In this work we develop some aspects of the theory of Hopf algebras to the context of autonomous map pseudomonoids. We concentrate in the Hopf modules and the Centre or Drinfel’d double. If A is a map pseudomonoid in a monoidal bicategory M , the analogue of the category of Hopf modules for A is an Eilenberg-Moore construction for a certain monad in Hom(M op,Cat). We study the existence of the ...

We present evidence on the global existence of solutions of De Gregorio’s equation, based on numerical computation and a mathematical criterion analogous to the Beale–Kato–Majda theorem. Its meaning in the context of a generalized Constantin–Lax–Majda equation will be discussed. We then argue that a convection term, if set in a proper form and in a proper magnitude, can deplete solutions of blo...

We show that a recently introduced Lax pair of the Sawada-Kotera equation is nota new one but is trivially related to the known old Lax pair. Using the so-called trivialcompositions of the old Lax pairs with a differentially constrained arbitrary operators,we give some examples of trivial Lax pairs of KdV and Sawada-Kotera equations.

Given a pseudomonad $mathcal{T} $ on a $2$-category $mathfrak{B} $, if a right biadjoint $mathfrak{A}tomathfrak{B} $ has a lifting to the pseudoalgebras $mathfrak{A}tomathsf{Ps}textrm{-}mathcal{T}textrm{-}mathsf{Alg} $ then this lifting is also right biadjoint provided that $mathfrak{A} $ has codescent objects. In this paper, we give  general results on lifting of biadjoints. As a consequence, ...

we show that a recently introduced lax pair of the sawada-kotera equation is nota new one but is trivially related to the known old lax pair. using the so-called trivialcompositions of the old lax pairs with a differentially constrained arbitrary operators,we give some examples of trivial lax pairs of kdv and sawada-kotera equations.

We prove separation of variables for the most general (Dn type) periodic Toda lattice with 2 × 2 Lax matrix. It is achieved by finding proper normalisation for the corresponding Baker-Akhiezer function. Separation of variables for all other periodic Toda lattices associated with infinite series of root systems follows by taking appropriate limits.

We consider the Lax representation of the new two-component coupled integrable system recently discovered by the author. Connection of the hierarchy of infinitely many Lax pairs with each other is presented.

It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quadgraphs has been recently discovered by A. Bobenko and one of the authors, and by F. Nijhoff. Introduction. In 1990 V.G. Drinfeld suggested the proble...