Exact Solution of a System of Generalized Hopf Equations

Abstract

In this paper we construct explicit solutions for initial value problem for a system of first order equations. When $n = 1$, this system is just the standard Hopf equation in conservative form. When $n > 1$, the system is non-conservative. We use the vanishing viscosity method to construct solutions. As the system is non-conservative we use Volpert product and the algebra of generalized Colombeau functions to make sense of the products which appear in the equations.