Long-time asymptotics for the massive Thirring model on the half-line

The asymptotic formula for the solution of the massive Thirring model on the half-line $x\ge 0$ is derived under the assumption that the initial and boundary values lie in Schwartz space. It is shown that the solution of the massive Thirring model can represented by the solution of the derived matrix Riemann–Hilbert problem. The relevant jump matrices are explicitly given in the form of the matrix-valued spectral functions that depend on the initial data and boundary values. By applying the nonlinear steepest descent analysis of two related Riemann–Hilbert problems, one for the spectral parameter at the origin and the other for the spectral parameter at infinity, the explicit long-time asymptotic formula and uniform error estimates for the solution of the massive Thirring model are given on the half-line.