Stability of standing periodic waves in the massive Thirring model

We analyze the spectral stability of the standing periodic waves in the massive Thirring model in laboratory coordinates. Since solutions of the linearized MTM equation are related to the squared eigenfunctions of the linear Lax system, the spectral stability of the standing periodic waves can be studied by using their Lax spectrum. Standing periodic waves are classified based on eight eigenvalues which coincide with the endpoints of the spectral bands of the Lax spectrum. Combining analytical and numerical methods, we show that the standing periodic waves are spectrally stable if and only if the eight eigenvalues are located either on the imaginary axis or along the diagonals of the complex plane.