Coherent structures are special solutions of a given wave equation which are usually localized and propagate without changing shape. They play an essential role not only in applications but also in the theory of wave propagation where it is believed that any initial data evolves into a superposition of coherent structures (asymptotic completeness conjecture). While the answer to the title question is still no, except for the rare integrable systems, I will present recent progress in this direction based on both local and global bifurcation theory.