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.
Citation:
A. V. Zhiber, S. Ya. Startsev, “Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems”, Mat. Zametki, 74:6 (2003), 848–857; Math. Notes, 74:6 (2003), 803–811