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Existence of dark solitons in discrete nonlinear Schrödinger equations with non-weak couplings

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Abstract

The anti-integrable limit for the discrete nonlinear Schrödinger equation is defined as the limit of vanishing couplings. There are an infinite number of trivial discrete dark solitons in this limit. The existence of discrete dark solitons continued from them has been proved only for sufficiently weak couplings. In the present paper, we focus on the case of non-weak couplings and prove the existence of discrete dark solitons over an explicitly given range of the coupling constant.

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Notes

  1. Dark soliton solutions in the one-dimensional continuous NLS have the form \(\varPsi (x,t)\,{=}\,A\tanh (Ax)\exp (-i\varOmega t)\), where A and \(\varOmega \) are constants and x is the spatial coordinate. Its envelop \(A\tanh (Ax)\) has different signs in the limits \(x\rightarrow \pm \infty \). In analogy with this, different signs of \(\phi _n\) in the limits \(n\rightarrow \pm \infty \) are required for discrete dark solitons.

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  1. Faculty of Engineering, Tottori University, 4-101 Koyama-Minami, Tottori, 680-8552, Japan

    Kazuyuki Yoshimura

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  1. Kazuyuki Yoshimura
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Correspondence to Kazuyuki Yoshimura.

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Yoshimura, K. Existence of dark solitons in discrete nonlinear Schrödinger equations with non-weak couplings. Japan J. Indust. Appl. Math. 36, 893–905 (2019). https://doi.org/10.1007/s13160-019-00371-5

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  • DOI: https://doi.org/10.1007/s13160-019-00371-5

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