Transition from weak turbulence to collapse turbulence regimes in the MMT model

Understanding the role of coherent structures emerging from a field of random waves is a topic of great interest in the nonlinear wave community. While there has been extensive research on the topic of strongly nonlinear localized coherent structures (e.g., solitons and breathers) in integrable systems, the behavior of such structures in nonintegrable systems is not yet as well understood. We study the forced-dissipated focusing one-dimensional (1D) Majda-McLaughlin-Tabak (MMT) with localized wave collapses as a result of soliton instability. Our results show that when the forcing perturbation strength is weak, there are few wave collapses in the field and there is good agreement with weak wave turbulence (WTT) predictions. As the forcing perturbation strength increases, we see an increase in high amplitude collapses, intermittency, and the departure from a power-law spectrum to an exponentially decaying spectrum resembling that of a two-species gas (comprised of waves and collapses). This is a novel discovery in the context of the MMT model and can be thought of as an analogy to a soliton gas in integrable turbulence. The transition from a weak turbulence regime to a strongly nonlinear “collapse” turbulence regime is also a new feature identified for the MMT model.