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The reverse space-time (RST) Sine-Gordon, Sinh-Gordon and nonlinearSchr\"odinger equations were recently introduced and shown to be integrableinfinite-dimensional dynamical systems. The inverse scattering transform (IST)for rapidly decaying data was also constructed. In this paper, IST for theseequations with nonzero boundary conditions (NZBCs) at infinity is presented.The NZBC problem is more complicated due to the associated branching structureof the associated linear eigenfunctions. With constant amplitude at infinity,four cases are analyzed; they correspond to two different signs of nonlinearityand two different values of the phase at infinity. Special soliton solutionsare discussed and explicit 1-soliton and 2-soliton solutions are found. Interms of IST, the difference between the RST Sine-Gordon/Sinh-Gordon equationsand the RST NLS equation is the time dependence of the scattering data.Spatially dependent boundary conditions are also briefly considered.

Inverse scattering transform for the nonlocal reverse space-time  Sine-Gordon, Sinh-Gordon and nonlinear Schr\"{o}dinger equations with nonzero  boundary conditions