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Mathematical Notes

, Volume 74, Issue 5–6, pp 803–811 | Cite as

Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems

  • A. V. Zhiber
  • S. Ya. Startsev
Article

Abstract

We generalize the notions of Laplace transformations and Laplace invariants for systems of hyperbolic equations and study conditions for their existence. We prove that a hyperbolic system admits the Laplace transformation if and only if there exists a matrix of rank k mapping any vector whose components are functions of one of the independent variables into a solution of this system, where k is the defect of the corresponding Laplace invariant. We show that a chain of Laplace invariants exists only if the hyperbolic system has a entire collection of integrals and the dual system has a entire collection of solutions depending on arbitrary functions. An example is given showing that these conditions are not sufficient for the existence of a Laplace transformation.

Laplace transform Laplace invariant hyperbolic system 

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Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • A. V. Zhiber
    • 1
  • S. Ya. Startsev
    • 1
  1. 1.Institute of Mathematics and Computer CenterRussian Academy of SciencesUfa

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