wavementioning

confidence: 99%

Dispersive shock waves in colloids with temperature dependent compressibility

Azmi

Marchant

2014

J. Nonlinear Optic. Phys. Mat.

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The formation of a dispersive shock wave in a colloidal medium, due to an initial jump in the light intensity, is studied. The compressibility of the colloidal particles is modeled using a series in the particle density, or packing fraction, where the virial coefficients depend on the particle interaction model. Both the theoretical hard disk and sphere repulsive models, and a model with temperature dependent compressibility, are considered. Experimental results for the second virial coefficient show that it is temperature dependent and that the particle interactions can be either repulsive or attractive. These effects are modeled using a power-law relationship. The governing equation is a focusing nonlinear Schrödinger-type equation with an implicit nonlinearity. The initial jump is resolved via a dispersive shock wave which forms before the onset of modulational instability. A semi-analytical solution is developed for the one-dimensional line bore case which predicts the amplitude of the solitary waves which form in the dispersive shock wave. The solitary wave amplitude versus jump height diagrams can exhibit three different kinds of behaviors. A unique solution, an Sshaped solution curve and multiple solution branches where the upper branch has separated from the lower branches. A bifurcation from the low to the high power branch can occur for many parameter choices as the amplitude of the initial jump increases. The effect of temperature on the evolution of the bore, the amplitude of the solitary waves and the bifurcation patterns are all discussed and the semi-analytical solutions are found to be very accurate. AbstractThe formation of a dispersive shock wave in a colloidal medium, due to an initial jump in the light intensity, is studied. The compressibility of the colloidal particles is modelled using a series in the particle density, or packing fraction, where the virial coefficients depend on the particle interaction model. Both the theoretical hard disk and sphere repulsive models, and a model with temperature dependent compressibility, are considered. Experimental results for the second virial coefficient show that it is temperature dependent and that the particle interactions can be either repulsive or attractive; these effects are modelled using a power law relationship. The governing equation is a focusing nonlinear Schrödinger-type equation with an implicit nonlinearity. The initial jump is resolved via a dispersive shock wave which forms before the onset of modulational instability. A semi-analytical solution is developed for the one dimensional line bore case which predicts the amplitude of the solitary waves which form in the dispersive shock wave. The solitary wave amplitude versus jump height diagrams can exhibit three different kinds of behaviours; an unique solution, an S-shaped solution curve and multiple solution branches where the upper branch has separated from the lower branches. A bifurcation from the low to the high power branch, can occur for many parameter choices, as the ...

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“…In order to obtain an approximation for the amplitude of the solitary waves, we use uniform soliton theory [15,16]. The mass conservation equation for (1) is…”

Section: Semi-analytical Solutionsmentioning

confidence: 99%

Circular dispersive shock waves in colloidal media

Azmi

Marchant

2016

J. Nonlinear Optic. Phys. Mat.

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A dispersive shock wave (DSW), with a circular geometry, is studied in a colloidal medium. The colloidal particle interaction is based on the repulsive theoretical hard sphere model, where a series in the particle density, or packing fraction is used for the compressibility. Experimental results show that the particle interactions are temperature dependent and can be either repulsive or attractive, so the second term in the compressibility series is modified to allow for temperature dependent effects, using a power-law relationship. The governing equation is a focusing nonlinear Schrödinger-type equation with an implicit nonlinearity. The initial jump in electric field amplitude is resolved via a DSW, which forms before the onset of modulational instability. A semi-analytical solution for the amplitude of the solitary waves in a DSW of large radius, is derived based on a combination of conservation laws and geometrical considerations. The effect of temperature and background packing fraction on the evolution of the DSW and the amplitude of the solitary waves is discussed and the semi-analytical solutions are found to be very accurate, when compared with numerical solutions. Abstract. A dispersive shock wave (DSW), with a circular geometry, is studied in a colloidal medium. The colloidal particle interaction is based on the repulsive theoretical hard sphere model, where a series in the particle density, or packing fraction is used for the compressibility. Experimental results show that the particle interactions are temperature dependent and can be either repulsive or attractive, so the second term in the compressibility series is modified to allow for temperature dependent effects, using a power law relationship. The governing equation is a focusing nonlinear Schrödinger-type equation with an implicit nonlinearity. The initial jump in electric field amplitude is resolved via a DSW, which forms before the onset of modulational instability. A semianalytical solution for the amplitude of the solitary waves in a DSW of large radius, is derived based on a combination of conservation laws and geometrical considerations. The effect of temperature and background packing fraction on the evolution of the DSW and the amplitude of the solitary waves is discussed and the semi-analytical solutions are found to be very accurate, when compared with numerical solutions.

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Approximate Techniques for Dispersive Shock Waves in Nonlinear Media

Cited by 21 publications

References 39 publications

Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals

Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals

Dispersive shock waves in colloids with temperature dependent compressibility

Circular dispersive shock waves in colloidal media

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