In this talk, we consider the k-dispersion generalized Benjamin–Ono equation

∂ₜu + ∂ₓ(D^α u + u^{k+1}) = 0, (t, x) ∈ ℝ × ℝ.

We prove that, for any even integer k ≥ 4 and α in (1, 2), solutions with initial data in the energy space H^{α/2}(ℝ) exist globally in time and scatter. Our approach combines the concentration–compactness–rigidity method introduced by Kenig and Merle with monotonicity formulas developed by Tao for the KdV equation (cf. Kim and Kwon for the Benjamin–Ono equation), together with the Caffarelli–Silvestre fractional extension and commutator estimates. This work is based on joint research with F. Linares and T. S. R. Santos.