Implicit Quiescent Optical Soliton Perturbation Having Nonlinear Chromatic Dispersion and Linear Temporal Evolution with Kudryashov’s Forms of Self–Phase Modulation Structure by Lie Symmetry
Abstract
The paper retrieves implicit quiescent optical solitons for the nonlinear Schr¨odinger’s equation that is taken up with nonlinear chromatic dispersion and linear temporal evolution. Using a stationary or quiescent approach combined with Lie symmetry analysis, the study systematically examines six distinct self–phase–modulation structures proposed by Kudryashov. The analytical procedure reduces the governing equation to amplitude forms whose solutions are obtained through quadratures, leading to both implicit solitary–wave profiles and one explicit periodic case. The six forms of self–phase modulation structures, as proposed by Kudryashov, yielded solutions in terms of quadratures, periodic solutions as well as in terms of elliptic functions. The existence of each family of solutions is discussed in terms of the admissible parameter relations that ensure physically meaningful solitary profiles. The approach provides a unified framework compared with earlier methods based on direct elliptic–function expansions, highlighting how Lie symmetry facilitates a compact treatment of multiple nonlinear dispersion laws. The results are relevant to understanding stationary optical pulses in nonlinear fibers and photonic crystal fibers, and they establish a foundation for future numerical and experimental studies on nonlinear–dispersion–driven pulse propagation.
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References
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