Riemann–Hilbert problem associated with the vector Lakshmanan–Porsezian–Daniel model in the birefringent optical fibers

Abstract

In this paper, we investigate vector Lakshmanan-Porsezian-Daniel (VLPD) model which can be used to describe the ultrashort pulses in the birefringent optical fiber. Based on the unified transformation method, the Riemann-Hilbert problem is introduced and initial-boundary value problems of the VLPD model are studied. By solving the formulated matrix Riemann-Hilbert problem, the potential function solutions of the VLPD model can be reconstructed. Moreover, that the spectral functions are not independent but meet the so-called global relation is shown.

General multi-soliton and higher-order soliton solutions for a novel nonlocal Lakshmanan–Porsezian–Daniel equation

“…In addition, transform t → −t and take conjugation on the kernel equation and combine the symmetry relation (17), the corresponding eigenvector for the eigenvalue…”

“…Due to the important application in nonlinear optics, abundant optical solitons [6][7][8][9][10][11] and some new exact solutions [12] for LPD equation were explored. Besides, the rich dynamic behaviors of solitons, rogue waves and the interactional solutions for a multi-component LPD equation were studied by Darboux transformation [13][14][15][16] and inverse scattering method [17].…”

Lump-Type Solutions, Mixed Solutions and Rogue Waves for a (3+1)-Dimensional Variable-Coefficients Burgers Equation

“…The most important of these studies are meant to obtain soliton solutions. Many methods have been developed, including the inverse scattering transformation [7,8], the Darboux transformation [9], Bäcklund transformations [10], and Hirota's bilinear technique [11].…”

The exact solutions to a new type space reverse nonlocal Lakshmanan–Porserzian–Daniel equation

“…which can be obtained from the coupled LPD system with a space reduction, also has been given in [31], while the solutions of the nonlocal equation have not been studied. The coupled LPD system can depict the simultaneous propagation of the nonlinear waves in the inhomogeneous optical fiber [32][33][34][35][36][37][38][39][40][41]. Thus, the nonlocal LPD equation (1.3) governs the nonlinear wave propagation in the physical system that the coupled LPD system is related to a parity symmetry.…”