Bilinear structure and Schlesinger transforms of the qq-PIII_{\rm III} and qq-PVI_{\rm VI} equations

We show that the recently derived ($q$-) discrete form of the Painlev\'e VI equation can be related to the discrete P$_{\rm III}$, in particular if one uses the full freedom in the implementation of the singularity confinement criterion. This observation is used here in order to derive the bilinear forms and the Schlesinger transformations of both $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$.Comment: 10 pages, Plain Te

Observability of Dark Matter Substructure with Pulsar Timing Correlations

Dark matter substructure on small scales is currently weakly constrained, and its study may shed light on the nature of the dark matter. In this work we study the gravitational effects of dark matter substructure on measured pulsar phases in pulsar timing arrays (PTAs). Due to the stability of pulse phases observed over several years, dark matter substructure around the Earth-pulsar system can imprint discernible signatures in gravitational Doppler and Shapiro delays. We compute pulsar phase correlations induced by general dark matter substructure, and project constraints for a few models such as monochromatic primordial black holes (PBHs), and Cold Dark Matter (CDM)-like NFW subhalos. This work extends our previous analysis, which focused on static or single transiting events, to a stochastic analysis of multiple transiting events. We find that stochastic correlations, in a PTA similar to the Square Kilometer Array (SKA), are uniquely powerful to constrain subhalos as light as $\sim 10^{-13}~M_\odot$, with concentrations as low as that predicted by standard CDM.Comment: 45 pages, 12 figure

Do All Integrable Evolution Equations Have the Painlev\'e Property?

We examine whether the Painleve property is necessary for the integrability of partial differential equations (PDEs). We show that in analogy to what happens in the case of ordinary differential equations (ODEs) there exists a class of PDEs, integrable through linearisation, which do not possess the Painleve property. The same question is addressed in a discrete setting where we show that there exist linearisable lattice equations which do not possess the singularity confinement property (again in analogy to the one-dimensional case).Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Integrable systems without the Painlev\'e property

We examine whether the Painlev\'e property is a necessary condition for the integrability of nonlinear ordinary differential equations. We show that for a large class of linearisable systems this is not the case. In the discrete domain, we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non-Painlev\'e continuous linearisable systems. We find that while these discrete systems are themselves linearisable, they possess nonconfined singularities

Nonintegrability of (2+1)-dimensional continuum isotropic Heisenberg spin system: Painlev\'e analysis

While many integrable spin systems are known to exist in (1+1) and (2+1) dimensions, the integrability property of the physically important (2+1) dimensional isotropic Heisenberg ferromagnetic spin system in the continuum limit has not been investigated in the literature. In this paper, we show through a careful singularity structure analysis of the underlying nonlinear evolution equation that the system admits logarithmic type singular manifolds and so is of non-Painlev\'e type and is expected to be nonintegrable.Comment: 11 pages. to be published in Phys. Lett. A (2006

Riccati Solutions of Discrete Painlev\'e Equations with Weyl Group Symmetry of Type E8(1)E_8^{(1)}

We present a special solutions of the discrete Painlev\'e equations associated with $A_0^{(1)}$, $A_0^{(1)*}$ and $A_0^{(1)**}$-surface. These solutions can be expressed by solutions of linear difference equations. Here the $A_0^{(1)}$-surface discrete Painlev\'e equation is the most generic difference equation, as all discrete Painlev\'e equations can be obtained by its degeneration limit. These special solutions exist when the parameters of the discrete Painlev\'e equation satisfy a particular constraint. We consider that these special functions belong to the hypergeometric family although they seems to go beyond the known discrete and $q$-discrete hypergeometric functions. We also discuss the degeneration scheme of these solutions.Comment: 22 page

Pulsar Timing Probes of Primordial Black Holes and Subhalos

Pulsars act as accurate clocks, sensitive to gravitational redshift and acceleration induced by transiting clumps of matter. We study the sensitivity of pulsar timing arrays (PTAs) to single transiting compact objects, focusing on primordial black holes and compact subhalos in the mass range from $10^{-12} M _{\odot}$ to well above $100~M_\odot$. We find that the Square Kilometer Array can constrain such objects to be a subdominant component of the dark matter over this entire mass range, with sensitivity to a dark matter sub-component reaching the sub-percent level over significant parts of this range. We also find that PTAs offer an opportunity to probe substantially less dense objects than lensing because of the large effective radius over which such objects can be observed, and we quantify the subhalo concentration parameters which can be constrained.Comment: 18 pages, 6 figure