Letters in Mathematical Physics

, Volume 96, Issue 1–3, pp 5–35 | Cite as

On Linear Degeneracy of Integrable Quasilinear Systems in Higher Dimensions

  • Evgeny V. Ferapontov
  • Karima R. Khusnutdinova
  • Christian Klein


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.

Mathematics Subject Classification (2000)

35L40 35L65 37K10 


multi-dimensional quasilinear systems hydrodynamic reductions integrability linear degeneracy Cauchy problem classical solutions 


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Copyright information

© Springer 2011

Authors and Affiliations

  • Evgeny V. Ferapontov
    • 1
  • Karima R. Khusnutdinova
    • 1
  • Christian Klein
    • 2
  1. 1.Department of Mathematical SciencesLoughborough UniversityLoughborough, LeicestershireUK
  2. 2.Institut de Mathématiques de BourgogneDijon CedexFrance

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