Instability of the split-step method for a signal with nonzero central frequency

Abstract

We obtain analytical conditions for the occurrence of numerical instability (NI) of a split-step method when the simulated solution of the nonlinear Schrödinger equation is close to a plane wave with nonzero carrier frequency. We also numerically study such an instability when the solution is a sequence of pulses rather than a plane wave. The plane-wave-based analysis gives reasonable predictions for the frequencies of the numerically unstable Fourier modes but overestimates the instability growth rate. The latter is found to be strongly influenced by the randomness of the signal's profile: The more randomly it varies during the propagation, the weaker is the NI. Using an example of a realistic transmission system, we demonstrate that our single-channel results can be used to predict occurrences of NI in multichannel simulations. We also give an estimate for the integration step size for which NI, while present, will not affect simulation results for such systems. Using that estimate may lead to a significant saving of computational time.

Instability of the finite-difference split-step method applied to the nonlinear Schrödinger equation. II. moving soliton

“…To answer the specific question above, let us start by noting two facts. First, as we showed in Sections III and IV (see also [5]), NI occurs via interaction of two groups of Fourier harmonics (p-and q-terms in (3.7)). Second, if an unstable mode were to propagate along with the soliton, it would have to have the same group velocity as the soliton.…”

“…6. On the other hand, we note that there is also an upper bound on S. Namely, we showed in [5] that K sol = S/(2|β|) is to be less than approximately 1/ √ |β| t in order for the FD-SSM to yield an accurate solution of the NLS (1.1). The main part of this work is organized as follows.…”

“…We emphasize that this equation is different from its counterparts in [1] and [4] and, therefore, requires a qualitatively different analysis. This analysis is also different from the von Neumann analysis for a moving plane wave [5], [6] in that it requires one to take into account spatially dependent coefficients in the equation for the error. Our analysis is presented in Section IV.…”

Effect of noise on extreme events probability in a one-dimensional nonlinear Schrödinger equation

“…We now comment on the time step t. By the stability condition of the split-step method (with α = 0 and no noise terms), one needs to have [26,27]:…”

Metastable energy strata in numerical discretizations of weakly nonlinear wave equations

“…In particular and in contrast to the nonlinear wave equation considered in the present paper, there is no formation of energy strata in the other modes. The stability in numerical discretizations of these plane wave solutions under small perturbations of the initial value is an old question [22] and has been studied on short time intervals [4,7,19,20,22] and on long time intervals [10].…”