Global well-posedness for intermediate NLS with nonvanishing conditions at infinity

The intermediate nonlinear Schr\"odinger equation (INLS) describes the dynamics of the envelope of weakly nonlinear internal waves in a stratified fluid of finite depth. While the INLS equation is known to admit dark soliton solutions, these solutions possess nonvanishing boundary conditions at spatial infinity and therefore fall outside the scope of existing well-posedness frameworks. This paper establishes the local and global well-posedness of a generalized INLS equation in Zhidkov-type spaces tailored to these nonvanishing boundary conditions. Furthermore, we rigorously justify the deep-water limit, proving that solutions of the generalized INLS converge to those of the generalized Calogero-Moser (CM) derivative NLS equation in Zhidkov-type spaces. Our well-posedness theory relies on the modified energy method combined with frequency envelopes, marking the first application of these techniques to Zhidkov-type spaces.