There is more than one wave equation: some remarks on KdV equation

Water waves are among the richest and most challenging problems of classical mechanics. Beyond nonlinear effects an interesting property of the water wave equation is the dispersion. Korteweg-deVries (KdV) equation is among the most studied one dimensional models of dispersive PDE and in the context of water waves arises a shallow water approximation. I will discuss the oscillatory effect of  dispersion with linear water wave and KdV equation as examples. These heuristic discussions will be complemented by some rigorous results for equations generalizing KdV.