Abstract
In ultrafast optics, optical pulses are generated to be of shorter pulse duration, which has enormous significance to industrial applications and scientific research. The ultrashort pulse evolution in fiber lasers can be described by the higher-order Ginzburg-Landau (GL) equation. However, analytic soliton solutions for this equation have not been obtained by use of existing methods. In this paper, a novel method is proposed to deal with this equation. The analytic soliton solution is obtained for the first time, and is proved to be stable against amplitude perturbations. Through the split-step Fourier method, the bright soliton solution is studied numerically. The analytic results here may extend the integrable methods, and could be used to study soliton dynamics for some equations in other disciplines. It may also provide the other way to obtain two-soliton solutions for higher-order GL equations.
Highlights
Investigations on solitons have been made great progress since the first report on inverse scattering transformation (IST) method for soliton solutions[1]
Soliton solutions have been obtained in such nonlinear partial differential equations as nonlinear Schrödinger (NLS) equation, Sine-Gordon equation, Gross-Pitaevskii equation, Korteweg-de Vries equation, Burgers equation, Kadomtsev-Petviashvili equation and so on[9,10,11,12,13]
The evolution of ultrashort pulses in fiber lasers can be described by the higher-order GL equation in the following form[18]:
Summary
Investigations on solitons have been made great progress since the first report on inverse scattering transformation (IST) method for soliton solutions[1]. In addition to the IST method, there are some other integrable methods, such as Backlünd transformations, bilinear method, separation variable method and Darboux transformation, can be used to solve those equations[15,16,17]. Among all those methods, the bilinear method may be more direct and effective to solve integrable equations. The physical parameters β2, g, Ω, β3, γ, α and TR correspond to the GVD, optical gain, gain bandwidth, third-order dispersion (TOD), Kerr nonlinearity, optical loss and intra-pulse Raman scattering, respectively. Owing to the modified bilinear method, one-soliton solutions for the standard form of the complex GL equation can be obtained[19,20]. The bilinear operator Dzm and Dtn are a trivial case of modified Hirota bilinear operators, which can be defined by[19]
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