## ] o c n m p [## Open Communications in Nonlinear Mathematical Physics |

Articles and Letters published during 2022.

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.

We search and classify two-component versions of the quad equations in theABS list, under certain assumptions. The independent variables will be called$y,z$ and in addition to multilinearity and irreducibility the equation pair isrequired to have the following specific properties: (1) The two equationsforming 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 inany corner direction is by a multilinear equation pair. One straightforward wayto construct such two-component pairs is by taking some particular equation inthe ABS list (in terms of $y$), using replacement $y \leftrightarrow z$ forsome particular shifts, after which the other equation of the pair is obtainedby property (1). This way we can get 8 pairs for each starting equation. One ofour 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 forthe CAC integrability test, for each choice of the bottom equations we could inprinciple have $8^2$ possible side-equations. However, we find that onlyequations constructed with an even number of $y \leftrightarrow z$ replacementsare possible, and for each such equation there are two sets of "side" equationpairs that produce (the same) genuine Bäcklund transformation and Lax pair.

We elaborate on a new methodology, which starting with an integrableevolution equation in one spatial dimension, constructs an integrable forcedversion of this equation. The forcing consists of terms involving quadraticproducts of certain eigenfunctions of the associated Lax pair. Remarkably, someof these forced equations arise in the modelling of important physicalphenomena. The initial value problem of these equations can be formulated as aRiemann-Hilbert problem, where the "jump matrix" has explicit x and tdependence and can be computed in terms of the initial data. Thus, theseequations can be solved as efficiently as the nonlinear integrable equationsfrom which they are generated. Details are given for the forced versions of thenonlinear Schrodinger.

It is known that knowledge of a symmetry of a scalar Ito stochasticdifferential equations leads, thanks to the Kozlov substitution, to itsintegration. In the present paper we provide a classification of scalarautonomous Ito stochastic differential equations with simple noise possessingsymmetries; 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 toa standard form via a well known transformation; for such standard forms wealso provide the integration of the symmetric equations. Our work extendsprevious classifications in that it also consider recently introduced types ofsymmetries, in particular standard random symmetries, not considered in those.

In this short communication we introduce a rather simple autonomous system of2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whoseinitial-values problem is explicitly solvable by algebraic operations. Its ODEsfeature 2 right-hand sides which are the ratios of 2 homogeneous polynomials offirst degree divided by the same homogeneous polynomial of second degree. Themodel features only 4 arbitrary parameters. We also report its isochronousvariant featuring 4 nonlinearly-coupled first-order ODEs in 4 dependentvariables, featuring 9 arbitrary parameters.

We propose a Hamiltonian formalism for $N$ periodic dressing chain with theeven number $N$. The formalism is based on Dirac reduction applied to the $N+1$periodic dressing chain with the odd number $N+1$ for which the Hamiltonianformalism is well known. The Hamilton dressing chain equations in the $N$ evencase depend explicitly on a pair of conjugated Dirac constraints and areequivalent to $A^{(1)}_{N-1}$ invariant symmetric Painlevé equations.

The classification of scalar Ito equations with a single noise source whichadmit a so called standard symmetry and hence are -- by the Kozlov construction-- integrable is by now complete. In this paper we study the situation,occurring in physical as well as biological applications, where there are twoindependent noise sources. We determine all such autonomous Ito equationsadmitting a standard symmetry; we then integrate them by means of the Kozlovconstruction. We also consider the case of three or more independent noises,showing no standard symmetry is present.

We study all five-, six-, and one eight-vertex type two-state solutions ofthe Yang-Baxter equations in the form $A_{12} B_{13} C_{23} = C_{23} B_{13}A_{12}$, and analyze the interplay of the `gauge' and `inversion' symmetries ofthese solution. Starting with algebraic solutions, whose parameters have nospecific interpretation, and then using these symmetries we can construct aparametrization where we can identify global, color and spectral parameters. Weshow in particular how the distribution of these parameters may be changed by achange of gauge.

We compute invariants for the two-variable Möbius transformation. Inparticular we are interested in partial differential equations in two dependentand two independent variables that are kept invariant under thistransformation.

Kontsevich's graph flows are -- universally for all finite-dimensional affinePoisson manifolds -- infinitesimal symmetries of the spaces of Poissonbrackets. We show that the previously known tetrahedral flow and the recentlyobtained pentagon-wheel flow preserve the class of Nambu-determinant Poissonbi-vectors $P=[\![\varrho(\boldsymbol{x})\,\partial_x\wedge\partial_y\wedge\partial_z,a]\!]$ on$\mathbb{R}^3\ni\boldsymbol{x}=(x,y,z)$ and $P=[\![[\![\varrho(\boldsymbol{y})\,\partial_{x^1}\wedge\ldots\wedge\partial_{x^4},a_1]\!],a_2]\!]$on $\mathbb{R}^4\ni\boldsymbol{y}$, including the general case $\varrho\not\equiv 1$. We detect that the Poisson bracket evolution $\dot{P} =Q_\gamma(P^{\otimes^{\# Vert(\gamma)}})$ is trivial in the second Poissoncohomology, $Q_\gamma = [\![ P, \vec{X}([\varrho],[a]) ]\!]$, for theNambu-determinant bi-vectors $P(\varrho,[a])$ on $\mathbb{R}^3$. For the globalCasimirs $\mathbf{a} = (a_1,\ldots,a_{d-2})$ and inverse density $\varrho$ on$\mathbb{R}^d$, we analyse the combinatorics of their evolution induced by theKontsevich graph flows, namely $\dot{\varrho} = \dot{\varrho}([\varrho],[\mathbf{a}])$ and $\dot{\mathbf{a}} =\dot{\mathbf{a}}([\varrho],[\mathbf{a}])$ with differential-polynomialright-hand sides. Besides the anticipated collapse of these formulas by usingthe Civita symbols (three for the tetrahedron $\gamma_3$ and five for thepentagon-wheel graph cocycle $\gamma_5$), as dictated by the behaviour$\varrho(\mathbf{x}') = \varrho(\mathbf{x}) […]

We report a class of symmetry-intergable third-order evolution equations in1+1 dimensions under the condition that the equations admit a second-orderrecursion operator that contains an adjoint symmetry (integrating factor) oforder six. The recursion operators are given explicitly.