Integrable interactions are synonymous with non-diffractive scattering, meaning that the set of incoming momenta for any scattering event coincides with the set of outgoing momenta.  A system is integrable if the two particle scattering matrix obeys a particular relation known as the Yang-Baxter equation.  Nonintegrable interactions correspond to diffractive scattering, where the set of outgoing momenta may take on all values consistent with energy and momentum conservation. Such processes play a vital role in the kinetics of one dimensional gases, where binary collisions are unable to alter the distribution function.
In this talk I'll discuss how diffractive scattering arises when the Yang-Baxter equation is violated, and show how to calculate the diffraction amplitude when this violation is small.