Integrable operators, ∂‾\overline{\partial}-Problems, KP and NLS hierarchy

We develop the theory of integrable operators $\mathcal{K}$ acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of a $\overline{\partial}$-problem in the complex plane. When such a $\overline{\partial}$-problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert-Carleman determinant of the operator $\mathcal{K}$. With suitable choices of the setup we show that the Hilbert-Carleman determinant is a $\tau$-function of the Kadomtsev-Petviashvili (KP) or nonlinear Schr\"odinger hierarchies.Comment: 24 pages, no figure

Soliton shielding of the focusing nonlinear Schrodinger equation

We consider a gas of $N$ solitons of the Focusing Nonlinear Schr\"odinger (FNLS) equation in the limit $N\to\infty$ with apoint spectrum chosen to interpolate a given spectral soliton density over a domain of the complex spectral plane. We call this class of initial data, deterministic soliton gas. We show that when the domain is a disc and the soliton density is an analytic function, then the corresponding deterministic soliton gas surprisingly yields the one-soliton solution with point spectrum the center of the disc. We call this effect {\it soliton shielding}. When the domain is an ellipse, the soliton shielding reduces the spectral data to the soliton density concentrating between the foci of the ellipse. The physical solution is asymptotically step-like oscillatory, namely, the initial profile is a periodic elliptic function in the negative $x$--direction while it vanishes exponentially fast in the opposite direction.Comment: 5 pages, add references and missprint correction

Topical natural-origin polynucleotides in radiation-induced skin and mucosal toxicity

Seventy to 90 percent of patients who have received radiation treatment struggle with radiation skin and mucosal toxicity. The inflicted damage to progenitor cells and local microcirculation makes it more likely that wounds, infections, and fibrosis may occur; lesions of variable severity often co-exist. Acute erythema, hyperpigmentation, and mild desquamation usually wane in weeks and require only minor treatment. Conversely, the management of persistent radiation dermatitis and telangiectasia remains unsatisfactory; chronic lesions may progress to tissue atrophy and disfiguring fibrosis. Protrophic, natural-origin polynucleotides, formulated as Class III medical devices, have long shown to be a reliable topical option to stop the progression of radiation-related lesions. The present review illustrates the rationale of polynucleotides in skin and mucosal radiodermatitis management. It also illustrates the clinical results in a series of exploratory clinical studies carried out with polynucleotide devices over the last decade. The examined studies open the way to the high-level clinical research program, which will develop over the next years

Multi-wavelength polarimetric study towards the open cluster NGC 1893

We present multi-wavelength linear polarimetric observations for 44 stars of the NGC 1893 young open cluster region along with V-band polarimetric observations of stars of other four open clusters located between l ~160 to ~175 degree. We found evidence for the presence of two dust layers located at a distance of ~170 pc and ~360 pc. The dust layers produce a polarization Pv ~2.2%. It is evident from the clusters studied in the present work that, in the Galactic longitude range l ~160 to 175 degree and within the Galactic plane (|b| < 2 degree), the polarization angles remain almost constant, with a mean ~163 degree and a dispersion of 6 degree. The small dispersion in polarization angle could be due to the presence of uniform dust layer beyond 1 kpc. Present observations reveal that in case of NGC 1893, the foreground two dust layers, in addition to the intracluster medium, seems to be responsible for the polarization effects. It is also found that towards the direction of NGC 1893, the dust layer that exists between 2-3 kpc has a negligible contribution towards the total observed polarization. The weighted mean for percentage of polarization (Pmax) and the wavelength at maximum polarization ({\lambda}max) are found to be 2.59 \pm 0.02% and 0.55 \pm 0.01 \mum respectively. The estimated mean value of {\lambda}max indicates that the average size of the dust grains within the cluster is similar to that in the general interstellar medium. The spatial variation of the polarization is found to decrease towards the outer region of the cluster. In the present work, we support the notion, as already has been shown in previous studies, that polarimetry, in combination with (U-B)/(B-V) colour-colour diagram, is a useful tool for identifying non-members in a cluster.Comment: 36 pages, 12 figures, 10 tables, accepted for the publication in MNRA

Integrable operators, dbar-problems, KP and NLS hierarchy

We develop the theory of integrable operators K acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of adbar-problem in the complex plane. When such adbar-problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert–Carleman determinant of the operator . With suitable choices of the setup we show that the Hilbert–Carleman determinant is a τ-function of the Kadomtsev–Petviashvili (KP) or nonlinear Schrödinger hierarchies