Abstract

We construct nonlinear extensions of Dirac relativistic electron equation that preserve its other desirable properties such as locality reparability conservation of probability and Poincare invariance we determine the constraint that the nonlinear term must obey and classify the resultant non polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. The aims of this paper is to solve the nonlinear Dirac equation using the analysis method. We followed the historical analysis mathematical method and we found the following some results: Possibility of solving the nonlinear Dirac equation for many –bodies system with may bring about more insight into physical system.