Phaselike branch

This shows the anticipated fact that the instability of the uniform oscillation to long wavelength fluctuations corresponds precisely to the negative sign of the phase diffusion constant. The equality = —y may also be confirmed, where y is the quantity which appeared in (4.2.36) and is the abbreviation of —a> defined in (4.2.35). More generally, it is possible to prove that the dispersion curve of the phaselike branch has an exact correspondence to the linearized form of the phase diffusion equation (4.2.36), or one may possibly have... 

Note that Ap- 0 and Aa-> -2 as 0. The spectrum of Ap may be called the phaselike branch, and that of Aa the amplitudelike branch, since they are associated, respectively, with phase fluctuations and amplitude fluctuations in the limit of vanishing q. We may say that the instability of the uniform oscillation is related to the unstable growth of some phaselike fluctuations. 

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