Rotating Wave Solution of the Ginzburg-Landau Equation

6 Rotating Wave Solution of the Ginzburg-Landau Equation 

Before going into analytical theory, we show some results of the computer simulation carried out for the Ginzburg-Landau equation in the form of (2.4.18). From the nice symmetry of this model, we expect that the center of steady rotation is in the state of vanishing R, i.e., (X, Y) = (0,0), and hence the phase 0 of W cannot be defined there. Let this phase singularity be situated at r = 0 using polar coordinates (r, 6). Further, the rotation number / is assumed to be 1 as before  

One may imagine a more general circumstance where a number of such phase singularities coexist in the system. Then the sum of the associated rotation numbers // must be conserved as long as none of them happen to be absorbed by the wall. A convenient initial distribution satisfying (6.6.1) is shown in Fig. 6.5 where X and y have constant slopes in directions making 90° to each other. Consequently, the zero-level contours of X and Y intersect vertically at r = 0. It is clear that 

Let us try to develop an analytical theory which could explain some aspects of the above numerical results. First, let (2.4.18) or (2.4.13) be expressed in terms of R and 0  

Note that (6.6.3) reduces to the nonlinear phase diffusion equation if we neglect the space dependence of R, The previous numerical simulation suggests that R 

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