On the Indication from Pioneer 10/11 Data of an Anomalous Acceleration

Hubble's law, which states a linear increase in velocities with distances, can physically be understood in terms of an acceleration cH. This work proposes a connection between this "universal" acceleration seen in the solar system and the anomalous acceleration acting on the Pioneer 10/11 spacecraft, in which the Hubble constant inferred from Pioneer 10/11 data is ~ 87 km/s/Mpc. Its physical implication is discussed in relation with Mach's principle.Comment: Written originally in two part

The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions

In this paper we apply the formal Inverse Spectral Transform for integrable dispersionless PDEs arising from the commutation condition of pairs of one-parameter families of vector fields, recently developed by S. V. Manakov and one of the authors, to one distinguished class of equations, the so-called Dunajski hierarchy. We concentrate, for concreteness, i) on the system of PDEs characterizing a general anti-self-dual conformal structure in neutral signature, ii) on its first commuting flow, and iii) on some of their basic and novel reductions. We formally solve their Cauchy problem and we use it to construct the longtime behavior of solutions, showing, in particular, that unlike the case of soliton PDEs, different dispersionless PDEs belonging to the same hierarchy of commuting flows evolve in time in very different ways, exhibiting either a smooth dynamics or a gradient catastrophe at finite time