We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D^(1)_4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered
. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.
We have derived a family of equations related to the untwisted affine Lie algebras $A^{(1)}_{r}$ using a Coxeter $mathbb{Z}_{r+1}$ reduction. They represent the third member of the hierarchy of soliton equations related to the algebra. We also give s
ome particular examples and impose additional reductions.
In the present note new effects concerned the dynamics of a charged spin-1/2 particle in a strong monochromatic plane wave background are discussed beyond the conventional dipole approximation. Namely using the semiclassical approach in the semi-rela
tivistic regime it is shown that the magnetic forces associated with the laser field have been retained and alter the spin evolution appreciably.
The fate of entanglement of spins for two heavy constituents of a bound state moving in a strong laser field is analyzed within the semiclassical approach. The bound state motion as a whole is considered classically beyond the dipole approximation an
d taking into account the magnetic field effect by using the exact solution to the Newton equation. At the same time the evolution of constituent spins under the laser influence is studied quantum mechanically. The spin density matrix is determined as solution to the von Neumann equation with the effective Hamiltonian, describing spin-laser interaction along the bound state classical trajectory. Based on the solution, the dynamics of concurrence of spins is calculated for the maximally entangled Werner states as well as for an initially uncorrelated state.
