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Discontinuous progressing waves for semilinear systems. (English) Zbl 0598.35069
This paper is concerned with the existence, propagation and interaction of piecewise smooth solutions to semilinear systems of partial differential equations $$P(x,D)u+f(x,u)=0$$. Here P is a first order system and the jump discontinuities of the solutions, u, occur across a simply characteristic surface $$\Sigma$$. Transport equations along rays describe the propagation of the jumps. The interaction problem is harder. When P is a $$2\times 2$$ system the interaction of such waves and the production of a pair of waves from piecewise smooth discontinuous Cauchy data is studied. Related recent results are reported by M. Beals and G. Métivier [Duke Math. J. 53, 125-138 (1986)] and by the authors in the Proceedings of the College de France Seminar, 1985-1986.

##### MSC:
 35L67 Shocks and singularities for hyperbolic equations 35L60 First-order nonlinear hyperbolic equations 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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