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If G k k0 d k k0 , Equations (3.103) and (3.104) are ordinary nonlinear equations which can be solved analytically. This case corresponds to a covariance function in r-space that is constant.
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There are basically two possible iterations methods for the random coupling model equation: a) by using Equation (3.101)
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we can iterate it, considering the second term on the random homogeneous space as a perturbation. This gives
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which cannot be used for long times because of the singular form. b) A more interesting method is to write Equation (3.101) as hGiK where L is the linear operator G 0 1 G 0 L hGi
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Equation (3.106) is then iterated giving G 0  : G 0 0 L 1 G 1 G 0 L . . .
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This is the operator analogue of a continued fraction. For the 1D-model and the covariance function $ exp jxj= , it was possible to show that this iteration process converges for je z=cj < 1=2 and to nd its analytic continuation. The proof is somewhat arti cial because we used the fact that for any function f k is bounded and analytic in the half plane Im z < 0 G k k0 f k0 dk0 f k i= 3:108
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Hence we get a nonlinear nite difference equation, which is solved by means of a continued fraction. Unfortunately, it is not possible to extend this method to the scalar wave equation. 3.5. RANDOM TAYLOR EXPANSION AT SHORT WAVELENGTHS In Section 3.4, we have found that in the limiting case ek0 ) 1, the random refractive index behaves as a mere random value and not as a random function. This is easily understood when is very large ) l , each realization (or sample) of the random index is a very slowly varying function that can be approximated by a constant. At an intermediate level, between the general random function and random variable, we could try to approximate a random function by a linear function or a quadratic function with random variables as coef cients. For example, constructing a limited random Taylor expansion of the random function. The random equation for this model is: @C x ik0 1 em x C x d x @x 3:109
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where m x is a random function, d x is Dirac s distribution at the origin, and the wave number k0 2p=l 2pf =c is taken, as in Section 3.3, to be positive. Here l is the wavelength, f is the radiated frequency, and c is the velocity of light. We want to approximate the random function m x by its random Taylor expansion [7,22] m x m 0 xm0 0 x2 00 m 0 . . . 2 3:110
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where m 0 , m0 0 , m00 0 ; . . . are not independent random variables. We cannot keep the covariance function expf jxj= g because the corresponding random function is not mean square differentiable (see Reference [31]). As we do not need to specify the covariance G, we shall only assume that it has derivatives of all orders at x 0 and that G 0 1. That leads us to an approximation for C x , that is, ! @C x x2 ik0 1 em 0 exm0 0 e m00 0 C x d x : 2 @x It is solved for the mean wave function hC x i Y x e
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As m x is a Gaussian random function, the multivariate distribution of m 0 , m0 0 , m00 0 is also Gaussian; it is thus determined by its second order moment such as h m 0 2 i; h m0 0 2 i; hm 0 m0 0 i, and so forth. They are easily calculated in terms