Abstract:
In this paper the system (0.1) of $N$ differential equations with random coefficients $\eta_{kj}(z)$ is considered. This system of coupled mode propagation is a mathematical model for wave-guides with random imperfections. The sum
\begin{equation}
\sum_{j=1}^N |E_j(z)|^2
\end{equation}
is the power flow at the output of the wave-guide ($z$ is its lehgth). The physical considerations justify the investigation of an asymptotic problem when $N\to\infty$, $\eta_{kj}(z)\to 0$, $\alpha_j\to 0$, $z\to\infty$. Under some conditions the variance of the sum (1) converges to 0, while its expectation remains positive.
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