Nonlinear Schrodinger waves in a disordered potential: Branched flow, spectrum diffusion, and rogue waves

The problem of nonlinear Schrodinger (NLS) waves in a disordered potential arises in many physical occasions, such as hydrodynamics, optics, and cold atoms. It provides a paradigm for studying the interaction between nonlinearity and random effect, but the current results are far from perfect. In this paper, we systematically simulate the turbulent waves for the focusing NLS equation with dynamical (time-dependent) random potentials, where the enhanced branching structures evolve into branched soliton flows as the nonlinearity increases. In this process, the occurrence of rogue waves for short times results from the interplay of linear random focusing and modulation instability. While the nonlinear spectral analysis reveals that for longer times, it is due to a self-organization of larger solitons competing with breakup of intermediate solitons. On the other hand, we found that the strong nonlinearity can significantly increase the width of the linear (Fourier) spectrum for several time scales, but its spreading rate becomes suppressed, which has a dependence on the correlation length of the potential. We hope that our findings will facilitate a deeper understanding of the nonlinear waves interacting with disordered media.

Automated summary

In this paper, we systematically simulate turbulent waves in the focusing NLS equation with dynamical random potentials and study their properties. We observe that enhanced branching structures evolve into branched soliton flows as the nonlinearity increases. Short-lived rogue waves result from the interplay of linear random focusing and modulation instability, while longer-lived rogue waves are due to a self-organization of larger solitons competing with the breakup of intermediate solitons. We also find that strong nonlinearity can significantly increase the width of the linear spectrum but with a suppressed spreading rate depending on the correlation length of the potential.