A generalized nonlinear Schr\"odinger equation as model for turbulence,   collapse, and inverse cascade
  A two-dimensional generalized cubic nonlinear Schr\"odinger equation with complex coefficients for the group dispersion and nonlinear terms is used to investigate the evolution of a finite-amplitude localized initial perturbation. It is found that modulation of the latter can lead to side-band formation, wave condensation, collapse, turbulence, and inverse cascade, although not all together nor in that order. 
Dian Zhao, M. Y. Yu
physics.plasm-ph math-ph math.MP physics.flu-dyn