  We consider the periodic standing waves in the derivative nonlinear
Schrodinger (DNLS) equation arising in plasma physics. By using a newly
developed algebraic method with two eigenvalues, we classify all periodic
standing waves in terms of eight eigenvalues of the Kaup-Newell spectral
problem located at the end points of the spectral bands outside the real line.
The analytical work is complemented with the numerical approximation of the
spectral bands, this enables us to fully characterize the modulational
instability of the periodic standing waves in the DNLS equation.
