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The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type

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We obtain the necessary and sufficient conditions for a two-component (2+1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions. These conditions are in involution, implying that the systems in question are locally parametrized by 15 arbitrary constants. It is proved that all such systems possess three conservation laws of hydrodynamic type and, therefore, are symmetrizable in Godunov’s sense. Moreover, all such systems are proved to possess a scalar pseudopotential which plays the role of the ‘dispersionless Lax pair’. We demonstrate that the class of two-component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions, thus establishing the equivalence of the two possible definitions of the integrability. Explicit linearly degenerate examples are constructed.

History

School

  • Science

Department

  • Mathematical Sciences

Pages

192959 bytes

Publication date

2003

Notes

This pre-print has been submitted, and accepted, to the journal, Journal of Physics A- Mathematical and General [© Institute of Physics]. The definitive version: FERAPONTOV, E.V. and KHUSNUTDINOVA,K.R., 2004. The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type. Journal of Physics A- Mathematical and General, 37(8), pp. 2949-2963, is available at: http://www.iop.org/EJ/journal/JPhysA.

Language

en