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__Instability of the split-step and related methods near localized solutions of nonlinear Schrödinger equations__

T. Lakoba

**Abstract**

The split-step method (SSM) is widely used
for numerical solution of nonlinear evolution equations,
e.g., the generalized NLS:
$
i u_t + u_{xx} + (V(x)+2|u|^2)u = 0.
$
We will show how SSM's instability
is analyzed using a modified linearized NLS
for a high-wavenumber numerical error. For example,
for the SSM which uses the finite-difference solution of its
dispersive step, this modified linearized NLS is:
$
i\psi_t + \delta\psi - \alpha \psi_{\chi\chi} +
V(\epsilon\chi)\psi +
2|u_{\rm sol}(\epsilon\chi)|^2 (2\psi + \psi^*) =0,
$
where $\psi$ is proportional to the numerical error. We will show
how the instability differs for a stationary, moving,
and oscillating background pulse.
We will also highlight differences between instabilities of the
finite-difference and Fourier implementations of the SSM,
and then discuss how these results can be used to predict the
instability of the integrating factor and
exponential time differencing methods.