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A global formalism for nonlinear waves in conservation laws

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Abstract

We introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws. Fundamental waves are represented as pairs of states statisfying the Rankine-Hugoniot conditions; after trivial solutions have been eliminated by means of a blow-up procedure, these pairs form an (n+1)-dimensional manifold

, the fundamental wave manifold. There is a distinguishedn-dimensional submanifold of

containing a single one-dimensonal foliation that represents the rarefaction curves for all families. Similarly, there is a foliation of

itself that represent shock curves. We identify othern-dimensional submanifolds of

that are naturally interpreted as boundaries of regions of admissible shock waves. These submanifolds also have one-dimensional foliations, which represent curves of composite waves. This geometric framework promises to simplify greatly the study of the stability and bifurcation properties

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Authors and Affiliations

  1. Department of Mathematics, University of Wyoming, P.O. Box 3036 Univ. Station, 82071, Laramie, WY, USA

    Eli L. Issacson

  2. Instituto de Matemática Pura e Aplicada and Department of Mathematics, Pontifícia Universidade Católica, 22460, Rio de Janeiro, RJ, Brazil

    Dan Marchesin

  3. Department of Mathematics, Pontifícia Universidade Católica, 22453, Rio de Janeiro, RJ, Brazil

    C. Frederico Palmeira

  4. Department of Mathematics and of Applied Mathematics and Statistics, State University of New York, 11794-3651, Stony Brook, NY, USA

    Bradley J. Plohr

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  1. Eli L. Issacson
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  2. Dan Marchesin
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  4. Bradley J. Plohr
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Communicated by A. Jaffe

This work was supported in part by: the NSF/CNPq U.S.-Latin America Cooperative Science Program under Grant INT-8612605; the Institute for Mathematics and its Applications with funds provided by the National Science Foundation; the Air Force Office of Scientific Research under Grant AFOSR 90-0075; the National Science Foundation under Grant 8901884; the U.S. Department of Energy under Grant DE-FG02-90ER25084; the U.S. Army Research Office under Grant DAAL03-89-K-0017; the Financiadora de Estudos e Projetos; the Conselho Nacional de Desenvolvimento Científico e Tecnológica (CNPq); the Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ); the Coordenação de Aperfeiçamento de Pessoal de Ensino Superior (CAPES); and the Sociedade Brasileira de Matemática (SBM)

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Issacson, E.L., Marchesin, D., Frederico Palmeira, C. et al. A global formalism for nonlinear waves in conservation laws. Commun.Math. Phys. 146, 505–552 (1992). https://doi.org/10.1007/BF02097015

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  • DOI: https://doi.org/10.1007/BF02097015

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