Global solution for the one-dimensional relativistic short wave-long wave interaction for a compressible flow

In this paper, we analyze a model of relativistic short wave-long wave interaction where the short waves are described by the massless $$1+1$$ 1 + 1 -dimensional Thirring model of nonlinear Dirac equation and the long waves are described by the $$1+1$$ 1 + 1 -dimensional relativistic Euler equations with γ =1 γ = 1 . The interaction coupling terms are modeled by a potential proportional to the relativistic specific volume in the Dirac equation and an external force proportional to the square modulus of the Dirac wave function in the relativistic Euler equation. An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow .