Steady states in dual-cascade wave turbulence

We study stationary solutions in the differential kinetic equation, which was introduced in Dyachenko A et al (1992 Physica D 57 96–160) for description of a local dual cascade wave turbulence. We give a full classification of single-cascade states in which there is a finite flux of only one conserved quantity. Analysis of the steady-state spectrum is based on a phase-space analysis of orbits of the underlying dynamical system. The orbits of the dynamical system demonstrate the blowup behaviour which corresponds to a 'sharp front' where the spectrum vanishes at a finite wave number. The roles of the Kolmogorov–Zakharov and thermodynamic scaling as intermediate asymptotic, as well as of singular solutions, are discussed.