## Specific Analytic Solutions

As a simple application of the moment or variational method, consider first the case of a low-energy pulse propagating in a constant-dispersion fiber with negligible nonlinear effects. Recalling that (1 +C2)/T2 is related to the spectral width of the pulse that does not change in a linear medium, we can replace this quantity with its initial value (1 +Cq)/Tq, where 7o and Co are input values at z = 0. Since the second term is negligible in Eq. (4.6.7), it can be integrated easily and provides the solution

where 5 = sgnQ^) and Lq = Tq/is the dispersion length. Using this solution in Eq. (4.6.8), we find that the pulse width changes as

T2(z) = T2[l+2sC0(z/LD) + (1 +Cl){z/LD)2}. (4.6.16)

It is easy to verify that these expressions agree with those obtained in Section 3.3.1 by solving the pulse propagation equation directly.

To solve Eqs. (4.6.6) and (4.6.7) in the nonlinear case, we make two approximations. First, we assume that fiber losses are compensated such that p(z) = 1 (ideal distributed amplification). Second, the nonlinear effects are weak enough that the chirp at any distance z can be written as C = CL + C', where the nonlinear part C' < CL. It is easy to see that the linear part is given by Eq. (4.6.15), while the nonlinear part satisfies dC yP0 7b

Dividing Eqs. (4.6.6) and (4.6.17), we obtain dC" jP{)T() yP0T()

where we replaced C with CL as C' <C C/.. This equation is now easy to solve, and the result can be written as

Once C = Ci + C' is known, the pulse width can be found from Eq. (4.6.8).

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