Small Data Scattering for the Global Solutions of the Supercritical Generalized KdV Equation

We consider the scattering problem for the global solutions of the supercritical generalized KdV equation ∂tu+∂xxxu+μ∂x(uk+1) = 0, where k > 4 is an integer, initial data u0 belongs to H1(ℝ), and μ = ±1. To solve the scattering problem, a scattering criteria is established firstly, and then a new inequality is introduced to obtain uniformly bounded solutions in H1(ℝ). Finally, we further clarify the conditions for the equation to have a global solution scattering in H1(ℝ). Our method is mainly inspired by the works of Farah, Linares, Pastor, and Visciglia.