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As recently discovered [PRL ${\bf 109}$ 190601(2012)], Anderson localizationin a bulk disordered system triggers the emergence of a coherent forwardscattering (CFS) peak in momentum space, which twins the well-known coherentbackscattering (CBS) peak observed in weak localization experiments. Goingbeyond the perturbative regime, we address here the long-time dynamics of theCFS peak in a 1D random system and we relate this novel interference effect tothe statistical properties of the eigenfunctions and eigenspectrum of thecorresponding random Hamiltonian. Our numerical results show that the dynamicsof the CFS peak is governed by the logarithmic level repulsion betweenlocalized states, with a time scale that is, with good accuracy, twice theHeisenberg time. This is in perfect agreement with recent findings based on thenonlinear $\sigma$-model. In the stationary regime, the width of the CFS peakin momentum space is inversely proportional to the localization length,reflecting the exponential decay of the eigenfunctions in real space, while itsheight is exactly twice the background, reflecting the Poisson statisticalproperties of the eigenfunctions. Our results should be easily extended tohigher dimensional systems and other symmetry classes.

Dynamics of localized waves in 1D random potentials: statistical theory of the coherent forward scattering peak