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Numerical Computation of Dark Solitons of a Nonlocal Nonlinear Schrödinger Equation

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Abstract

The existence and decay properties of dark solitons for a large class of nonlinear nonlocal Gross–Pitaevskii equations with nonzero boundary conditions in dimension one has been established recently (de Laire and López-Martínez in Commun Partial Differ Equ 47(9):1732–1794, 2022). Mathematically, these solitons correspond to minimizers of the energy at fixed momentum and are orbitally stable. This paper provides a numerical method to compute approximations of such solitons for these types of equations, and provides actual numerical experiments for several types of physically relevant nonlocal potentials. These simulations allow us to obtain a variety of dark solitons, and to comment on their shapes in terms of the parameters of the nonlocal potential. In particular, they suggest that, given the dispersion relation, the speed of sound and the Landau speed are important values to understand the properties of these dark solitons. They also allow us to test the necessity of some sufficient conditions in the theoretical result proving existence of the dark solitons.

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Acknowledgements

The authors acknowledge support from the Labex CEMPI (ANR-11-LABX-0007-01). A. de Laire was also supported by the ANR project ODA (ANR-18-CE40-0020-01). S. López-Martínez was supported by the Madrid Government (Comunidad de Madrid - Spain) under the multiannual Agreement with UAM in the line for the Excellence of the University Research Staff in the context of the V PRICIT (Regional Program of Research and Technological Innovation).

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Authors and Affiliations

  1. Univ. Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul Painlevé, Inria, 59000, Lille, France

    André de Laire & Guillaume Dujardin

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049, Madrid, Spain

    Salvador López-Martínez

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  1. André de Laire
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  2. Guillaume Dujardin
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  3. Salvador López-Martínez
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Correspondence to Guillaume Dujardin.

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Communicated by Anthony Bloch.

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Appendix

Appendix

Proof of Theorem 2

Let \(m=\inf _{\xi \in {{\mathbb {R}}}}({\widehat{{\mathcal {W}}}}(\xi )^-)\), where \(s^-=\min \{s,0\}\), and let \(R=\sqrt{2(\sigma -m)}\). For any \(\tilde{\sigma }\in (0,\sigma )\), we choose \(k=(\tilde{\sigma }-m)/R^2=(\tilde{\sigma }-m)/(2(\sigma -m))\). Clearly, \(k\in (0,1/2)\).

On the one hand, for a.e. \(|\xi |\ge R\), one trivially has

$$\begin{aligned} {\widehat{{\mathcal {W}}}}(\xi )\ge m={\tilde{\sigma }} - kR^2\ge {\tilde{\sigma }}-k\xi ^2. \end{aligned}$$

On the other hand, observe that \(\sigma -\xi ^2/2\ge {\tilde{\sigma }}-k\xi ^2\) if, and only if, \(|\xi |\le R\). Therefore, for a.e. \(|\xi |\le R\), we get

$$\begin{aligned} {\widehat{{\mathcal {W}}}}(\xi )\ge \sigma -\frac{\xi ^2}{2}\ge {\tilde{\sigma }}-k\xi ^2. \end{aligned}$$

We may now apply Theorem 1.1 in de Laire and López-Martínez (2022) and get a solution to \((TW _c)\) for a.e. \(c\in (0,\sqrt{2{\tilde{\sigma }}})\). Taking \({\tilde{\sigma }}\) arbitrarily close to \(\sigma \) we obtain a solution for a.e. \(c\in (0,\sqrt{2\sigma })\), which completes the proof. \(\square \)

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de Laire, A., Dujardin, G. & López-Martínez, S. Numerical Computation of Dark Solitons of a Nonlocal Nonlinear Schrödinger Equation. J Nonlinear Sci 34, 23 (2024). https://doi.org/10.1007/s00332-023-10001-7

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