Strong Photon Antibunching in Weakly Nonlinear Two-Dimensional Exciton-Polaritons

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

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Directional solidification of a binary alloy into a cellular convective flow: localized morphologies

Journal of Fluid Mechanics ◽

10.1017/s0022112099005856 ◽

1999 ◽

Vol 395 ◽

pp. 253-270 ◽

Cited By ~ 8

Author(s):

Y.-J. CHEN ◽

S. H. DAVIS

Keyword(s):

Directional Solidification ◽

Binary Alloy ◽

Temporal Dynamics ◽

Three Dimensional ◽

Multiple Scale ◽

Solute Diffusion ◽

Two Dimensional ◽

Morphological Instability ◽

Flow Strength ◽

Weakly Nonlinear

A steady, two-dimensional cellular convection modifies the morphological instability of a binary alloy that undergoes directional solidification. When the convection wavelength is far longer than that of the morphological cells, the behaviour of the moving front is described by a slow, spatial–temporal dynamics obtained through a multiple-scale analysis. The resulting system has a parametric-excitation structure in space, with complex parameters characterizing the interactions between flow, solute diffusion, and rejection. The convection in general stabilizes two-dimensional disturbances, but destabilizes three-dimensional disturbances. When the flow is weak, the morphological instability is incommensurate with the flow wavelength, but as the flow gets stronger, the instability becomes quantized and forced to fit into the flow box. At large flow strength the instability is localized, confined in narrow envelopes. In this case the solutions are discrete eigenstates in an unbounded space. Their stability boundaries and asymptotics are obtained by a WKB analysis. The weakly nonlinear interaction is delivered through the Lyapunov–Schmidt method.

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Out-of-plane buckling in two-dimensional glass drawing

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.221 ◽

2019 ◽

Vol 869 ◽

pp. 587-609

Author(s):

D. O’Kiely ◽

C. J. W. Breward ◽

I. M. Griffiths ◽

P. D. Howell ◽

U. Lange

Keyword(s):

Large Time ◽

Thin Sheet ◽

Stress Profile ◽

Two Dimensional ◽

Nonlinear Buckling ◽

Weakly Nonlinear ◽

Out Of Plane ◽

Long Time ◽

Glass Drawing ◽

Logarithmic Series

We derive a mathematical model for the drawing of a two-dimensional thin sheet of viscous fluid in the direction of gravity. If the gravitational field is sufficiently strong, then a portion of the sheet experiences a compressive stress and is thus unstable to transverse buckling. We analyse the dependence of the instability and the subsequent evolution on the process parameters, and the mutual coupling between the weakly nonlinear buckling and the stress profile in the sheet. Over long time scales, the sheet centreline ultimately adopts a universal profile, with the bulk of the sheet under tension and a single large bulge caused by a small compressive region near the bottom, and we derive a canonical inner problem that describes this behaviour. The large-time analysis involves a logarithmic asymptotic expansion, and we devise a hybrid asymptotic–numerical scheme that effectively sums the logarithmic series.

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