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Given two tensor fields of type (1,1) on a smooth n-dimensional manifold M, such that all their Fr\"olicher-Nijenhuis brackets vanish, the algebra of differential forms on M becomes a bi-differential graded algebra. As a consequence, there are partial differential equation (PDE) systems associated with it, which arise as the integrability condition of a system of linear equations and possess a binary Darboux transformation to generate exact solutions. We recover chiral models and potential forms of the self-dual Yang-Mills, as well as corresponding generalizations to higher than four dimensions, and obtain new integrable non-autonomous nonlinear matrix PDEs and corresponding systems., Comment: 20 pages

In 2005 Lorenzoni and Magri showed that a hydrodynamic-type hierarchy determined by the powers of a type (1,1) tensor field (on a smooth manifold) with vanishing Nijenhuis torsion can be deformed to a more general hierarchy, with the help of a chain of conservation laws of the new hierarchy. We review this construction. The (1,1) tensor fields of the resulting hierarchy have non-vanishing Nijenhuis torsion, in general, but their Haantjes tensor vanishes., Comment: 12 pages, third version: Proposition 3.1 in previous versions was incorrect and has been deleted, Theorem 4.5 follows directly from a result of Bogoyavlenskij in 2004, corresponding note added at the end of the paper

Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory associated with oriented simplices. Whereas the $(N-1)$-simplex equation can be regarded as a realization of the higher Bruhat order $B(N,N-2)$, the $N$-gon equation is a realization of the higher Tamari order $T(N,N-2)$. The latter and its dual $\tilde T(N,N-2)$, associated with which is the dual $N$-gon equation, have been shown to arise as suborders of $B(N,N-2)$ via a ''three-color decomposition''. There are two different reductions of $T(N,N-2)$ and $\tilde T(N,N-2)$, to ${T(N-1,N-3)}$, respectively $\tilde T(N-1,N-3)$. In this work, we explore the corresponding reductions of (dual) polygon equations, which lead to relations between solutions of neighboring (dual) polygon equations. We also elaborate (dual) polygon equations in this respect explicitly up to the octagon equation.

Using bidifferential calculus, we derive a vectorial binary Darboux transformation for the first member of the "negative" part of the AKNS hierarchy. A reduction leads to the first "negative flow" of the NLS hierarchy, which in turn is a reduction of a rather simple nonlinear complex PDE in two dimensions, with a leading mixed third derivative. This PDE may be regarded as describing geometric dynamics of a complex scalar field in one dimension, since it is invariant under coordinate transformations in one of the two independent variables. We exploit the correspondingly reduced vectorial binary Darboux transformation to generate multi-soliton solutions of the PDE, also with additional rational dependence on the independent variables, and on a plane wave background. This includes rogue waves., Comment: 19 pages, 5 figures, Second version: substantial changes. Third version: Section 3 substantially expanded. Fourth and fifth version: small amendments in Abstract, Introduction, first part of Section 3, Conclusion and references. To appear in Journal of Physics A: Mathematical and Theoretical

The theoretical physicist and mathematician Aristophanes Dimakis passed away on July 8, 2021, at the age of 68, in Athens, Greece. We briefly review his life, career and scientific achievements., Comment: 13 pages, 4 figures

We consider a matrix refactorization problem, i.e., a "Lax representation", for the Yang-Baxter map that originated as the map of polarizations from the "pure" 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang-Baxter equation, but satisfies a mixed version of the Yang-Baxter equation together with the Yang-Baxter map. Such maps have been called "entwining Yang-Baxter maps" in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is NOT in general a Yang-Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang-Baxter equation holds, by exploring the pure 3-soliton solution in the "tropical limit", where the 3-soliton interaction decomposes into 2-soliton interactions. Here this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang-Baxter maps as in the KP case, indicating a kind of universality., Comment: 30 pages, 10 figures, second version: two more references, corrections of typos in Appendix B, Remark B.3 added, third version: minor amendments. To appear in Letters in Mathematical Physics

