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One dimensional Dirac equation with quadratic nonlinearities
The local well-posedness for the nonlinear Dirac
equation with special forms of quadratic nonlinearities
in one space dimension is obtained by two approaches.
One is to apply the Fourier restriction norm method of
Bourgain [2, 3] by showing
the bilinear estimates for the nonlinearities.
Another is to study the explicit solutions for wave equations
and derive another bilinear estimates similar with
Bournaveas [4].