Mixed-type solutions and rogue wave on various backgrounds for the reverse-space-time higher-order modified Gerdjikov-Ivanov equation

In this paper, we firstly propose a reverse-space-time higher-order modified Gerdjikov-Ivanov equation which can be deduced by a simple but highly significant symmetry reduction of the corresponding local equation. According to the Lax pair, we elaborate the N-fold Darboux transformation in terms of the determinant representations. Next, we present several kinds of new mixed-type solutions. In particular, through the Taylor expansion, we construct the semi-degenerate Darboux transformation to derive rogue wave on various backgrounds. Specifically, these findings make a unique contribution to the study of the nonlocal equation, and we categorize these solutions, as well as to delineate the precise conditions under which they are generated. It should be emphasized that solutions of the reverse-space-time higher-order modified Gerdjikov-Ivanov equation have new properties different from those of the local case. Subsequently, dynamic behaviors of these solutions are illustrated in great detail through numerous images. We hope our investigation can provide a contribution that will facilitate the generation of additional novel localized waves on various backgrounds for other nonlocal equations.