A two-parameter nonlinear dispersive wave equation proposed by Majda, McLaughlin and Tabak (1997) is studied analytically and numerically as a model for testing the validity of weak turbulence theory. Kolmogorov-type solutions for the energy spectrum are determined explicitly and compared with numerical results. These show a strong dependence on the sign of the nonlinear term. In one case, there is agreement with the theory. In the other, there is disagreement. Possible explanations for this discrepancy will be given such as the emergence of coherent structures: quasi-solitons and wave collapses.