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The current paper recovers cubic–quartic optical solitons in fiber Bragg gratings having polynomial law of nonlinear refractive index structures. Lie symmetry analysis is carried out, starting with the basic analysis. Then, it is followed through with improved Kudryashov and generalized Arnous schemes. The parameter constraints are also identified for the existence of such solitons. Numerical surface plots support the adopted applied analysis. [ABSTRACT FROM AUTHOR]

The enhanced Kudryashov's approach retrieves quiescent bright, dark, and singular solitons to the governing model that is considered with cubic–quartic form of self-phase modulation. The algorithm however fails to retrieve stationary solitons when the nonlinearity is the generalized version of the cubic–quartic form. The current analysis is conducted with a direct approach without an intermediary phase-portrait analysis as in the past. [ABSTRACT FROM AUTHOR]

We provide a detailed description of a numerical approach that makes use of the shifted Chebyshev polynomials of the sixth kind to approximate the solution of some fractional order differential equations. Specifically, we choose the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) to describe this method. We write our approximate solution in the product form, which consists of unknown coefficients and shifted Chebyshev polynomials. To compute the numerical values of coefficients, we use the initial and boundary conditions and the collocation technique to create a system of equations whose number matches the unknowns. We test the applicability and accuracy of this numerical approach using two examples. [ABSTRACT FROM AUTHOR]