Articles and Letters published during 2022.
Emil Novruzov.
We consider the propagation of equatorial waves of small amplitude, in a flow with an underlying non-uniform current. Without making the too restrictive rigid-lid approximation, by exploiting the available Hamiltonian structure of the problem, we derive the dispersion relation for the propagation of coupled long-waves: a surface wave and an internal wave. Also, we investigate the above-mentioned model of wave-current interactions in the general case with arbitrary vorticities.
Jarmo Hietarinta.
We search and classify two-component versions of the quad equations in the
ABS list, under certain assumptions. The independent variables will be called
$y,z$ and in addition to multilinearity and irreducibility the equation pair is
required to have the following specific properties: (1) The two equations
forming the pair are related by $y\leftrightarrow z$ exchange. (2) When $z=y$
both equations reduce to one of the equations in the ABS list. (3) Evolution in
any corner direction is by a multilinear equation pair. One straightforward way
to construct such two-component pairs is by taking some particular equation in
the ABS list (in terms of $y$), using replacement $y \leftrightarrow z$ for
some particular shifts, after which the other equation of the pair is obtained
by property (1). This way we can get 8 pairs for each starting equation. One of
our main results is that due to condition (3) this is in fact complete for H1,
H3, Q1, Q3. (For H2 we have a further case, Q2, Q4 we did not check.) As for
the CAC integrability test, for each choice of the bottom equations we could in
principle have $8^2$ possible side-equations. However, we find that only
equations constructed with an even number of $y \leftrightarrow z$ replacements
are possible, and for each such equation there are two sets of "side" equation
pairs that produce (the same) genuine BĂ¤cklund transformation and Lax pair.
A. S. Fokas ; A. Latifi.
We elaborate on a new methodology, which starting with an integrable
evolution equation in one spatial dimension, constructs an integrable forced
version of this equation. The forcing consists of terms involving quadratic
products of certain eigenfunctions of the associated Lax pair. Remarkably, some
of these forced equations arise in the modelling of important physical
phenomena. The initial value problem of these equations can be formulated as a
Riemann-Hilbert problem, where the "jump matrix" has explicit x and t
dependence and can be computed in terms of the initial data. Thus, these
equations can be solved as efficiently as the nonlinear integrable equations
from which they are generated. Details are given for the forced versions of the
nonlinear Schrodinger.
Giuseppe Gaeta ; Miguel Angel Rodriguez.
It is known that knowledge of a symmetry of a scalar Ito stochastic
differential equations leads, thanks to the Kozlov substitution, to its
integration. In the present paper we provide a classification of scalar
autonomous Ito stochastic differential equations with simple noise possessing
symmetries; here "simple noise" means the noise coefficient is of the form $\s
(x,t) = s x^k$, with $s$ and $k$ real constants. Such equations can be taken to
a standard form via a well known transformation; for such standard forms we
also provide the integration of the symmetric equations. Our work extends
previous classifications in that it also consider recently introduced types of
symmetries, in particular standard random symmetries, not considered in those.
Francesco Calogero ; Farrin Payandeh.
In this short communication we introduce a rather simple autonomous system of
2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose
initial-values problem is explicitly solvable by algebraic operations. Its ODEs
feature 2 right-hand sides which are the ratios of 2 homogeneous polynomials of
first degree divided by the same homogeneous polynomial of second degree. The
model features only 4 arbitrary parameters. We also report its isochronous
variant featuring 4 nonlinearly-coupled first-order ODEs in 4 dependent
variables, featuring 9 arbitrary parameters.