We study the linear problem associated with modified affine Toda field
equation for the Langlands dual g^​∨, where
g^​ is an untwisted affine Lie algebra. The connection
coefficients for the asymptotic solutions of the linear problem are found to
correspond to the Q-functions for g-type quantum integrable
models. The ψ-system for the solutions associated with the fundamental
representations of g leads to Bethe ansatz equations associated
with the affine Lie algebra g^​. We also study the
A2r(2)​ affine Toda field equation in massless limit in detail and find
its Bethe ansatz equations as well as T-Q relations.Comment: 22 page