Linearly implicit methods for the nonlinear Schrödinger equation in nonhomogeneous media.

*(English)*Zbl 1024.65083Summary: The nonlinear Schrödinger equation in one-dimensional Cartesian coordinates is studied numerically by means of three linearly implicit finite difference methods that result in block tridiagonal matrices at each time level, as a function of the damping, pumping and changes in the refraction index. The numerical experiments performed to determine the effects of the time step, spatial step size, and implicitness parameters of the dispersion and nonlinear terms show that the most accurate results are obtained when both the dispersion and nonlinear terms are treated by means of a second-order-accurate trapezoidal discretization and full linearization of the nonlinear terms.

It is also shown that if either the dispersion or the nonlinear terms are treated with first-order-accurate approximation, substantial errors are found, and these errors are largest when the nonlinear terms are discretized by means of a first-order difference and do not decrease substantially as the number of grid points is increased. It has been observed that when solitons move from a medium to another one characterized by a higher refraction index, the amplitude and speed of the soliton increase whereas its width decreases. Upon approaching the interface, the amplitude of the soliton was found to increase downstream while some radiation was observed upstream; both the increase in amplitude and radiation were found to increase with the change in the refraction index.

The effect of the distance where the index of refraction changes was found to be small if this distance is smaller than the width of the soliton. For solitons propagating from a medium to another one of lower refraction index, the amplitude and speed of the soliton decrease whereas its width increases, the amplitude of the soliton increases downstream and spreads towards the boundary where it is reflected. When the index of refraction decreases below a threshold, it was found that the soliton may become trapped at the interface.

The results presented in this paper indicate that theories based on treating solitons as particles, perturbation methods, inverse scattering, and variational formulations can provide accurate results only when the change in the refraction index is sufficiently small.

It is also shown that if either the dispersion or the nonlinear terms are treated with first-order-accurate approximation, substantial errors are found, and these errors are largest when the nonlinear terms are discretized by means of a first-order difference and do not decrease substantially as the number of grid points is increased. It has been observed that when solitons move from a medium to another one characterized by a higher refraction index, the amplitude and speed of the soliton increase whereas its width decreases. Upon approaching the interface, the amplitude of the soliton was found to increase downstream while some radiation was observed upstream; both the increase in amplitude and radiation were found to increase with the change in the refraction index.

The effect of the distance where the index of refraction changes was found to be small if this distance is smaller than the width of the soliton. For solitons propagating from a medium to another one of lower refraction index, the amplitude and speed of the soliton decrease whereas its width increases, the amplitude of the soliton increases downstream and spreads towards the boundary where it is reflected. When the index of refraction decreases below a threshold, it was found that the soliton may become trapped at the interface.

The results presented in this paper indicate that theories based on treating solitons as particles, perturbation methods, inverse scattering, and variational formulations can provide accurate results only when the change in the refraction index is sufficiently small.

##### MSC:

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

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35Q51 | Soliton equations |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

##### Keywords:

error bounds; nonlinear Schrödinger equation; linearly implicit finite difference methods; damping and pumping; variable refraction index; solutions in finite lines; numerical experiments; perturbation methods; inverse scattering
PDF
BibTeX
XML
Cite

\textit{J. I. Ramos}, Appl. Math. Comput. 133, No. 1, 1--28 (2002; Zbl 1024.65083)

Full Text:
DOI

##### References:

[1] | Ramos, J.I., Linearization methods for reaction – diffusion equations: 1-d problems, Appl. math. comput., 88, 199-224, (1997) · Zbl 0904.65088 |

[2] | Ramos, J.I., Linearization methods for reaction – diffusion equations: multidimensional problems, Appl. math. comput., 88, 225-254, (1997) · Zbl 0904.65089 |

[3] | Herbst, B.M.; Morris, J.Ll.; Mitchell, A.R., Numerical experience with the nonlinear Schrödinger equation, J. comput. phys., 60, 282-305, (1985) · Zbl 0589.65084 |

[4] | Sanz-Serna, J.M., An explicit finite-difference scheme with exact conservation properties, J. comput. phys., 47, 199-210, (1982) · Zbl 0484.65062 |

[5] | Delfour, M.; Fortin, M.; Payne, G., Finite-difference solution of a nonlinear Schrödinger equation, J. comput. phys., 44, 277-288, (1981) · Zbl 0477.65086 |

[6] | Sanz-Serna, J.M., Methods for the numerical solution of the nonlinear Schrödinger equation, Math. comput., 43, 21-27, (1984) · Zbl 0555.65061 |

[7] | Strauss, W.; Vázquez, L., Numerical solution of a nonlinear klein – gordon equation, J. comput. phys., 28, 271-278, (1978) · Zbl 0387.65076 |

[8] | Akrivis, G.D.; Dougalis, V.A.; Karakashian, O.A., On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. math., 59, 31-53, (1991) · Zbl 0739.65096 |