We study soliton solutions of matrix "good" Boussinesq equations, generated via a binary Darboux transformation. Essential features of these solutions are revealed via their "tropical limit", as exploited in previous work about the KP equation. This limit associates a point particle interaction picture with a soliton (wave) solution., Comment: 24 pages, 11 figures, second version: some minor amendments

After an introduction to some aspects of bidifferential calculus on associative algebras, we focus on the notion of a "symmetry" of a generalized zero curvature equation and derive Backlund and (forward, backward and binary) Darboux transformations from it. We also recall a matrix version of the binary Darboux transformation and, inspired by the so-called Cauchy matrix approach, present an infinite system of equations solved by it. Finally, we sketch recent work on a deformation of the matrix binary Darboux transformation in bidifferential calculus, leading to a treatment of integrable equations with sources., Comment: 19 pages, to appear in "Algebraic Structures and Applications", S. Silvestrov et al (eds.), Springer Proceedings in Mathematics & Statistics, 2020

For the first two equations of the Volterra lattice hierarchy and the first two equations of its non-autonomous (non-isospectral) extension, we present Riccati systems for functions c_j(t), j=0,1,..., such that an expression in terms of Hankel determinants built from them solves these equations on the right half of the lattice. This actually achieves a complete linearization of these equations of the extended Volterra lattice hierarchy., Comment: 31 pages, 3rd version: introduction extended, part of Section 2 moved there, Appendix D added, additional references, to appear in Nonlinearity

In the tropical limit of matrix KP-II solitons, their support at fixed time is a planar graph with "polarizations" attached to its linear parts. In this work we explore a subclass of soliton solutions whose tropical limit graph has the form of a rooted and generically binary tree, as well as solutions with a limit graph consisting of two relatively inverted such rooted tree graphs. The distribution of polarizations over the constituting lines of the graph is fully determined by a parameter-dependent binary operation and a (in general non-linear) Yang-Baxter map, which in the vector KP case becomes linear, hence is given by an R-matrix. The parameter-dependence of the binary operation leads to a solution of the pentagon equation, which exhibits a certain relation with the Rogers dilogarithm via a solution of the hexagon equation, the next member in the family of polygon equations. A generalization of the R-matrix, obtained in the vector KP case, is found to also solve a pentagon equation. A corresponding local version of the latter then leads to a new solution of the hexagon equation., Comment: 10 pages, 5 figures, second version: correction of a typo in the B-R consistency condition, and q_1=1 in Example 3.3

We study soliton solutions of matrix Kadomtsev-Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to its constituting lines. There is a subclass of "pure line soliton solutions" for which we find that, in this limit, the distribution of polarizations is fully determined by a Yang-Baxter map. For a vector KP equation, this map is given by an R-matrix, whereas it is a non-linear map in case of a more general matrix KP equation. We also consider the corresponding Korteweg-deVries (KdV) reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain a new solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution of the functional tetrahedron equation arises from the parameter-dependence of the vector KP R-matrix., Comment: 23 pages, 9 figures, second version: some minor amendments, reformulations in Section 4, additional references [10] and [18]

The infinite families of Peregrine, Akhmediev and Kuznetsov-Ma breather solutions of the focusing Nonlinear Schroedinger (NLS) equation are obtained via a matrix version of the Darboux transformation, with a spectral matrix of the form of a Jordan block. The structure of these solutions is essentially determined by the corresponding solution of the Lyapunov equation. In particular, regularity follows from properties of the Lyapunov equation., Comment: 18 pages, 5 figures, second and third version: Remark 2.3 added, some minor corrections. To appear in J. Phys. A: Math. Theor

We study two families of (matrix versions of) generalized Volterra (or Bogoyavlensky) lattice equations. For each family, the equations arise as reductions of a partial differential-difference equation in one continuous and two discrete variables, which is a realization of a general integrable equation in bidifferential calculus. This allows to derive a binary Darboux transformation and also self-consistent source extensions via general results of bidifferential calculus. Exact solutions are constructed from the simplest seed solutions., Comment: 2nd version: several corrections and additions, 24 pages, 4 figures, to appear in Journal of Geometry and Physics

We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the sine-Gordon, nonlinear Schrodinger, KdV, Boussinesq, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP systems. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus., Comment: 35 pages, 1 figure, second version: some amendments, additional references, section 5 added

It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a "mixed order." We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of "polygon equations" realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the $N$-simplex equation to the $(N+1)$-gon equation, its dual, and a compatibility equation.

