REMARKS ON NONLINEAR DIRAC EQUATIONS IN ONE SPACE DIMENSION

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Abstract. This paper reviews recent mathematical progresses made onthe study of the initial-value problem for nonlinear Dirac equations in onespace dimension. We also prove the global existence of solutions to somenonlinear Dirac equations and propose a model problem (3.6). 1. IntroductionWe are interested in the following initial value problem for the one dimen-sional nonlinear Dirac equationsi(∂ t U 1 +∂ x U 1 ) +mU 2 = ∂ U¯ 1 W(U 1 , U 2 ),i(∂ t U 2 −∂ x U 2 ) +mU 1 = ∂ U¯ 2 W(U 1 , U 2 ),U j (x, 0) = u j (x),(1.1)where U j : R 1+1 → C for j = 1, 2 and m(≥ 0) is a mass. U¯ is a complexconjugate of U. The potential W satisfies the following properties:1. Symmetry: W(U 1 , U 2 ) = W(U 2 , U 1 ).2. Gauge invariance: W(e iθ U 1 , e iθ U 2 ) = W(U 1 , U 2 ) for any θ ∈ R.3. Polynomial in (U 1 , U 2 ) and (U¯ 1 , U¯ 2 ).It is known [11] that fourth order homogeneous polynomial satisfying theabove properties takes the formW = a 1 |U 1 | 2 |U 2 | 2 +a 2 (U¯ 1 U 2 +U¯ 2 U 1 ) 2 +a 3 (|U 1 | 4