] o c n m p [Open Communications in Nonlinear Mathematical Physics |
Edited by Norbert Euler, Rafael Hernández Heredero and Da-jun Zhang. Note: This issue is not complete: deadline for submissions has been June 14, 2026.
A phenomenon of "algebraic self-similarity" on 3d cubic lattice, providing what can be called an algebraic analogue of Kadanoff--Wilson theory, is shown to possess a 4d version as well. Namely, if there is a $4\times 4$ matrix $A$ whose entries are indeterminates over the field $\mathbb F_2$, then the $2\times 2\times 2\times 2$ block made of sixteen copies of $A$ reveals the existence of four direct "block spin" summands corresponding to the same matrix $A$. Moreover, these summands can be written out in quite an elegant way. Somewhat strikingly, if the entries of $A$ are just zeros and ones -- elements of $\mathbb F_2$ -- then there are examples where two more "block spins" split out, and this time with different $A$'s.
We derive a class of discrete Painlevé equations associated with the affine Weyl group E$_7^{(1)}$. The method used is the deautonomisation of a QRT mapping belonging to the canonical form VI (according to the classification of said mappings). An equation of such a form was the first instance of a symmetric -- in QRT parlance -- discrete analogue of the Painlevé VI equation. In this paper we present an exhaustive derivation of all the discrete Painlevé equations of this class. This is made possible thanks to previous studies that established the proper lengths of singularity patterns that are compatible with integrablity, and which were already successfully applied to the study of discrete Painlevé equations associated to the affine Weyl group E$_8^{(1)}$. Given that, from the latter, one can obtain by degeneration the equations related to E$_7^{(1)}$, we decided to link the results of the present study to those of the aforementioned ones. It turns out that a bridge from E$_8^{(1)}$ to E$_7^{(1)}$ exists in almost all cases, with one exception where, while in the former case a discrete Painlevé equation does exist, in the latter we find a mapping with only periodic coefficients, devoid of secular dependence.
We investigate the integrable structure and soliton dynamics of a coupled modified Korteweg-de Vries (cmKdV) system with a real symmetric coupling matrix. We introduce a vector reformulation of Hirota's bilinear formalism in which both the bilinear equations and their solutions are expressed directly at the vector level, rather than through a component-wise construction. This formulation preserves the intrinsic structure of the coupled system and provides a compact framework for multi-component nonlinear wave dynamics. Within this approach, we construct explicit one-, two-, and three-soliton solutions in closed vector form and recover the three-soliton condition directly at the vector level, confirming consistency with integrability. The method enables a unified treatment of focusing, defocusing, and mixed-sign regimes. In particular, for indefinite coupling, it reveals the existence of nontrivial vector ground states, leading to soliton solutions on non-zero backgrounds. These results highlight the structural advantages of the vector bilinear approach and open perspectives for the study of more general nonlinear excitations in multi-component integrable systems.
We develop the basic formulae of hyperspherical trigonometry in multidimensional Euclidean space, using multidimensional vector products, and their conversion to identities for elliptic functions. We show that the basic addition formulae for functions on the 3-sphere embedded in 4-dimensional space lead to addition formulae for elliptic functions, associated with algebraic curves, which have two distinct moduli. We give an application of these formulae to the cases of a multidimensional Euler top, using them to provide a link to the Double Elliptic model.
Here classes of moving boundary problems of Stefan-type for both an established non-linear evolution equation of cuspon theory and novel reciprocally linked solitonic equations are shown to be solvable via Painleve' II symmetry reduction.
We present $\frac{m^{2}}{4}+\frac{m}{2}+\frac{1-\left(-1\right)^{m}}{8}$ homogeneous $(3m-2)$-parameter families of Liouville integrable $(2m)$- and $(2m-1)$-dimensional Lotka-Volterra systems. We also study inhomogeneous versions of these systems.
