Articles and Letters published during April 2021 till the end of December 2021
Francesco Calogero ; Farrin Payandeh.
After tersely reviewing the various meanings that can be given to the
property of a system of nonlinear ODEs to be solvable, we identify a special
case of the system of two first-order ODEs with homogeneous quadratic
right-hand sides which is explicitly solvable. It is identified by 2 explicit
algebraic constraints on the 6 a priori arbitrary parameters that characterize
this system. Simple extensions of this model to cases with nonhomogeneous
quadratic right-hand sides are also identified, including isochronous cases.
G Nakamura ; B Grammaticos ; M Badoual.
We model the effect of vaccination on an epidemic which, like the current one, exhibits a climate-induced seasonality. Our study is carried out using a simple SIR model. One important feature of our approach is that of recruitment: by gradually introducing susceptible individuals we mimic the spatial evolution of the epidemic, which is absent in the classic SIR. We focus on the effect of vaccination on the number of hospital admissions. We show that any delay in the vaccination campaign results in an increase of hospitalisations, and if one tries to palliate for the delay by increasing the vaccination rate, this results in an inflation of the number of necessary doses. Considering a multi-agegroup population we show that it is advantageous to prioritise the vaccination of the older groups (upholding thus the current practice). Finally, we investigate whether a vaccination of the younger population based on awareness can be an optimal strategy, concluding by a negative.
C. Muriel ; M. C. Nucci.
New integrability properties of a family of sequences of ordinary
differential equations, which contains the Riccati and Abel chains as the most
simple sequences, are studied. The determination of n generalized symmetries of
the nth-order equation in each chain provides, without any kind of integration,
n-1 functionally independent first integrals of the equation. A remaining first
integral arises by a quadrature by using a Jacobi last multiplier that is
expressed in terms of the preceding equation in the corresponding sequence. The
complete set of n first integrals is used to obtain the exact general solution
of the nth-order equation of each sequence. The results are applied to derive
directly the exact general solution of any equation in the Riccati and Abel
chains.
Sergey V. Meleshko ; Colin Rogers.
Reciprocal transformations associated with admitted conservation laws were
originally used to derive invariance properties in non-relativistic gasdynamics
and applied to obtain reduction to tractable canonical forms. They have
subsequently been shown to have diverse physical applications to nonlinear
systems, notably in the analytic treatment of Stefan-type moving boundary
problem and in linking inverse scattering systems and integrable hierarchies in
soliton theory. Here,invariance under classes of reciprocal transformations in
relativistic gasdynamics is shown to be linked to a Lie group procedure.
Maciej Błaszak.
In this letter we consider three nonhomogeneous deformations of Dispersive
Water Wave (DWW) soliton equation and prove that their stationary flows are
equivalent to three famous Painlevé equations, i.e. $P_{II}$, $P_{III}$ and
$P_{IV},$ respectively.
Valentin Ovsienko.
This work is a first step towards a theory of "$q$-deformed complex numbers".
Assuming the invariance of the $q$-deformation under the action of the modular
group I prove the existence and uniqueness of the operator of translations
by~$i$ compatible with this action. Obtained in such a way $q$-deformed
Gaussian integers have interesting properties and are related to the Chebyshev
polynomials.
Mats Vermeeren.
Many integrable hierarchies of differential equations allow a variational
description, called a Lagrangian multiform or a pluri-Lagrangian structure. The
fundamental object in this theory is not a Lagrange function but a differential
$d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All
such action integrals must be stationary for a field to be a solution to the
pluri-Lagrangian problem. In this paper we present a procedure to obtain
Hamiltonian structures from the pluri-Lagrangian formulation of an integrable
hierarchy of PDEs. As a prelude, we review a similar procedure for integrable
ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely
related to the Poisson brackets between the corresponding Hamilton functions.
In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy
and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we
present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as
the Boussinesq hierarchy.
Giuseppe Gaeta.
We classify simple symmetries for an Ornstein-Uhlenbeck process, describing a
particle in an external force field $f(x)$. It turns out that for sufficiently
regular (in a sense to be defined) forces there are nontrivial symmetries only
if $f(x)$ is at most linear. We fully discuss the isotropic case, while for the
non-isotropic we only deal with a generic situation (defined in detail in the
text).