Abstract
The purpose of this paper is to classify the solutions of Riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws. Hyperbolic singularities play the role in the theory of Riemann problems that rest points play in the theory of ordinary differential equations: Indeed, generically, only a finite number of structures can appear in a neighborhood of such a singularity. In this, the first of three papers, the program of classification is discussed in general and the simplest structure that occurs is characterized.
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