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

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