Cubic nonlinear Dirac equation in a quarter plane

We study the initial-boundary value problem (IBV) for the cubic nonlinear Dirac equation in one space dimension{i(∂t+α∂x)ψ+βψ=〈βψ,ψ〉βψ,x>0,t>0,ψ(x,0)=ψ0(x),ψ(0,t)=h0(t), where ψ=ψ(t,x)∈C2 is a two-spinor field, α, β are hermitian (2×2)-matrices satisfying β2=α2=I, αβ+βα=0, 〈⋅,⋅〉 denotes the C2-scalar product. We prove the global in time existence of solutions of IBV problem for cubic Dirac equations with inhomogeneous Dirichlet boundary conditions. We obtain the sharp time decay of solutions in the uniform norm.