In this paper, we investigate the localized wave and mixed localized wave solutions of the Lakshmanan-Porsezian-Daniel (LPD) equation, which describes the energy storage and transfer processes in \(\alpha \) -helical proteins with interspine. First, we construct the generalized (m,N-m)-fold Darboux transformation (DT) of the LPD equation by two different kinds of Taylor series expansions. Then using the generalized DT formula, we analytically and graphically present the Nth-order multi-localized wave solutions including soliton solutions, breather solutions, rogue wave (RW) solutions and their mixed interaction cases in detail. To summarize the various mathematical structures of multi-localized wave solutions, we classify the different wave patterns by table. Finally, the dynamical behaviors of these localized wave and mixed localized wave interactions are carefully studied by numerical simulations. These results may be useful to understand the interaction phenomena in nonlinear localized waves and other physically relevant systems.
