One dimensional Dirac equation with quadratic nonlinearities

Shuji Machihara

doi:10.3934/dcds.2005.13.277

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2005, Volume 13, Issue 2: 277-290. Doi: 10.3934/dcds.2005.13.277

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One dimensional Dirac equation with quadratic nonlinearities

Shuji Machihara^1,

1.
Department of Mathematics, Shimane University, Matsue 690-8504

Received: July 31, 2004

Revised: March 31, 2005

Published: January 2005

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Abstract

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].

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Mathematics Subject Classification: 35L05, 35L60.

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