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Linearly implicit methods for the nonlinear Schrödinger equation in nonhomogeneous media. (English) Zbl 1024.65083
Summary: 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.

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)
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