In this paper we construct explicit solutions for initial value problem for a system of first order equations. When , this system is just the standard Hopf equation in conservative form. When , 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.
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K.T. Joseph, Exact Solution of a System of Generalized Hopf Equations. Z. Anal. Anwend. 21 (2002), no. 3, pp. 669–680DOI 10.4171/ZAA/1101