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1. Bounds on eigenfunctions of quantum cat maps

   https://doi.org/10.1088/1402-4896/ad0331  

   Kim, Elena; Koirala, Robert (October 2023, Physica Scripta)

   Abstract We studyℓ^∞norms ofℓ²-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Biévre in F Bonechi and S De Bièvre (2000,Communications in Mathematical Physics,211, 659–686)) we show that there exists a sequence of eigenfunctionsuwith $\parallel u{\parallel }_{\infty }\gtrsim {\left(\log N\right)}^{-1/2}$. For general eigenfunctions we show the upper bound $\parallel u{\parallel }_{\infty }\lesssim {\left(\log N\right)}^{-1/2}$. Here the semiclassical parameter is $h={\left(2\pi N\right)}^{-1}$. Our upper bound is analogous to the one proved by Bérard in P Bérard (1977,Mathematische Zeitschrift,155, 249-276) for compact Riemannian manifolds without conjugate points. 

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2. Existence and stability for the travelling waves of the Benjamin equation

   https://doi.org/10.1088/1361-6544/adabaa  

   Hakkaev, Sevdzhan; Stanislavova, Milena; Stefanov, Atanas G (January 2025, Nonlinearity)

   Abstract In the seminal work of Benjamin (1974Nonlinear Wave Motion(American Mathematical Society)), in the late 70s, he has derived the ubiquitous Benjamin model, which is a reduced model in the theory of water waves. Notably, it contains two parameters in its dispersion part and under some special circumstances, it turns into the celebrated KdV or the Benjamin–Ono equation, During the 90s, there was renewed interest in it. Benjamin (1992J. Fluid Mech.245401–11; 1996Phil. Trans. R. Soc.A3541775–806) studied the problem for existence of solitary waves, followed by works of Bona–Chen (1998Adv. Differ. Equ.351–84), Albert–Bona–Restrepo (1999SIAM J. Appl. Math.592139–61), Pava (1999J. Differ. Equ.152136–59), who have showed the existence of travelling waves, mostly by variational, but also bifurcation methods. Some results about the stability became available, but unfortunately, those were restricted to either small waves or Benjamin model, close to a distinguished (i.e. KdV or BO) limit. Quite recently, in 2024 (arXiv:2404.04711 [math.AP]), Abdallahet al, proved existence, orbital stability and uniqueness results for these waves, but only for large values of $\frac{c}{{\gamma }^2}\gg 1$. In this article, we present an alternative constrained maximization procedure for the construction of these waves, for the full range of the parameters, which allows us to ascertain their spectral stability. Moreover, we extend this construction to allL²subcritical cases (i.e. power nonlinearities $(|u{|}^{p-2}u{)}_x$, $2<p\text{⩽}6$). Finally, we propose a different procedure, based on a specific form of the Sobolev embedding inequality, which works for all powers $2<p<\mathrm{\infty }$, but produces some unstable waves, for largep. Some open questions and a conjecture regarding this last result are proposed for further investigation. 

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3. Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

   https://doi.org/10.1088/1361-6544/ac50bb  

   Chen, Qinbo; de la Llave, Rafael (March 2022, Nonlinearity)

   Abstract The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the followinga prioriunstable Hamiltonian system with a time-periodic perturbation ${\mathcal{H}}_{\epsilon }\left(p,q,I,\phi ,t\right)=h\left(I\right)+\sum _{i=1}^n\pm \left(\frac{1}{2}{p}_i^2+V_i\left(q_i\right)\right)+\epsilon H_1\left(p,q,I,\phi ,t\right),$where $\left(p,q\right)\in {\mathbb{R}}^n\times {\mathbb{T}}^n$, $\left(I,\phi \right)\in {\mathbb{R}}^d\times {\mathbb{T}}^d$withn,d⩾ 1,V_iare Morse potentials, andɛis a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbationsH₁. Indeed, the set of admissibleH₁isC^ωdense andC³open (a fortiori,C^ωopen). Our perturbative technique for the genericity is valid in theC^ktopology for allk∈ [3, ∞) ∪ {∞,ω}. 

