Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers

Abstract

We report the existence and properties of vector breather and semirational rogue-wave solutions for the coupled higher-order nonlinear Schrödinger equations, which describe the propagation of ultrashort optical pulses in birefringent optical fibers. Analytic vector breather and semirational rogue-wave solutions are obtained with Darboux dressing transformation. We observe that the superposition of the dark and bright contributions in each of the two wave components can give rise to complicated breather and semirational rogue-wave dynamics. We show that the bright-dark type vector solitons (or breather-like vector solitons) with nonconstant speed interplay with Akhmediev breathers, Kuznetsov-Ma solitons, and rogue waves. By adjusting parameters, we note that the rogue wave and bright-dark soliton merge, generating the boomeron-type bright-dark solitons. We prove that the rogue wave can be excited in the baseband modulation instability regime. These results may provide evidence of the collision between the mixed ultrashort soliton and rogue wave.

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

“…Similar to the nonlinear Schrödinger (NLS) model, generally speaking, there are two types of coupled LPD models. One is the vector LPD (VLPD) model 10 ; it dimensionless form as follows 11,12 : $$$$ {\displaystyle \begin{array}{cc}\hfill i{q}_{1t}& +\frac{1}{2}{q}_{1 zz}+{q}_1R+\varepsilon \left[{q}_{1 zz zz}+4{q}_{1 zz}R+2{q}_{1z}\left({R}_z+2P\right)+2{q}_1\left(3{R}^2+{\overline{P}}_z+2S\right)\right]=0,\hfill \\ {}\hfill i{q}_{2t}& +\frac{1}{2}{q}_{2 zz}+{q}_2R+\varepsilon \left[{q}_{2 zz zz}+4{q}_{2 zz}R+2{q}_{2z}\left({R}_z+2P\right)+2{q}_2\left(3{R}^2+{\overline{P}}_z+2S\right)\right]=0,\hfill \end{array}} $$$$ where $$$ R=\left({\left|{q}_1\right|}^2+{\left|{q}_2\right|}^2\right) $$$…”

“…9 Similar to the nonlinear Schrödinger (NLS) model, generally speaking, there are two types of coupled LPD models. One is the vector LPD (VLPD) model 10 ; it dimensionless form as follows 11,12 :…”

“…From the practical viewpoint, some of these higher‐order effects can be used for a particular purpose; for example, the fourth‐order dispersion and the quintic nonlinearity may help stabilize the propagation of soliton‐like pulses or beams, while the self‐steepening effect may play an active role in the formation of shock waves or the generation of giant extreme waves. Recently, many mathematical physicists have done a lot of research on the VLPD model (), such as the Lax pair, bound‐state solutions, and conservation laws, 13,14 the vector semirational RWs and modulation instability, 10 the higher‐order interactional solutions and RW pairs, 15 and other soliton solutions 11,12 . Also, the integrability of the three‐component LPD model has been studied, such as, the Lax pair, 16 the soliton solution obtained by the Hirota bilinear method 17 and by the generalized DT method 18 .…”

Interactions of Localized Wave Structures on Periodic Backgrounds for the Coupled Lakshmanan–Porsezian–Daniel Equations in Birefringent Optical Fibers

“…Recently, many studies have investigated System (1.2). [31][32][33][34][35][36][37][38] Four different types of breathers on plane wave backgrounds were obtained via Nth-order Darboux transformation (DT), which was constructed using the loop method. [32] Breather-tosoliton conversion constrains, multi-peak, W-shaped, M-shaped, and anti-dark solutions have been obtained.…”

“…where q 1 and q 2 represent the complex envelopes of the twocomponent electric fields; x and t denote the distance along the direction of the propagation and retarded time, 𝛽 is a real parameter that indicates the strength of higher-order linear and nonlinear effects, * represents the complex conjugate, [31][32][33][34][35][36][37][38] respectively. Recently, many studies have investigated System (1.2).…”