Normal Self-intersections of the Characteristic Variety

Let P = PtP2 + 2 be a linear partial differential operator on R^ with Pt and P2, of orders mx and ra2, respectively, strictly hyperbolic with respect to the first variable and Q of order ml + m2 2. Although the characteristic variety of P may have self-intersections, the hyperbolicity of Px and P2 implies local solvability for Pu = ƒ; indeed the Cauchy problem for P is locally solvable. In this note we shall consider the propagation of singularities near the simplest type of point zQ G T*R \0 where the principal symbol p = plp2 of P has a multiple zero. We shall suppose that the characteristic varieties A(PX) and A(P2) of Px and P2 intersect normally at z0 , that is, dpx(z0) and dp2(z0) are linear independent. In addition, it will be assumed that the Poisson bracket {pv p2}(z0) i= 0. This latter assumption means that the Hamiltonian vector fields H and H are not tangent to A{Pl) n A(P2) at z0 . So, the two forward pointing bicharacteristics (of px and p2) through z0 consist, near z0 , of nonsingular points of A(P), except for z0 itself. Let these curves be denoted by ci ; R D I 3 p —> ct(p) where I is an open interval containing 0, ^.(0) = z0 and (c^id/dp) = H . It will be assumed that I is chosen so small that