Lax Pair and new exact solutions of the nonlinear Dirac equation

We derive, using two different methods, exact analytic stationary solutions of the massive nonlinear Dirac equation (NLDE) in (1+1) dimensions as described by Thirring and Gross-Neveu models. In the first method, the equations are analyzed and some mathematical and physical properties of the solutions are inferred including continuity and wave equations of the current density. These properties are then used to derive the exact stationary solutions. The same solutions of Thirring model are rederived by finding a Lax pair representing the model and applying the Darboux transformation. The significance of finding the Lax pair lies in the fact that it opens the possibility to find an infinite chain of solutions by successive applications of the Darboux transformation. Within the range of physical interest, the derived solutions are localized. They show solitonic behavior of the spinor fields of a bounded fermions. Basic conserved quantities are calculated from the momentum energy tensor and some conclusions are drawn.