Reverse Space-Time Nonlocal Sine-Gordon/Sinh-Gordon Equations with Nonzero Boundary Conditions

Nonlocal reverse space-time Sine/Sinh-Gordon type equations were recently introduced. They arise from a remarkably simple nonlocal reduction of the well-known AKNS scattering problem, hence, they constitute an integrable evolution equations. Furthermore, the inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, the IST for these novel nonlocal equations corresponding to nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complex due to the intricate branching structure of the associated linear eigenfunctions. Two cases are analyzed, which correspond to two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1-soliton and 2-soliton solutions are found. Both spatially independent and spatially dependent boundary conditions are considered.