We study the gauge-invariant dynamically conserved charges, and their corresponding densities, for instanton solutions of Yang-Mills theories in four dimensional Euclidean space, for the gauge group $SU(2)$. Those charges were constructed in [1,2] through the integral equations of Yang-Mills theory, using techniques on generalized loop spaces. We use the integral non-Abelian Gauss law to evaluate the gauge-invariant flux of the magnetic and electric non-Abelian fields through spherical surfaces centered at the origin of the instanton solution. From such a flux, we define gauge-invariant charge densities by considering the charge within an infinitesimal spherical shell of radius $r\equiv\sqrt{x_i \, x^i}/λ$, with $λ$ being the parameter of the instanton solution, defining its size, and $x_i \, x^i = (x^1)^2 + (x^{2})^2 + (x^{3})^2$. We discuss the issue of the reparameterization invariance of the charges and densities, and show that the magnetic and electric fluxes for the instanton and anti-instanton, at $r=1$ and $x^4 = 0$, $x^4$ being the Euclidean time, are non-zero and observable. Our results give an interesting picture of the internal structure of the instanton, and may be important for the properties of the Yang-Mills $θ$-vacuum.
This paper investigates the pole dynamics and perturbation theory of algebraic soliton solutions associated with the Satsuma--Mimura (SM) equation. First, we give a qualitative analysis of the pole system associated with algebraic soliton solutions, thereby completing a point left open in \cite{Yan_2025}. We then examine three perturbations of the SM equation. The $u_x$ perturbation preserves exact linearizability and leads to an explicit shifted algebraic soliton solution. The $u_{xx}$ perturbation can be reduced to the unperturbed SM equation by a scaling transformation, which yields the corresponding pole asymptotics and soliton profile. For the genuinely nontrivial $Hu_{xx}$ perturbation, we derive the first-order perturbation equations and present a short-time numerical simulation based on an explicit Euler discretization. These results clarify how algebraic solitons of the SM equation respond to different perturbative mechanisms.
This is a partial review of the bilinear method, focusing on the integrability based on the 3-soliton-solution condition and the transformations between $τ$ functions. \textit{Dedicated to Jarmo Hietarinta's 80th birthday}.
Recently, a family of unconventional integrators for higher order ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for first order ODEs with quadratic vector fields. All these integrators possess remarkable conservation properties. In particular, for the first and the second order Hamiltonian ODEs, the discretization by polarization possesses an integral of motion and an invariant volume form. In this note, we extend our previously proposed algebraic approach to derivation of these integrals to discretizations of ODEs of an arbitrary order. For all orders $\ge 3$, these integrals are new.
The generalized Yang-Baxter equations are multi-site versions of the standard Yang-Baxter equation. When spectral parameters are included, such equations are expected to lead to integrable Hamiltonians with local interactions involving multiple degrees of freedom. In this work we characterize both the continuous and discrete symmetries of these equations required to establish an equivalence class of solutions. We find that the set of such symmetries depends on the number of sites on which the equation is supported. In several cases there are more symmetries than the standard Yang-Baxter equation, thus placing heavy constraints on the number of inequivalent solutions and the associated integrable models. As an application we show how to restrict the generic $16\times16$ ansatz to the 30-dimensional subspace invariant under a chosen discrete symmetry subgroup. An explicit $16\times 16$ solution is also written down.
The objective of this work is to develop a framework that exploits the lattice structure of the $k$-th Volterra--Bogoyavlensky equations ($k\in\mathbb N$, $k>1$) to generate rational solutions of higher symmetric Painlevé equations. For $k=2$, we show that the Volterra lattice, equipped with suitable initial conditions, exactly models the one- and two-dimensional orbits generated by half-translation operators of the $A_2^{(1)}$ symmetric Painlevé IV equations. This correspondence yields explicit closed-form expressions for all solution components in terms of generalized Okamoto polynomials and leads to new algebraic recurrence relations among these polynomials. We present two generalizations of the above Volterra lattice. One is derived from a fractional translation of the $A_{4}^{(1)}$ symmetric Painlevé equations. It generalizes Volterra lattice structure in the multi-component setup of the affine $A_{4}^{(1)}$ group and it is shown to generate solutions of the $A_{4}^{(1)}$ symmetric Painlevé equations from the seed solution invariant under dihedral group $D_{5}$. The other is the $k=3$ Bogoyavlensky lattice structure. It satisfies recurrence relations that naturally extend recurrence relations of the Volterra lattice. These results shed light on connection between Volterra--Bogoyavlensky lattices, dihedral symmetries, and rational solutions of higher Painlevé systems.