Abundant solutions for the Lakshmanan–Porsezian–Daniel equation in an optical fiber through Riemann–Hilbert approach

The integrable Lakshmanan–Porsezian–Daniel (LPD) equation originating in nonlinear fiber is studied in this work via the Riemann–Hilbert (RH) approach. First, we give the spectral analysis of the Lax pair, from which an RH problem is formulated. Afterwards, by solving the special RH problem with reflectionless under the conditions of irregularity, the formula of general [Formula: see text]-soliton solutions can be obtained. In addition, the localized structures and dynamic behaviors of the breathers and solitons corresponding to the real part, imaginary part and modulus of the resulting solution [Formula: see text] are shown graphically and discussed in detail. Unlike 1- or 2-order breathers and solitons, 3-order breathers and soliton solutions rapidly collapse when they interact with each other. This phenomenon results in unbounded amplitudes which imply that higher-order solitons are not a simple nonlinear superposition of basic soliton solutions.