Abstract
We study the dynamics of one-dimensional nonlinear waves with a square-root dispersion. This dispersion allows strong interactions of distant modes in wave-number space, and it leads to a modulational instability of a carrier wave interacting with distant sidebands. Weak wave turbulence is found when the system is damped and weakly driven. A driving force that exceeds a critical strength leads to wave collapses coexisting with weak wave turbulence. We explain this transition behavior with the modulational instability of waves with the highest power: Below the threshold the instability is suppressed by the external long-wave damping force. Above the threshold the instability initiates wave collapses.
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