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4. Acceleration and Spectral Redistribution of Cosmic Rays in Radio-jet Shear Flows

   https://doi.org/10.3847/1538-4357/acfda9  

   Webb, G. M.; Xu, Y.; Biermann, P. L.; Al-Nussirat, S.; Mostafavi, P.; Li, G.; Barghouty, A. F.; Zank, G. P. (November 2023, The Astrophysical Journal)

   Abstract A steady-state, semi-analytical model of energetic particle acceleration in radio-jet shear flows due to cosmic-ray viscosity obtained by Webb et al. is generalized to take into account more general cosmic-ray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution functionf₀(r,p) at cylindrical radiusrfrom the jet axis (assumed to lie along thez-axis) is given by convolving the particle momentum spectrum $f_0(\infty ,p\prime )$with the Green’s function $G(r,p;p\prime )$, which describes the monoenergetic spectrum solution in which $f_0\to \delta (p-p\prime )$asr→ ∞ . Previous work by Webb et al. studied only the Green’s function solution for $G(r,p;p\prime )$. In this paper, we explore for the first time, solutions for more general and realistic forms for $f_0(\infty ,p\prime )$. The flow velocityu=u(r)e_zis along the axis of the jet (thez-axis).uis independent ofz, andu(r) is a monotonic decreasing function ofr. The scattering time $\tau {(r,p)={\tau }_0\left(p/p_0\right)}^{\alpha }$in the shear flow region 0 <r<r₂, and $\tau {(r,p)={\tau }_0\left(p/p_0\right)}^{\alpha }{\left(r/r_2\right)}^s$, wheres> 0 in the regionr>r₂is outside the jet. Other original aspects of the analysis are (i) the use of cosmic ray flow lines in (r,p) space to clarify the particle spatial transport and momentum changes and (ii) the determination of the probability distribution ${\psi }_p(r,p;p\prime )$that particles observed at (r,p) originated fromr→ ∞ with momentum $p\prime$. The acceleration of ultrahigh-energy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described. 

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5. Magellan/IMACS Spectroscopy of Grus I: A Low Metallicity Ultra-faint Dwarf Galaxy*

   https://doi.org/10.3847/1538-4357/ac96ed  

   Chiti, Anirudh; Simon, Joshua D.; Frebel, Anna; Pace, Andrew B.; Ji, Alexander P.; Li, Ting S. (November 2022, The Astrophysical Journal)

   Abstract We present a chemodynamical study of the Grus I ultra-faint dwarf galaxy (UFD) from medium-resolution (R∼ 11,000) Magellan/IMACS spectra of its individual member stars. We identify eight confirmed members of Grus I, based on their low metallicities and coherent radial velocities, and four candidate members for which only velocities are derived. In contrast to previous work, we find that Grus I has a very low mean metallicity of 〈[Fe/H]〉 = −2.62 ± 0.11 dex, making it one of the most metal-poor UFDs. Grus I has a systemic radial velocity of −143.5 ± 1.2 km s⁻¹and a velocity dispersion of ${\sigma }_{\mathrm{rv}}={2.5}_{-0.8}^{+1.3}$km s⁻¹, which results in a dynamical mass of $M_{1/2}\left(r_h\right)=8_{-4}^{+12}\times {10}^5$M_⊙and a mass-to-light ratio ofM/L_V= ${440}_{-250}^{+650}$M_⊙/L_⊙. Under the assumption of dynamical equilibrium, our analysis confirms that Grus I is a dark-matter-dominated UFD (M/L> 80M_⊙/L_⊙). However, we do not resolve a metallicity dispersion (σ_[Fe/H]< 0.44 dex). Our results indicate that Grus I is a fairly typical UFD with parameters that agree with mass–metallicity and metallicity-luminosity trends for faint galaxies. This agreement suggests that Grus I has not lost an especially significant amount of mass from tidal encounters with the Milky Way, in line with its orbital parameters. Intriguingly, Grus I has among the lowest central densities ( ${\rho }_{1/2}\sim {3.5}_{-2.1}^{+5.7}\times {10}^7$M_⊙kpc⁻³) of the UFDs that are not known to be tidally disrupting. Models of the formation and evolution of UFDs will need to explain the diversity of these central densities, in addition to any diversity in the outer regions of these relic galaxies. 

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