We present a numerical approach for generalised Korteweg-de Vries (KdV) equations on the real line. In the spatial dimension we compactify the real line and apply a Chebyshev collocation method. The time integration is performed with an implicit RungeKutta method of fourth order. Several examples are discussed: initial data bounded but not vanishing at infinity as well as data not satisfying the Faddeev condition, i.e. with a slow decay towards infinity. (C) 2022 IMACS.