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On Linear Degeneracy of Integrable Quasilinear Systems in Higher Dimensions

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Abstract

We investigate (d + 1)-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case d ≥ 3 we formulate a conjecture that any such system with an irreducible dispersion relation must be linearly degenerate. We prove this conjecture in the 2-component case, providing a complete classification of multi- dimensional integrable systems in question. In particular, our results imply the non- existence of 2-component integrable systems of hydrodynamic type for d ≥ 6. In the second half of the paper we discuss a numerical and analytical evidence for the impossibility of the breakdown of smooth initial data for linearly degenerate systems in 2 + 1 dimensions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK

    Evgeny V. Ferapontov & Karima R. Khusnutdinova

  2. Institut de Mathématiques de Bourgogne, 9 Avenue Alain Savary, BP 47870, 21078, Dijon Cedex, France

    Christian Klein

Authors
  1. Evgeny V. Ferapontov
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  2. Karima R. Khusnutdinova
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  3. Christian Klein
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Correspondence to Christian Klein.

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This paper is dedicated to V.B. Matveev on the occasion of his 65th birthday

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Ferapontov, E.V., Khusnutdinova, K.R. & Klein, C. On Linear Degeneracy of Integrable Quasilinear Systems in Higher Dimensions. Lett Math Phys 96, 5–35 (2011). https://doi.org/10.1007/s11005-011-0462-4

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  • DOI: https://doi.org/10.1007/s11005-011-0462-4

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