Scattering at a reflectionless potential

This smooth potential well has reflection coefficient zero for all energies. Hence the scattered wave has no reflected part.

Reflectionless potentials are constructed via inverse scattering theory. It turns out that there is a deep connection between the Korteweg-deVries equation (describing shallow water waves) and the Schrödinger equation. The potential function above has the shape of a soliton (solitary wave), that is, a solution of the Korteweg-deVries equation that moves with constant shape.