The correspondence between the real-complex transition in energy and delocalization-localization transition is well established in a class of Aubry-André-Harper model with exponential non-Hermitian on-site potentials. In this paper, we study a generalized Aubry-André model with off-diagonal modulation and non-Hermitian on-site potential. We find that, when there exists an incommensurate off-diagonal modulation, the correspondence breaks down, although the extended phase is maintained in a wide parameter range of the strengths of the on-site potential and the off-diagonal hoppings. An additional intermediate phase with a non-Hermitian mobility edge emerges when the off-diagonal hoppings become commensurate. This phase is characterized by the real and complex sections of the energy spectrum corresponding to the extended and localized states. In this case, the aforementioned correspondence reappears due to the recovery of the $\mathcal{PT}$ symmetry.

An extended (2+1)‐dimensional Calogero‐Bogoyavlenskii‐Schiff‐like equation is proposed by using the generalized bilinear operators based on a prime number p = 3. By combining multiexponential functions with a quadratic function, the interaction between lumps and multikink soliton is generated. In the meanwhile, the interaction of lump with periodic waves and the interaction among lumps, periodic waves, and multikink soliton can be obtained by introducing the ansätz forms. The dynamics of these interaction solutions are analyzed graphically by selecting appropriate parameters.