Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials

Abstract

Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schrödinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than one scalar potential, or vector potential, or both. We also obtain the associated Lax pairs. We discuss various interesting reductions in (2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear Schrödinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schrödinger-Hirota-Maxwell-Bloch equations, along with their Lax pairs.

Darboux transformation and exact solutions of the integrable Heisenberg ferromagnetic equation with self-consistent potentials

“…The most famous example might be the nonlinear Schrodinger -Maxwell -Bloch equation, which now is one of the most important equations in theory of pulse propagation along the optical fiber. Another interesting fundamental soliton equation is the Heisenberg ferromagnetic equation with self-consistent sources [23]- [24].…”

“…The Myrzakulov-XCIX equation (M-XCIX equation) reads as [23]- [24] (about our notation and definitons, see e.g. [11]- [22]) 4) where the matrix operators U and V have the form…”

Solitons for the (2+1)-dimensional nonlinear Schrödinger-Maxwell-Bloch equations in an erbium-doped fibre

“…Ref. [21] has investigated the MyrzakulovLakshmanan equations, and through certain gauge or geometric equivalence, given Equation (6) with the Lax pairs. The N -fold DT of and one-soliton solutions Equation (6) has been given as well [22].…”

“…[21,22] have presented the reduction r = δq * , k = δp * , where * means complex conjugate and δ = ±1, so that Equation (5) stake the forms [23] iq t + q xy − vq + 2i p = 0,…”

“…[17][18][19][20]. We consider the interaction of the spin vector with the scalar potential [21], the (2 + 1)-dimensional NLS-MB equations have been generated as [21][22][23] iq t + q xy − vq − 2i p = 0, (5a)…”