The Singularities of Solutions of the Cauchy Problems for Systems Whose Characteristic Roots Are Non-Uniform Multiple

R. Courant and P. D. Lax [1] and D. Ludwig [4] investigated the singularities of the solutions of the Cauchy problems for diagonalizable linear hyperbolic systems whose characteristic roots are real and uniform multiple. They constructed a uniform asymptotic solution and proved that the singularities of the solutions propagate only along the characteristic surfaces on which the singularities of the initial data lie. Secondly, D. Ludwig and B. Granoff [2] dropped the condition that the characteristic roots are uniform multiple. They defined their hyperbolicity for systems with constant coefficient in the principal part whose normal surface has self-intersection points and discussed the propagation of singularities by constructing a uniform asymptotic solution. An important feature of their results is that the singularities of the solutions propagate also along the characteristic surface which generally does not carry the singular support of the initial data. Geometrically, this is an enveloping surface generated by a family of surfaces which connect the two characteristic surfaces with intersection points. The complex versions corresponding to the results of [1] and [4] were done by Y. Hamada [3], C. Wagschal [6], especially for meromorphic Cauchy data. The aim of this paper is to extend the results of [2] for a certain type of systems with variable coefficients in the complex domain. Our results include as a corollary the exactness of the asymptotic solution constructed by D. Ludwig and B. Granoff [2] in the real analytic case.