In this paper we provide an algebraic construction for the negative even mKdV hierarchy which gives rise to time evolutions associated to even graded Lie algebraic structure. We propose a modification of the dressing method, in order to incorporate a non-trivial vacuum configuration and construct a deformed vertex operator forŝl(2), that enable us to obtain explicit and systematic solutions for the whole negative even grade equations.

We investigate the non-relativistic Schrödinger and Pauli-Dirac oscillators in noncommutative phase space using the five-dimensional Galilean covariant framework. The Schrödinger oscillator presented the correct energy spectrum whose non isotropy is caused by the noncommutativity with an expected similarity between this system and the particle in a magnetic field. A general Hamiltonian for the 3-dimensional Galilean covariant PauliDirac oscillator was obtained and it presents the usual terms that appears in commutative space, like Zeeman effect and spin-orbit terms. We find that the Hamiltonian also possesses terms involving the noncommutative parameters that are related to a type of magnetic moment and an electric dipole moment.

The algebraic structure of the integrable mixed mKdV/sinh-Gordon model is discussed and extended to the AKNS/Lund-Regge model and to its corresponding supersymmetric versions. The integrability of the models is guaranteed from the zero curvature representation and some soliton solutions are discussed.

We study the Galilean Dirac oscillator in a non-commutative situation, with space-space and momentum-momentum non-commutativity. The wave equation is obtained via a 'Galilean covariant' approach, which consists in projecting the covariant equations from a (4, 1)-dimensional manifold with light-cone coordinates, to a (3, 1)-dimensional Galilean space-time. We obtain the exact wave functions and their energy levels for the plane and discuss the effects of non-commutativity.

We investigate the spin of the electron in a non-relativistic context by using the Galilean covariant Pauli–Dirac equation. From a non-relativistic Lagrangian density, we find an appropriate Dirac-like Hamiltonian in the momentum representation, which includes the spin operator in the Galilean covariant framework. Within this formalism, we show that the total angular momentum appears as a constant of motion. Additionally, we propose a non-minimal coupling that describes the Galilean interaction between an electron and the electromagnetic field. Thereby, we obtain, in a natural way, the Hamiltonian including all the essential interaction terms for the electron in a general vector field.