KdV and BO equations with bore-like data (Q1854052)

The authors prove the global well-posedness of the initial value problem for the Korteweg-de Vries (KdV) equation with initial conditions of a special type. These are functions \(g\) satisfying { (i)} \( g(x) \to C_\pm \) as \(x\to \pm \infty \), { (ii)} \( g'\in H^s\) for some \(s\geq 0\), { (iii)} \( (g-C_+)\in L^2(0,\infty)\), \((g-C_-)\in L^2(-\infty ,0)\). Similar conclusions are derived for the solutions of the Benjamin-Ono (BO) equation.