  In this work, we extend the Riemann-Hilbert (RH) method in order to study the
coupled modified Korteweg-de Vries equation (cmKdV) under nonzero boundary
conditions (NZBCs), and successfully find its solutions with their various
dynamic propagation behaviors. In the process of spectral analysis, it is
necessary to introduce Riemann surface to avoid the discussion of multi-valued
functions, and to obtain the analytical and asymptotic properties needed to
establish the RH problem. The eigenfunction have a column that is not analytic
in a given region, so we introduce the auxiliary eigenfunction and the adjoint
matrix, which is necessary to derive the analytical eigenfunctions. The
eigenfunctions have three kinds of symmetry, which leads to three kinds of
symmetry of the scattering matrix, and the discrete spectrum is also divided
into three categories by us. The asymptoticity of the modified eigenfunction is
derived. Based on the analysis, the RH problem with four jump matrices in a
given area is established, and the relationship between the cmKdV equation and
the solution of the RH problem is revealed. The residue condition of reflection
coefficient with simple pole is established. According to the classification of
discrete spectrum, we discuss the soliton solutions corresponding to three
kinds of discrete spectrum classification and their propagation behaviors in
detail.
