] 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 is 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.