[9] | Tourigny, Y., Optimal H^1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation, IMA J. numer. anal., 11, 509-523, (1991) · Zbl 0737.65095 |

[10] | Shamardan, A.B., The numerical treatment of the nonlinear Schrödinger equation, Comput. math. appl., 19, 67-73, (1990) · Zbl 0702.65096 |

[11] | Akrivis, G.D., Finite difference discretization of the cubic Schrödinger equation, IMA J. numer. anal., 13, 115-124, (1993) · Zbl 0762.65070 |

[12] | Sanz-Serna, J.M.; Verwer, J.G., Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. numer. anal., 6, 25-42, (1986) · Zbl 0593.65087 |

[13] | Ablowitz, M.J.; Herbst, B.M.; Weideman, J.A.C., Dynamics of semidiscretizations of the defocusing nonlinear Schrödinger equation, IMA J. numer. anal., 11, 539-552, (1991) · Zbl 0737.65088 |

[14] | Ablowitz, M.J.; Ladik, J., A nonlinear difference scheme and inverse scattering, Stud. appl. math., 55, 213-229, (1976) · Zbl 0338.35002 |

[15] | Ablowitz, M.J.; Segur, H., Solitons and the inverse scattering transform, (1978), SIAM Philadelphia, PA · Zbl 0299.35076 |

[16] | Ablowitz, M.J.; Taha, T.R., Analytical and numerical aspects of certain nonlinear evolution equations - IV, J. comput. phys., 77, 540-548, (1988) · Zbl 0646.65087 |

[17] | Taha, T.R., A numerical scheme for the nonlinear Schrödinger equation, Comput. math. appl., 22, 77-84, (1991) · Zbl 0755.65130 |

[18] | Fornberg, B.; Whitham, G.B., A numerical and theoretical study of certain nonlinear wave phenomena, Phil. trans. R. soc. London, 289, 373-404, (1978) · Zbl 0384.65049 |

[19] | Weideman, J.A.C.; Herbst, B.M., Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. numer. anal., 23, 485-507, (1986) · Zbl 0597.76012 |

[20] | Sanz-Serna, J.M.; Calvo, M.P., Numerical Hamiltonian problems, (1994), Chapman & Hall London · Zbl 0816.65042 |

[21] | Tang, Y.-F.; Vázquez, L.; Zhang, F.; Pérez-Garcı́a, V.M., Symplectic methods for the nonlinear Schrödinger equation, Comput. math. appl., 32, 73-83, (1996) · Zbl 0858.65124 |

[22] | Zhang, F.; Pérez-Garcı́a, V.M.; Vázquez, L., Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. math. comput., 75, 165-177, (1995) · Zbl 0832.65136 |

[23] | Aceves, A.B.; Maloney, J.V.; Newell, A.C., Theory of light-beam propagation at nonlinear interfaces. I. equivalent-particle theory for a single interface, Phys. rev. A, 39, 1809-1827, (1989) |

[24] | Aceves, A.B.; Maloney, J.V.; Newell, A.C., Theory of light-beam propagation at nonlinear interfaces. II. multiple-particle and multiple-interface extensions, Phys. rev. A, 39, 1828-1840, (1989) |

[25] | Knapp, R., Transmission of solitons through random media, Physica D, 85, 496-508, (1995) · Zbl 0887.34074 |

[26] | Abdullaev, F.Kh.; Baizakov, B.B.; Umarov, B.A., Resonance phenomena in interaction of a spatial soliton with the modulated interface of two nonlinear media, Opt. commun., 156, 341-346, (1998) |

[27] | Abdullaev, F.Kh.; Caputo, J.G., Propagation of an envelope soliton in a medium with spatially varying dispersion, Phys. rev. E, 55, 6061-6071, (1997) |

[28] | Pérez-Garcı́a, V.M.; Porras, M.A.; Vázquez, L., The nonlinear Schrödinger equation with dissipation and the moment method, Phys. lett. A, 202, 176-182, (1995) |

[29] | Aceves, A.B.; Maloney, J.V., Effect of two-photon absorption on bright spatial soliton switches, Opt. lett., 17, 1488-1490, (1992) |

[30] | Karlsson, M.; Anderson, D., Super-Gaussian approximation of the fundamental radial mode in nonlinear parabolic-index optical fibers, J. opt. soc. am. B, 9, 1558-1562, (1992) |

[31] | Kodama, Y.; Ablowitz, M.J., Perturbations of solitons and solitary waves, Stud. appl. math., 64, 225-245, (1981) · Zbl 0486.76029 |

[32] | Karpman, V.I., Soliton evolution in the presence of perturbation, Phys. scr., 20, 462-478, (1979) · Zbl 1063.35531 |

[33] | Lantz, E.; Métin, D.; Cornet, H.; Lacourt, A., Transmission of a Gaussian beam through a nonlinear thin film near the total reflection state, J. opt. soc. am. B, 11, 347-354, (1994) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.