We have derived a family of equations related to the untwisted affine Lie algebras $A_r^{\left(1\right)}$ using a Coxeter ℤ_r+1 reduction. They represent the third member of the hierarchy of soliton equations related to the algebra. Applying additional ℤ₂-involutions, we also obtain the real Hamiltonian forms of these equations. We also give some particular examples and impose additional reductions.

We demonstrate that nonlinear magnetic solitary excitations (solitons) traveling through a Heisenberg spin chain may be used as a robust tool capable of coherent control of the qubit's state. The physical problem is described by a Hamiltonian involving the interaction between the soliton and the qubit. We show that under certain conditions the generic Hamiltonian may be mapped on that of a qubit two-level system with matrix elements depending on the soliton parameters. We considered the action of a bright and a dark soliton depending on the driving nonlinear wave function. We considered a local interaction restricted the closest to the qubit spin in the chain. We computed the expressions of the physical quantities of interest for all cases and analyzed their behavior in some special limits.

We derive a new exactly solvable multi-component system of non-linear evolution equations (NLEEs). The system consists of three $1+1$-dimensional evolution equations—one first-order and two second-order in the spatial variable. We review their Lax representation, formulate the scattering problem, and derive the soliton-like solutions of the system.