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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. MĂ©tivier [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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