The Gould-Hopper Polynomials in the Novikov-Veselov equation

We use the Gould-Hopper (GH) polynomials to investigate the Novikov-Veselov (NV) equation. The root dynamics of the $\sigma$-flow in the NV equation is studied using the GH polynomials and then the Lax pair is found. In particulr, when $N=3,4,5$, one can get the Gold-fish model. The smooth rational solutions of the NV equation are also constructed via the extended Moutard transformation and the GH polynomials. The asymptotic behavior is discussed and then the smooth rational solution of the Liouville equation is obtained.Comment: 22 pages, no figur

Darboux transformations for a 6-point scheme

We introduce (binary) Darboux transformation for general differential equation of the second order in two independent variables. We present a discrete version of the transformation for a 6-point difference scheme. The scheme is appropriate to solving a hyperbolic type initial-boundary value problem. We discuss several reductions and specifications of the transformations as well as construction of other Darboux covariant schemes by means of existing ones. In particular we introduce a 10-point scheme which can be regarded as the discretization of self-adjoint hyperbolic equation

Uncovering the stellar structure of the dusty star-forming galaxy GN20 at z=4.055 with MIRI/JWST

Luminous infrared galaxies at high redshifts ($z$>4) include extreme starbursts that build their stellar mass over short periods of time (>100 Myr). These galaxies are considered to be the progenitors of massive quiescent galaxies at intermediate redshifts ($z\sim$2) but their stellar structure and buildup is unknown. Here, we present the first spatially resolved near-infrared imaging of GN20, one of the most luminous dusty star-forming galaxies known to date, observed at an epoch when the Universe was only 1.5 Gyr old. The 5.6$\mu$m image taken with the JWST Mid-Infrared Instrument (MIRI/JWST) shows that GN20 is a very luminous galaxy (M$_\mathrm{1.1\mu m,AB}$=$-$25.01), with a stellar structure composed of a conspicuous central source and an extended envelope. The central source is an unresolved nucleus that carries 9% of the total flux. The nucleus is co-aligned with the peak of the cold dust emission, and offset by 3.9 kpc from the ultraviolet stellar emission. The diffuse stellar envelope is similar in size to the clumpy CO molecular gas distribution. The centroid of the stellar envelope is offset by 1 kpc from the unresolved nucleus, suggesting GN20 is involved in an interaction or merger event supported by its location as the brightest galaxy in a proto-cluster. The stellar size of GN20 is larger by a factor of about 3-5 than known spheroids, disks, and irregulars at $z\sim$4, while its size and low S\'ersic index are similar to those measured in dusty, infrared luminous galaxies at $z\sim$2 of the same mass. GN20 has all the ingredients necessary for evolving into a massive spheroidal quiescent galaxy at intermediate $z$: it is a large, luminous galaxy at $z$=4.05 involved in a short and massive starburst centred in the stellar nucleus and extended over the entire galaxy, out to radii of 4 kpc, and likely induced by the interaction or merger with a member of the proto-cluster.Comment: 7 pages, 4 figure

Strong (Hb + [OIII]) and Ha emitters at redshift z ~ 7-8 unveiled with JWST/NIRCam and MIRI imaging in the Hubble eXtreme Deep Field (XDF)

The JWST is revolutionizing the study of high-redshift galaxies by providing for the first time a high-sensitivity view of the early Universe at infrared wavelengths, both with its Near Infrared Camera (NIRCam) and Mid Infrared Instrument (MIRI). In this paper, we make use of medium and broad-band NIRCam imaging, as well as ultra-deep MIRI 5.6 microns imaging, in the Hubble eXtreme Deep Field (XDF) to identify prominent line emitters at z ~ 7-8. Out of a total of 58 galaxies at z ~ 7-8, we find 18 robust candidates (~31%) for prominent (Hb + [OIII]) emitters, based on their enhanced fluxes in the F430M and F444W filters, with rest-frame EW(Hb + [OIII]) ~ 87 - 2100 A. Among these emitters, 16 lie on the MIRI coverage area and 12 show a clear flux excess at 5.6 microns, indicating the simultaneous presence of a prominent Ha emission line with rest-frame EW(Ha) ~ 200 - 3000 A. This is the first time that Ha emission can be detected in individual galaxies at z>7. The Ha line, when present, allows us to separate the contributions of the Hb and [OIII] emission lines to the (Hb + [OIII]) complex and derive Ha-based star formation rates (SFRs). We find that in some cases [OIII]/Hb > 1, suggesting low metallicities, but a few have [OIII]/Hb < 1, so the NIRCam flux excess is mainly driven by Hb. The vast majority of prominent line emitters are very young starbursts or galaxies on their way to/from the starburst cloud. They make for a cosmic SFR density log10(SFRD_Ha / Msun yr^-1 Mpc^-3) ~ 2.35, which is about a third of the total value at z ~ 7-8. Therefore, the strong Ha emitters likely had an important role in reionization.Comment: 15 pages, 9 figures. Submitted to Ap

Darboux transformations for linear operators on two dimensional regular lattices

Darboux transformations for linear operators on regular two dimensional lattices are reviewed. The six point scheme is considered as the master linear problem, whose various specifications, reductions, and their sublattice combinations lead to other linear operators together with the corresponding Darboux transformations. The second part of the review deals with multidimensional aspects of (basic reductions of) the four point scheme, as well as the three point scheme.Comment: 23 pages, 3 figures, presentation improve

Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations

Nonlinear SUSY approach to preparation of quantum systems with pre-planned spectral properties is reviewed. Possible multidimensional extensions of Nonlinear SUSY are described. The full classification of ladder-reducible and irreducible chains of SUSY algebras in one-dimensional QM is given. Emergence of hidden symmetries and spectrum generating algebras is elucidated in the context of Nonlinear SUSY in one- and two-dimensional QM.Comment: 75 pages, Minor corrections, Version published in Journal of Physics