A KdV-like advection-dispersion equation with some remarkable properties

We discuss a new non-linear advection-dispersion equation u_t + (2 u_{xx}/u) u_x = epsilon u_{xxx}, invariant under scaling of dependent variable and referred to here as SIdV. This PDE (with dispersion coefficient unity) was discovered in a genetic programming search for equations sharing the Korteweg-de Vries (KdV) solitary wave solution. Indeed, there is a one-parameter family of first order advection equations with cubic dispersion sharing the KdV solitary wave, that interpolate between SIdV and KdV. SIdV is one of the two simplest such translation and space-time reflection-symmetric equations invariant under rescaling of wave amplitude u. The scale-invariant advection in SIdV is reminiscent of the (E x B)/B^2 velocity of plasma physics. We identify two conservation laws, though initial investigations indicate that SIdV does not follow from a polynomial Lagrangian of the KdV sort. Nevertheless, SIdV possesses solitary and periodic travelling waves and recurrence properties usually associated with integrable systems. At a special value of dispersion coefficient, it is exactly solvable, while in the dispersionless limit it is related to a non-linear diffusive equation. This novel equation (and its higher-dimensional and complex generalizations) may find applications in physics and engineering and also be a rich source for further numerical and analytical explorations.