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

Lakoba, Taras I.

doi:10.1002/num.22039

Cited by 4 publications

(12 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[8‐15] in and Ref. in . For convenience of analysis, we assume periodic boundary conditions everywhere except in Section VI.C:

u (- L / 2, t) = u (L / 2, t), u_{x} (- L / 2, t) = u_{x} (L / 2, t) .

This is consistent with the fact that the pulses whose dynamics we simulate are meant to have vanishing asymptotics at

| x | \to \infty

.…”

Section: Introductionmentioning

confidence: 60%

“…This latter statement may at first appear counter-intuitive. However, we emphasize, as we have done and demonstrated in Parts I and II of this study [1,2], that properties of the NI depend not only on the numerical method and the underlying equation, but also on the simulated solution.…”

Section: Introductionmentioning

confidence: 60%

“…This work extends our study of numerical instability (NI) that may occur when the cubic nonlinear Schrödinger equation (NLS) is simulated by the finite‐difference split‐step method (fd‐SSM). For the reader's convenience, we will briefly summarize the setup and main conclusions of , while also introducing modifications that will pertain to this work.…”

Section: Introductionmentioning

confidence: 66%

“…In other words, the weaker the NI, the longer it takes the NI to become observable. In we also explained why this time depends on the initial noise.…”

Section: “Sluggish” Nimentioning

confidence: 95%

“…[8][9][10][11][12][13][14][15] in [1] and Ref. [2] in [2]. For convenience of analysis, we assume periodic boundary conditions everywhere except in Section VI.C: u(−L/2, t) = u(L/2, t), u x (−L/2, t) = u x (L/2, t).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Instability of the finite‐difference split‐step method applied to the generalized nonlinear Schrödinger equation. III. external potential and oscillating pulse solutions

Lakoba

2016

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…[8‐15] in and Ref. in . For convenience of analysis, we assume periodic boundary conditions everywhere except in Section VI.C:

u (- L / 2, t) = u (L / 2, t), u_{x} (- L / 2, t) = u_{x} (L / 2, t) .

This is consistent with the fact that the pulses whose dynamics we simulate are meant to have vanishing asymptotics at

| x | \to \infty

.…”

Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 66%

“…In other words, the weaker the NI, the longer it takes the NI to become observable. In we also explained why this time depends on the initial noise.…”

Section: “Sluggish” Nimentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Instability of the finite‐difference split‐step method applied to the generalized nonlinear Schrödinger equation. III. external potential and oscillating pulse solutions

Lakoba

2016

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

show abstract

Stability analysis of the numerical Method of characteristics applied to a class of energy-preserving hyperbolic systems. Part II: Nonreflecting boundary conditions

Lakoba

Deng

2019

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Study of instability of the Fourier split-step method for the massive Gross–Neveu model

Lakoba¹

2020

Journal of Computational Physics

View full text Add to dashboard Cite

Stability properties of the well-known Fourier split-step method used to simulate a soliton and similar solutions of the nonlinear Dirac equations, known as the Gross-Neveu model, are studied numerically and analytically. Three distinct types of numerical instability that can occur in this case, are revealed and explained. While one of these types can be viewed as being related to the numerical instability occurring in simulations of the nonlinear Schrödinger equation, the other two types have not been studied or identified before, to the best of our knowledge. These two types of instability are unconditional, i.e. occur for arbitrarily small values of the time step. They also persist in the continuum limit, i.e. for arbitrarily fine spatial discretization. Moreover, one of them persists in the limit of an infinitely large computational domain. It is further demonstrated that similar instabilities also occur for other numerical methods applied to the Gross-Neveu soliton, as well as to certain solitons of another relativistic field theory model, the massive Thirring.

show abstract

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

Cited by 4 publications

References 5 publications

Instability of the finite‐difference split‐step method applied to the generalized nonlinear Schrödinger equation. III. external potential and oscillating pulse solutions

Instability of the finite‐difference split‐step method applied to the generalized nonlinear Schrödinger equation. III. external potential and oscillating pulse solutions

Stability analysis of the numerical Method of characteristics applied to a class of energy-preserving hyperbolic systems. Part II: Nonreflecting boundary conditions

Study of instability of the Fourier split-step method for the massive Gross–Neveu model

Contact Info

Product

Resources

About