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