skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.


Title: Global existence and singularity formation for the generalized Constantin–Lax–Majda equation with dissipation: the real line vs. periodic domains
Abstract The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation Λ σ , where Λ σ ˆ = | k | σ , both for the problem on the circle x [ π , π ] and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for σ 1 and all real values of an advection parameterawhen the data is small. We also derive new analytical solutions in both geometries whena = 0, and on the real line when a = 1 / 2 , for various values ofσ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet fora = 0 andσ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values ofσthat are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory.  more » « less
Award ID(s):
1907684 1909407
PAR ID:
10483358
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
Institute of Physics
Date Published:
Journal Name:
Nonlinearity
Volume:
37
Issue:
2
ISSN:
0951-7715
Page Range / eLocation ID:
025004
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract Polyatomic molecules have been identified as sensitive probes of charge-parity violating and parity violating physics beyond the Standard Model (BSM). For example, many linear triatomic molecules are both laser-coolable and have parity doublets in the ground electronic X ˜ 2 Σ + ( 010 ) state arising from the bending vibration, both features that can greatly aid BSM searches. Understanding the X ˜ 2 Σ + ( 010 ) state is a crucial prerequisite to precision measurements with linear polyatomic molecules. Here, we characterize the fundamental bending vibration of 174 YbOH using high-resolution optical spectroscopy on the nominally forbidden X ˜ 2 Σ + ( 010 ) A ˜ 2 Π 1 / 2 ( 000 ) transition at 588 nm. We assign 39 transitions originating from the lowest rotational levels of the X ˜ 2 Σ + ( 010 ) state, and accurately model the state’s structure with an effective Hamiltonian using best-fit parameters. Additionally, we perform Stark and Zeeman spectroscopy on the X ˜ 2 Σ + ( 010 ) state and fit the molecule-frame dipole moment to D m o l = 2.16 ( 1 ) Dand the effective electrong-factor to g S = 2.07 ( 2 ) . Further, we use an empirical model to explain observed anomalous line intensities in terms of interference from spin–orbit and vibronic perturbations in the excited A ˜ 2 Π 1 / 2 ( 000 ) state. Our work is an essential step toward searches for BSM physics in YbOH and other linear polyatomic molecules. 
    more » « less
  2. Abstract We combine our dynamical modeling black-hole mass measurements from the Lick AGN Monitoring Project 2016 sample with measured cross-correlation time lags and line widths to recover individual scale factors,f, used in traditional reverberation-mapping analyses. We extend our sample by including prior results from Code for AGN Reverberation and Modeling of Emission Lines (caramel) studies that have utilized our methods. Aiming to improve the precision of black-hole mass estimates, as well as uncover any regularities in the behavior of the broad-line region (BLR), we search for correlations betweenfand other AGN/BLR parameters. We find (i) evidence for a correlation between the virial coefficient log 10 ( f mean , σ ) and black-hole mass, (ii) marginal evidence for a similar correlation between log 10 ( f rms , σ ) and black-hole mass, (iii) marginal evidence for an anticorrelation of BLR disk thickness with log 10 ( f mean , FWHM ) and log 10 ( f rms , FWHM ) , and (iv) marginal evidence for an anticorrelation of inclination angle with log 10 ( f mean , FWHM ) , log 10 ( f rms , σ ) , and log 10 ( f mean , σ ) . Last, we find marginal evidence for a correlation between line-profile shape, when using the root-mean-square spectrum, log 10 ( FWHM / σ ) rms , and the virial coefficient, log 10 ( f rms , σ ) , and investigate how BLR properties might be related to line-profile shape usingcaramelmodels. 
    more » « less
  3. Abstract We present13CO(J= 1 → 0) observations for the EDGE-CALIFA survey, which is a mapping survey of 126 nearby galaxies at a typical spatial resolution of 1.5 kpc. Using detected12CO emission as a prior, we detect13CO in 41 galaxies via integrated line flux over the entire galaxy and in 30 galaxies via integrated line intensity in resolved synthesized beams. Incorporating our CO observations and optical IFU spectroscopy, we perform a systematic comparison between the line ratio 12 / 13 I [ 12 CO ( J = 1 0 ) ] / I [ 13 CO ( J = 1 0 ) ] and the properties of the stars and ionized gas. Higher 12 / 13 values are found in interacting galaxies compared to those in noninteracting galaxies. The global 12 / 13 slightly increases with infrared colorF60/F100but appears insensitive to other host-galaxy properties such as morphology, stellar mass, or galaxy size. We also present azimuthally averaged 12 / 13 profiles for our sample up to a galactocentric radius of 0.4r25(∼6 kpc), taking into account the13CO nondetections by spectral stacking. The radial profiles of 12 / 13 are quite flat across our sample. Within galactocentric distances of 0.2r25, the azimuthally averaged 12 / 13 increases with the star formation rate. However, Spearman rank correlation tests show the azimuthally averaged 12 / 13 does not strongly correlate with any other gas or stellar properties in general, especially beyond 0.2r25from the galaxy centers. Our findings suggest that in the complex environments in galaxy disks, 12 / 13 is not a sensitive tracer for ISM properties. Dynamical disturbances, like galaxy interactions or the presence of a bar, also have an overall impact on 12 / 13 , which further complicates the interpretations of 12 / 13 variations. 
