Exact Soliton Solution Of The Spin Chain In A Plane Wave Background And Quantum Phase Transition Of The Dipolar Bosons In Optical Lattices

In the first section of this paper, spin wave interaction for anisotropic Heisenberg spin chain driven by an external magnetic field are studied in terms of the Holstein-Primakoff transformation. At a low temperature and continuum limitation the equation of motion can reduce to the Nonlinear-Schr?dinger equation. Employing Darboux transformation we construct exact N-soliton solution for anisotropic spin chain driven by an external magnetic field in linear wave background. As a special case the explicit one-and two-soliton solutions in spin wave background is obtained. The dispersion law, effective soliton mass, and the energy of each soliton are investigated detailedly. Our result show that the stability criterion of soliton is related with anisotropic parameter and the amplitude of the linear wave. In the second part of this paper, using Green function method we study the energy spectrum of dipolar bose atoms with the dipole-dipole interaction in an optical lattice. The Superfluid-Mott-Insulator phase transition condition of the dipolar bosons is determined from the energy-band structure of the excitation spectrum as a function of interatomic repulsion, dipolar bosons interaction and the tunnel coupling constants. Furthermore, the superfluid phase is explained explicitly from the energy spectrum derived in terms of Bogoliubov approach.