A compact split step Padé scheme for higher-order nonlinear Schrödinger equation (HNLS) with power law nonlinearity and fourth order dispersion.

*(English)*Zbl 1217.65177Summary: We propose a compact split step Padé scheme (CSSPS) to solve the scalar higher-order nonlinear Schrödinger equation (HNLS) with higher-order linear and nonlinear effects such as the third and fourth order dispersion effects, Kerr dispersion, stimulated Raman scattering and power law nonlinearity. The stability of this method is proved. It is shown as well that the CSSPS method gives the same results as classical numerical methods like the split step Fourier method and Crank-Nicolson method but it presents many advantages over theme. It is more efficient. This proposed scheme is well suited to higher-order dispersion effects and readily generalized for nonlinear and dispersion managed fibers. We tested this scheme for the case of the quintic nonlinearity and confirmed that this effect has no significant role on the propagation of single solitons.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35Q51 | Soliton equations |

##### Keywords:

optical solitons; compact Padé scheme; higher-order nonlinear Schrödinger equation; power law nonlinearity; higher order dispersion; numerical examples; comparison of methods; Kerr dispersion; Raman scattering; stability; Fourier method; Crank-Nicolson method
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\textit{M. Smadi} and \textit{D. Bahloul}, Comput. Phys. Commun. 182, No. 2, 366--371 (2011; Zbl 1217.65177)

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