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Laplace transformations of hydrodynamic-type systems in Riemann invariants

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Abstract

The conserved densities of hydrodynamic-type systems in Riemann invariants satisfy a system of linear second-order partial differential equations. For linear systems of this type, Darboux introduced Laplace transformations, which generalize the classical transformations of a second-order scalar equation. It is demonstrated that the Laplace transformations can be pulled back to transformations of the corresponding hydrodynamic-type systems. We discuss finite families of hydrodynamic-type systems that are closed under the entire set of Laplace transformations. For 3 × 3 systems in Riemann invariants, a complete description of closed quadruples is proposed. These quadruples appear to be related to a special quadratic reduction of the (2 + 1)-dimensional 3-wave system.

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Authors and Affiliations

  1. Institute of Mathematical Modeling, Russian Academy of Sciences, USSR

    E. V. Ferapontov

  2. Center of Nonlinear Investigations, L. D. Landau Institute of Theoretical Physics, USSR

    E. V. Ferapontov

Authors
  1. E. V. Ferapontov
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 86–97, January, 1997.

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Ferapontov, E.V. Laplace transformations of hydrodynamic-type systems in Riemann invariants. Theor Math Phys 110, 68–77 (1997). https://doi.org/10.1007/BF02630370

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  • DOI: https://doi.org/10.1007/BF02630370

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