Chaotic scattering in solitary wave interactions: A singular iterated-map description

We derive a family of singular iterated maps--closely related to Poincare maps--that describe chaotic interactions between colliding solitary waves. The chaotic behavior of such solitary wave collisions depends on the transfer of energy to a secondary mode of oscillation, often an internal mode of the pulse. Unlike previous analyses, this map allows one to understand the interactions in the case when this mode is excited prior to the first collision. The map is derived using Melnikov integrals and matched asymptotic expansions and generalizes a ``multi-pulse'' Melnikov integral and allows one to find not only multipulse heteroclinic orbits, but exotic periodic orbits. The family of maps derived exhibits singular behavior, including regions of infinite winding. This problem is shown to be a singular version of the conservative Ikeda map from laser physics and connections are made with problems from celestial mechanics and fluid mechanics.Comment: 29 pages, 17 figures, submitted to Chaos, higher-resolution figures available at author's website: http://m.njit.edu/goodman/publication

Stellar disks of Collisional Ring Galaxies I. New multiband images, Radial intensity and color profiles, and confrontation with N-body simulations

We present new multi-band imaging data in the optical (BVRI and Halpha) and near infrared bands (JHK) of 15 candidate ring galaxies from the sample of Appleton & Marston (1997). We use these data to obtain color composite images, global magnitudes and colors of both the ring galaxy and its companion(s), and radial profiles of intensity and colors. We find that only nine of the observed galaxies have multi-band morphologies expected for the classical collisional scenario of ring formation, indicating the high degree of contamination of the ring galaxy sample by galaxies without a clear ring morphology. The radial intensity profiles, obtained by masking the off-centered nucleus, peak at the position of the ring, with the profiles in the continuum bands broader than that in the Halpha line. The images as well as the radial intensity and color profiles clearly demonstrate the existence of the pre-collisional stellar disk outside the star-forming ring, which is in general bluer than the disk internal to the ring. The stellar disk seems to have retained its size, with the disk outside the ring having a shorter exponential scale length as compared to the values expected in normal spiral galaxies of comparable masses. The rings in our sample of galaxies are found to be located preferentially at around half-way through the stellar disk. The most likely reason for this preference is bias against detecting rings when they are close to the center (they would be confused with the resonant rings), and at the edge of the disk the gas surface density may be below the critical density required for star formation. Most of the observed characteristics point to relatively recent collisions (<80 Myr ago) according to the N-body simulations of Gerber et al. (1996).Comment: To appear in AJ issue of September 2008. High resolution color image of Figure 2 and other supplementary images are available at http://www.inaoep.mx/~ydm/rings

Singular solutions of a modified two-component Camassa-Holm equation

The Camassa-Holm equation (CH) is a well known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow dependence on average density as well as pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially-confined initial data. Numerical results for MCH2 are given and compared with the pure CH2 case. These numerics show that the modification in MCH2 to introduce average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for MCH2 shows a new asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, MCH2 also allows the phase shift of the peakon collision to diverge in certain parameter regimes.Comment: 25 pages, 11 figure

Occlusion resistant learning of intuitive physics from videos

To reach human performance on complex tasks, a key ability for artificial systems is to understand physical interactions between objects, and predict future outcomes of a situation. This ability, often referred to as intuitive physics, has recently received attention and several methods were proposed to learn these physical rules from video sequences. Yet, most of these methods are restricted to the case where no, or only limited, occlusions occur. In this work we propose a probabilistic formulation of learning intuitive physics in 3D scenes with significant inter-object occlusions. In our formulation, object positions are modeled as latent variables enabling the reconstruction of the scene. We then propose a series of approximations that make this problem tractable. Object proposals are linked across frames using a combination of a recurrent interaction network, modeling the physics in object space, and a compositional renderer, modeling the way in which objects project onto pixel space. We demonstrate significant improvements over state-of-the-art in the intuitive physics benchmark of IntPhys. We apply our method to a second dataset with increasing levels of occlusions, showing it realistically predicts segmentation masks up to 30 frames in the future. Finally, we also show results on predicting motion of objects in real videos

Finding the Needles in the Haystacks: High-Fidelity Models of the Modern and Archean Solar System for Simulating Exoplanet Observations

We present two state-of-the-art models of the solar system, one corresponding to the present day and one to the Archean Eon 3.5 billion years ago. Each model contains spatial and spectral information for the star, the planets, and the interplanetary dust, extending to 50 AU from the sun and covering the wavelength range 0.3 to 2.5 micron. In addition, we created a spectral image cube representative of the astronomical backgrounds that will be seen behind deep observations of extrasolar planetary systems, including galaxies and Milky Way stars. These models are intended as inputs to high-fidelity simulations of direct observations of exoplanetary systems using telescopes equipped with high-contrast capability. They will help improve the realism of observation and instrument parameters that are required inputs to statistical observatory yield calculations, as well as guide development of post-processing algorithms for telescopes capable of directly imaging Earth-like planets.Comment: Accepted for publication in PAS