Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H3, H4, and the dihedral group I2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the...

In additional to the parity ($\mathcal{P}$) symmetric, time reversal ($\mathcal{T}$) and $\mathcal{PT}$ symmetric nonlocal integrable systems, some other types of Klein-Gordon models with space-time exchange nonlocality moving nolocality are proposed. The Lax pairs established nonlinear equations explicitly given. A special soliton solution, composed $\mathcal{PT}$-symmetric part $\mathcal{PT}$...

Liftings of endofunctors on sets to relations are commonly used capture bisimulation coalgebras. Lax versions have been in those cases where strict lifting fails bisimilarity, as well modeling other notions simulation. This paper provides tools for defining and manipulating lax liftings. As a central result, we define notion distributive law functor over the powerset monad, show that there is a...

Two integrable lattice hierarchies associated with two discrete matrix spectral problems are derived, and the infinitely many conservation laws of the first integrable model is obtained. Moreover, the Darboux transformations based on different Darboux matrixes (2.3.4) and (3.2.4) for the above-mentioned integrable hierarchies are established with the help of two different gauge transformations ...

This work is dedicated to systems of matrix nonlinear evolution equations related Hermitian symmetric spaces the type $\mathbf{A.III}$. The under consideration generalize $1+1$ dimensional Heisenberg ferromagnet equation in sense that their Lax pairs are linear bundles pole gauge like for original model. Here we present certain local and nonlocal reductions. A integrable deformation some its re...

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) RmKdV mKdV), are connected through a transformation that combines both Miura transformations. Lax pairs straightforwardly obtained by means aforementioned have also identified classical Lie symmetries RmKdV. Non-trivial similari...

The cubic Szego equation has been studied as an integrable model for deterministic turbulence, starting with the foundational work of Gerard and Grellier. We introduce a truncated version this equation, wherein majority Fourier mode couplings are eliminated while signature features preserved, namely, Lax-pair structure nested hierarchy finite-dimensional dynamically invariant manifolds. Despite...

We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form. A system of discrete isomonodromic deformations is completely determined by commutation relations between the fact...