Creation, Annihilation, and Interaction of Delta-Waves in Nonlinear Models: a Distributional Product Approach

Using a solution concept defined in the setting of a product of distributions, we consider the nonlinear equation f(t)ut + (u2)x = 0, where f is a continuous function. This equation can be regarded a generalization of Burgers inviscid equation and allows to study several kinds of interaction of δ-waves. If f(t) ≠ 0 for all t, collisions of δ-waves cannot exist. If f(t) = 0 for certain values of t, collisions of δ-waves may arise. In certain cases, the evolution of δ-waves under collision is similar to classical solitons collisions, for instance, in the Korteweg—de Vries equation. Phenomena of scattering, merging, annihilation and creation of Dirac masses are also possible in this setting. These results are easy to obtain essentially because, in our approach, the product of distributions is a distribution which does not depend on approximation processes. We include the main ideas concerning such products and several results obtained within this framework.