Related papers: Noncommutative Boussinesq and NLS type 2- and 3-si…
We discuss mathematical methods to derive Nonlinear Schr\"odinger equations (NLS) in "low dimensional" settings, i.e. the 3-dimensional physical space e.g. to 2 or 1 space dimensions. Beside from the case the system exhibits an internal…
The method due to Nijhoff and Bobenko & Suris to derive Lax pairs for partial difference equations (PDeltaEs) is applied to edge constrained Boussinesq systems. These systems are defined on a quadrilateral. They are consistent around the…
Brief review of the methods for solving the multicomponent nonlinear Schrodinger (MNLS) equations and analysis of their Hamiltonian structures is given. Main attention is paid to the MNLS related to the C.II- and D.III-types symmetric…
We deal with nonlinear systems of parabolic type satisfying component-wise structural conditions. The nonlinear terms are Carath\'eodory maps having controlled growth with respect to the solution and the gradient and the data are in…
We consider the cubic nonlinear Schr\"odinger equation (NLS) in any spatial dimension, which is a well-known example of an infinite-dimensional Hamiltonian system. Inspired by the knowledge that the NLS is an effective equation for a system…
We associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse Kasteleyn matrix and hence all correlations. It may be seen as a quantization of the limit shape…
These notes provide an introduction to the noncommutative matrix geometry which arises within matrix models of Yang-Mills type. Starting from basic examples of compact fuzzy spaces, a general notion of embedded noncommutative spaces…
We show that the noncommutative Yang-Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The noncommutative Yang-Mills action is invariant under combined…
In this paper we formulate an integrable model on the simple cubic lattice. The $N$ -- valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex type Tetrahedron Equation. In the…
We study vector quadrirational Yang-Baxter maps representing the momentum-energy transformation of two particles after elastic relativistic collisions. The collision maps admit Lax representations compatible with an r-matrix Poisson…
We construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, $3$-Lie algebras and, in particular, ternary Nambu-Lie algebras. We show that the structures obtained satisfy an invertibility property…
It is widely believed that via the Seiberg-Witten map, the linearly realized BPS equation in the non-commutative space is related to the non-linearly realized BPS equation in the commutative space in the zero slope limit. We show that the…
We develop a theory of non-unitary set-theoretical solutions to the Quantum Yang-Baxter equation. Our results generalize those obtained by Etingof, Schedler and the author. We remark that some of our constructions are similar to…
Based on the completeness relation for the squared solutions of the Lax operator $L$ we show that a subset of nonlocal equations from the hierarchy of nonlocal nonlinear Schr\"odinger equations (NLS) is a completely integrable system. The…
We study the solutions of the local Zamolodhcikov tetrahedron equation in the form of correspondences derived by $3\times 3$ matrices. We present all the associated generators of 4-simplex maps satisfying the local tetrahedron equation.…
A three-by-three matrix spectral problem for AKNS soliton hierarchy is proposed and the corresponding Bargmann symmetry constraint involved in Lax pairs and adjoint Lax pairs is discussed. The resulting nonlinearized Lax systems possess…
We present two lists of multi-component systems of integrable difference equations defined on the edges of a $\mathbb{Z}^2$ graph. The integrability of these systems is manifested by their Lax formulation which is a consequence of the…
We construct the recursion operators for the non-commutative Burgers equations using their Lax operators. We investigate the existence of any integrable mixed version of left- and right-handed Burgers equations on higher symmetry grounds.
Relativistic complex Burgers-Schr\"odinger and Nonlinear Schr\"odinger equations are constructed. In the non-relativistic limit they reduce to the standard Burgers and NLS equations respectively and are integrable at any order of…
This paper concerns locally finite 2-complexes $X_{m,n}$ which are combinatorial models for the Baumslag-Solitar groups $BS(m,n)$. We show that, in many cases, the locally compact group Aut($X_{m,n}$) contains incommensurable uniform…