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Instability of the finite‐difference split‐step method applied to the generalized nonlinear Schrödinger equation. III. external potential and oscillating pulse solutions
Author(s) -
Lakoba Taras I.
Publication year - 2017
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22071
Subject(s) - mathematics , instability , pulse (music) , nonlinear system , bounded function , soliton , nonlinear schrödinger equation , mathematical analysis , partial differential equation , noise (video) , numerical analysis , stability (learning theory) , series (stratigraphy) , schrödinger equation , mathematical physics , physics , mechanics , quantum mechanics , voltage , paleontology , artificial intelligence , machine learning , computer science , image (mathematics) , biology
This is the final part of a series of articles where we have studied numerical instability (NI) of localized solutions of the generalized nonlinear Schrödinger equation (gNLS). It extends our earlier studies of this topic in two ways. First, it examines differences in the development of the NI between the case of the purely cubic NLS and the case where the gNLS has an external bounded potential. Second, it investigates how the NI is affected by the oscillatory dynamics of the simulated pulse. The latter situation is common when the initial condition is not an exact stationary soliton. We have found that in this case, the NI may remain weak when the time step exceeds the threshold quite significantly. This means that the corresponding numerical solution, while formally numerically unstable, can remain sufficiently accurate over long times, because the numerical noise will stay small. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 633–650, 2017