Travelling Waves for the Brio System

In the setting of a product of distributions, we define a concept of a solution for the Brio system ut+12(u2+v2)x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{t}+\frac{1}{2}(u^{2}+v^{2})_{x}=0$$\end{document}, vt+(uv-v)x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{t} +(uv-v)_{x}=0$$\end{document}, which extends the classical solution concept. New results about that product allow us to establish necessary and sufficient conditions for the propagation of distributional travelling waves. Within this framework, we prove that continuous travelling waves are necessarily constant functions. Thus, if we want to seek for travelling waves in the Brio system, we must seek them among distributions that are not continuous functions. Examples that include discontinuous functions, measures and distributions which are not measures are given explicitly. For the reader’s convenience and completeness, a survey of the main ideas and formulas needed for multiplying distributions is also provided.