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We present a numerical approach for generalised Korteweg-de Vries (KdV)equations on the real line. In the spatial dimension we compactify the realline and apply a Chebyshev collocation method. The time integration isperformed with an implicit Runge-Kutta method of fourth order. Several examplesare discussed: initial data bounded but not vanishing at infinity as well asdata not satisfying the Faddeev condition, i.e. with a slow decay towardsinfinity.

Spectral approach to Korteweg-de Vries equations on the compactified real line