Following these results, de Broglie developed the dynamical equations for his pilot wave theory.[9] Initially, de Broglie proposed a double solution approach, in which the quantum object consists of a physical wave (u-wave) in real space which has a spherical singular region that gives rise to particle-like behaviour; in this initial form of his theory he did not have to postulate the existence of a quantum particle.[10] He later formulated it as a theory in which a particle is accompanied by a pilot wave.