We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of `Riemann equations' are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional NLS, KP and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated `Riemann equations'. For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the `Riemann equations' associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes `breaking' multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations., Comment: 40 pages, 3 figures, revised version: presentation improved, extended introduction, several minor amendments

Via a "tropical limit" (Maslov dequantization), Korteweg-deVries (KdV) solitons correspond to piecewise linear graphs in two-dimensional space-time. We explore this limit., Comment: 10 pages, 4 figures, conference "Physics and Mathematics of Nonlinear Phenomena 2013"

We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the D-dimensional vacuum Einstein equations with D-2 commuting Killing vector fields. A large class of exact solutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski-Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and double Myers-Perry black holes, black saturn, bicycling black rings).

In a tropical approximation, any tree-shaped line soliton solution, a member of the simplest class of soliton solutions of the Kadomtsev-Petviashvili (KP-II) equation, determines a chain of planar rooted binary trees, connected by right rotation. More precisely, it determines a maximal chain of a Tamari lattice. We show that an analysis of these solutions naturally involves higher Bruhat and higher Tamari orders., Comment: 28 pages, 25 figures, 2nd version: some corrections and changes in section 4, Tamari Memorial Festschrift

The non-autonomous chiral model equation for an $m \times m$ matrix function on a two-dimensional space appears in particular in general relativity, where for $m=2$ a certain reduction of it determines stationary, axially symmetric solutions of Einstein's vacuum equations, and for $m=3$ solutions of the Einstein-Maxwell equations. Using a very simple and general result of the bidifferential calculus approach to integrable partial differential and difference equations, we generate a large class of exact solutions of this chiral model. The solutions are parametrized by a set of matrices, the size of which can be arbitrarily large. The matrices are subject to a Sylvester equation that has to be solved and generically admits a unique solution. By imposing the aforementioned reductions on the matrix data, we recover the Ernst potentials of multi-Kerr-NUT and multi-Demianski-Newman metrics.

We express a matrix version of the self-induced transparency (SIT) equations in the bidifferential calculus framework. An infinite family of exact solutions is then obtained by application of a general result that generates exact solutions from solutions of a linear system of arbitrary matrix size. A side result is a solution formula for the sine-Gordon equation., Comment: 7 pages, 2 figures, 19th International Colloquium on Integrable Systems and Quantum Symmetries (ISQS19), Prague, Czech Republic, June 2010

The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a transition to a different rooted binary tree takes place. We prove that these correspond to maximal chains in Tamari lattices (which are poset structures on associahedra). We further derive results that allow to compute details of the evolution, including the critical events. Moreover, we present some insights into the structure of the more general line soliton solutions. All this yields a characterization of possible evolutions of line soliton patterns on a shallow fluid surface (provided that the KP-II approximation applies)., Comment: 49 pages, 36 figures, second version: section 4 expanded

We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the two derivations of the bidifferential graded algebra, we recover "negative flows", leading to an extension of the respective hierarchy. In this way we also meet a matrix and vector version of the short pulse equation and also the sine-Gordon equation. For these equations corresponding solution formulas are also derived. In all these cases the solutions are parametrized in terms of matrix data that have to satisfy a certain Sylvester equation.

Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy leads to a multicomponent Burgers hierarchy. We show in particular that any solution of the latter also solves a corresponding multicomponent (potential) KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more general relation between the multicomponent linear heat hierarchy and the multicomponent KP hierarchy. From this results a construction of exact solutions of the latter via a matrix linear system., Comment: 18 pages, 4 figures

We show that there is an operator with a simple geometric significance which yields the ordinary geometry of a linear equidistant lattice via Connes' distance function., Comment: 4 pages, LATEX