Nonuniformities small amplitude

Bifurcations. In many situations the uniform solution of the reaction-diffusion equation exists, but is stable only for certain regimes of the parameters. Near to the transition zone between stability and instability of this uniform solution, other nonuniform, small-amplitude, solutions exist as well. If these solutions are stable, their appearance can be considered as bifurcation phenomena. The equation... 

Departures from the assumption of small-amplitude oscillations about a uniform, quiescent state can modify the modes of oscillation. Also, the boundary conditions influence the solutions [for example, if there were an open isobaric boundary, then p = 0 would be appropriate there, and equation (8) would be changed]. Nevertheless, the results that have been given form good first approximations for use in approaches to the analysis of acoustic instabilities. Frequencies and spatial dependences of amplitudes are less strongly influenced by flow nonuniformities that are the magnitudes of the amplitudes of the various modes. Nothing in what has been presented so far provides a basis for calculating the constants bk , and... 

In the absence of dissipation and with uniform surface tension, plane small-amplitude capillary waves will propagate undamped and unamplified. Viscosity and surface tension gradients lead to the damping of capillary waves. In the following section we shall discuss the damping due to the presence of surface-active substances, which because of the wave shape are not uniformly distributed, giving rise to a spatially nonuniform surface tension. Of interest in... 

Ray transit time 5-6 Example Clad power-law profiles 5-7 Small-amplitude nonuniformities 5-8 Example Slight core-radius variations 5-9 Example Slight exponent variations 5-10 Adiabatic invariant 5-11 Example Step profile 5-12 Example Clad power-law profiles 5-13 Radiation loss... 

In Chapter 3 we showed that on a uniform fiber, pulse spreading is proportional to distance z along the fiber. Aceordingly, the only influenee slowly varying nonuniformities can have on this result is to modify the coefficient multiplying z, in addition to changing the pulse shape and redueing pulse power by radiation. The latter is discussed in Section 5-13. However, as we show below, these effects tend to be small when the fiber variations are of small amplitude. 

A weakly guiding fiber of refractive-index profile n(x, y) carries a sinusoidal nonuniformity of small amplitude. The profile of the perturbed fiber is assumed to have the form... 

We assume that we can approximate the solution of Eq. (5-14) by expanding (z) in powers of a small dimensionless ratio k(z), which is proportional to the amplitude of the nonuniformity. For reasons given below, we retain terms correct to second order and set... 

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