American Journal of Mathematics

We construct examples of bounded solutions to a semilinear system Pu = f(z, u), z ∈ Ω ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], f smooth, P = P · Id, with P a strictly hyperbolic differential operator of second order with smooth coefficients, such that for a time function t of P, the following properties are satisfied: (1) u is conormal to a smooth characteristic hypersurface Σ in t < 0 such that Σ develops a swallowtail singularity at time t = 0; (2) For t > 0, u is singular, not only at Σ, but also on the forward characteristic cone over the swallowtail tip.