Spectral behavior and influence of collapses in the focusing MMT model

Determine how the wave action spectrum behaves in the one-dimensional focusing Majda-McLaughlin-Tabak (MMT) model and ascertain whether wave collapses affect the spectral properties of the wave field.

Background

The Majda–McLaughlin–Tabak (MMT) model was introduced as a testbed for wave turbulence theory (WTT). In defocusing settings, recent studies clarified that the width of the inertial range governs spectral slopes and that spectra can approach Kolmogorov–Zakharov predictions when the inertial range is sufficiently wide.

The focusing version of the MMT model admits modulational instability and finite-time wave collapses, which are coherent structures. Prior numerical investigations reported spectra flatter than WTT predictions and increased intermittency at moderate nonlinearity, but did not cover regimes where collapses dominate. As a result, the behavior of the spectrum in the focusing MMT model and the role of collapses remained explicitly uncertain.