Dynamic behaviors of abundant solutions for the Lakshmanan--Porsezian--Daniel equation in an optical fiber

The integrable Lakshmanan–Porsezian–Daniel (LPD) equation originating in nonlinear fiber is studied in this work via the Riemann–Hilbert (RH) approach. Firstly we perform the spectral analysis of the Lax pair along with LPD equation, from which a RH problem is formulated. Afterwards, using the symmetry relations of the potential matrix, the formula of N-soliton solutions can be obtained by solving the special RH problem with reflectionless under the conditions of irregularity. In particular, the localized structures and dynamic behaviors of the breathers and solitons corresponding to the real part, imaginary part and modulus of the resulting solution r(x,t) are shown graphically and discussed in detail. One of the innovations in the paper is that the higher-order linear and nonlinear term β has important impact on the velocity, phase, period, and wavewidth of wave dynamics. The other is that collisions of the high-order breathers and soliton solutions are elastic interaction which imply they remain bounded all the time.