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In the papers by Bücker et al. (Nature Physics 7, 608 (2011) and Phys. Rev. A 86, 013638), the decay of first transverse excited states into pairs ("twins") of transverse ground states with opposite longitudinal momenta, is experimentally observed and theoretically modelled. However, the effect of phase fluctuations is not considered adequately. I am trying to include them in the model, both analytically and computationally (the latter with greater success than the former), and calculate the density of states. One of the approaches I'm taking is transforming the problem into a (2D) Langevin equation, taking a first order-approximation, finding the stationary state of the corresponding Fokker-Planck equation, and using that to obtain the Lyapunov exponent and then the density of states. I'm also studying the Lyapunov exponent in related problems.