Binary Darboux transformation, solitons, periodic waves and modulation instability for a nonlocal Lakshmanan–Porsezian–Daniel equation

In this paper, a nonlocal Lakshmanan–Porsezian–Daniel equation is investigated with the help of the binary Darboux transformation method and asymptotic analysis. Nonlocality of that equation has been reflected in that the solutions of that equation at the location ς depend on both the local solution at ς and the nonlocal solution at −ς, where ς is the retarded time coordinate. We derive the formulas of the Nth-order solutions through the obtained binary Darboux transformation, where N is a positive integer. Under certain conditions, the first-order periodic waves and solitons are obtained, e.g., degenerate solitons, dark–dark solitons, bright–bright solitons and dark–bright solitons. Interactions between/among the dark solitons, bright solitons and periodic wave are discussed and graphically illustrated. We discuss the modulation instability of that equation.