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The massive ODE/IM correspondence is a relation between the linear problemassociated with modified affine Toda field equations and two-dimensionalmassive integrable models. We study the massive ODE/IM correspondence for the$A_r^{(1)}$-type modified affine Toda field equations. Based on the$\psi$-system satisfied by the solutions of the linear problem, we derive theBethe ansatz equations and determine the asymptotic behavior of the Q-functionsfor large value of the spectral parameter. We derive the non-linear integralequations for the Q-functions from the Bethe ansatz equations. We compute theeffective central charge in the UV limit, which is identified with the one ofthe non-unitary $WA_r$ minimal models when the solution has trivial monodromyaround the origin of the complex plane.

Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations