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Dirac equation with certain quadratic nonlinearities in one space dimension. (English) Zbl 1119.35034
Summary: We discuss the time local existence of solutions to the Dirac equation for special types of quadratic nonlinearities in one space dimension. Solutions with more rough data than those of the previous work [{\it S. Machihara}, Discrete Contin. Dyn. Syst. 13, No. 2, 277--290 (2005; Zbl 1077.35001)] are obtained. The Fourier transforms of solutions with respect to both variables $x$ and $t$ are investigated. Certain linear and bilinear estimates on solutions are derived, and a standard iteration argument gives the existence results.

35L60Nonlinear first-order hyperbolic equations
81Q05Closed and approximate solutions to quantum-mechanical equations
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