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Traveling wave solutions for explicit-time nonlinear photorefractive dynamics equation

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Abstract

We solve the traveling wave solution for the explicit-time nonlinear photorefractive dynamics equation. Nonlinearity comes from the support of linear and quadratic electro-optic effects. We investigate two cases, i.e., the wave assumes to be in low amplitude and without such an assumption. In this step, we apply the direct solution method by setting the ansatz of the wave solution from the start. The first case relies on the Taylor series expansion to integrate the equations of these results and get an exact solution for the traveling wave. In the second case, we thoroughly evaluate the original dynamic equation. It reduces to a first-order differential equation that provides the initial conditions for a numerical evaluation. The exact solution shows the propagation wave traveling in the positive and negative directions in the diffraction axis direction. An angle between the traveling wave and the center of the propagation plane decreases as the value of the displacement constant in the solution increases. In addition, we also study the power of the traveling wave, and the numerical solution gives a kink-like traveling wave.

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The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank Hanifah Azzaura Musyayy-adah for the helpful discussion.

Funding

This work supported by the Lembaga Penelitian dan Pengabdian Masyarakat (LPPM), Universitas Andalas under a fundamental research grant (No.Ref.T/66/UN.16.17. PT.01.03/IS-RD/2021).

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Authors and Affiliations

  1. Theoretical Physics Laboratory, