The competition between wave turbulence and coherent structures

Wave turbulence of weakly nonlinear dispersive waves is a disordered state in which energy or other conserved quantities are transferred from sources in wavenumber space (the driving range) to sinks (the dissipation range). The theory of wave turbulence provides an analytic derivation of all statistical quantities (most notably the Kolmogorov–Zakharov spectrum) from the underlying equations of motion. A competing and radically different turbulent process with a significant impact on the statistical properties is the formation of coherent structures. Under what conditions can we observe purely weak wave turbulence, and when is it superseded by coherent structures? We study this problem for an influential model of one-dimensional turbulent dynamics, the Majda–McLaughlin–Tabak equation. The formation of narrow radiating solitary waves (pulses) leads to spectra that are steeper than the Kolmogorov–Zakharov spectra. However, for sufficiently large box sizes, we find that wave turbulence prevails within a broad range of four orders of magnitude of the driving force.