Inverse scattering transform for the complex reverse space-time nonlocal modified Korteweg-de Vries equation with nonzero boundary conditions and constant phase shift

Recently, the complex reverse space-time (RST) nonlocal modified Korteweg-de Vries equation (mKdV) was introduced and shown to be an integrable infinite-dimensional dynamical system. The inverse scattering transform (IST) for zero boundary condition was studied by Ablowitz and Musslimani in 2016. In this paper, the IST for the complex RST nonlocal mKdV equation with nonzero boundary conditions at infinity is presented. The direct and inverse scattering problems are analyzed. The method to carry out the inverse problem employs two Riemann surfaces associated with square root branch points in the eigenfunctions/scattering data, which is more complicated than that for the zero boundary condition. Four cases are considered, special soliton solutions are discussed, and an explicit 1-soliton and two 2-soliton are found for three cases. In another case, there are no solitons.