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Newton’s second law is applied to study the motion of a particle subjected to a time dependent impulsive force containing a Dirac delta distribution. Within this setting, we prove that this problem can be rigorously solved neither by limit processes nor by using the theory of distributions (limited to the classical Schwartz products). However, using a distributional multiplication, not defined by a limit process, a rigorous solution emerges.

We study the possibility of collision of a δ-wave with a stationary δ-wave in a model ruled by equation f (t)u t + [u 2 − β (x − γ(t))u] x = 0, where f , β and γ are given real functions and u = u(x,t) is the state variable. We adopt a solution concept which is a consistent extension of the classical solution concept. This concept is defined in the setting of a distributional product, which is not constructed by approximation processes. By a convenient choice of f , β and γ, we are able to distinguish three distinct dynamics for that collision, to which correspond phenomena of solitonic behaviour, scattering, and merging. Also, as a particular case, taking f = 2 and β = 0 we prove that the referred collision is impossible to arise in the setting of the inviscid Burgers equation. To show how this framework can be applied to other physical models, we included several results already obtained.

The present paper concerns the study of a Riemann problem for the system u t + 1 2 u 2 +φ (v) x = 0, v t + uv x = 0, with a one dimensional space variable. We consider φ an entire function that takes real values on the real axis. Under certain conditions, this system provides solutions to the pressureless gas dynamics and the isentropic fluid dynamics systems. We get all solutions of this problem within a convenient space of distributions that contains discontinuous functions and Dirac measures. For this purpose, we use a solution concept defined in the setting of a distributional product. This concept consistently extends the classical solution concept and can also be considered as an extension of the weak solution concept for nonlinear evolution equations. Our product, not defined by approximation processes, can be applied to several physical models.

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