    more » « less
  4. Abstract M dwarfs are common host stars to exoplanets but often lack atmospheric abundance measurements. Late-M dwarfs are also good analogs to the youngest substellar companions, which share similarTeff∼ 2300–2800 K. We present atmospheric analyses for the M7.5 companion HIP 55507 B and its K6V primary star with Keck/KPIC high-resolution (R∼ 35,000)K-band spectroscopy. First, by including KPIC relative radial velocities between the primary and secondary in the orbit fit, we improve the dynamical mass precision by 60% and find M B = 88.0 3.2 + 3.4 M Jup , putting HIP 55507 B above the stellar–substellar boundary. We also find that HIP 55507 B orbits its K6V primary star with a = 38 3 + 4 au ande= 0.40 ± 0.04. From atmospheric retrievals of HIP 55507 B, we measure [C/H] = 0.24 ± 0.13, [O/H] = 0.15 ± 0.13, and C/O = 0.67 ± 0.04. Moreover, we strongly detect13CO (7.8σsignificance) and tentatively detect H 2 18 O (3.7σsignificance) in the companion’s atmosphere and measure 12 CO / 13 CO = 98 22 + 28 and H 2 16 O / H 2 18 O = 240 80 + 145 after accounting for systematic errors. From a simplified retrieval analysis of HIP 55507 A, we measure 12 CO / 13 CO = 79 16 + 21 and C 16 O / C 18 O = 288 70 + 125 for the primary star. These results demonstrate that HIP 55507 A and B have consistent12C/13C and16O/18O to the <1σlevel, as expected for a chemically homogeneous binary system. Given the similar flux ratios and separations between HIP 55507 AB and systems with young substellar companions, our results open the door to systematically measuring13CO and H 2 18 O abundances in the atmospheres of substellar or even planetary-mass companions with similar spectral types. 
    more » « less
  5. Abstract We measure the CO-to-H2conversion factor (αCO) in 37 galaxies at 2 kpc resolution, using the dust surface density inferred from far-infrared emission as a tracer of the gas surface density and assuming a constant dust-to-metal ratio. In total, we have ∼790 and ∼610 independent measurements ofαCOfor CO (2–1) and (1–0), respectively. The mean values forαCO (2–1)andαCO (1–0)are 9.3 5.4 + 4.6 and 4.2 2.0 + 1.9 M pc 2 ( K km s 1 ) 1 , respectively. The CO-intensity-weighted mean is 5.69 forαCO (2–1)and 3.33 forαCO (1–0). We examine howαCOscales with several physical quantities, e.g., the star formation rate (SFR), stellar mass, and dust-mass-weighted average interstellar radiation field strength ( U ¯ ). Among them, U ¯ , ΣSFR, and the integrated CO intensity (WCO) have the strongest anticorrelation with spatially resolvedαCO. We provide linear regression results toαCOfor all quantities tested. At galaxy-integrated scales, we observe significant correlations betweenαCOandWCO, metallicity, U ¯ , and ΣSFR. We also find thatαCOin each galaxy decreases with the stellar mass surface density (Σ) in high-surface-density regions (Σ≥ 100Mpc−2), following the power-law relations α CO ( 2 1 ) Σ 0.5 and α CO ( 1 0 ) Σ 0.2 . The power-law index is insensitive to the assumed dust-to-metal ratio. We interpret the decrease inαCOwith increasing Σas a result of higher velocity dispersion compared to isolated, self-gravitating clouds due to the additional gravitational force from stellar sources, which leads to the reduction inαCO. The decrease inαCOat high Σis important for accurately assessing molecular gas content and star formation efficiency in the centers of galaxies, which bridge “Milky Way–like” to “starburst-like” conversion factors. 
    more » « less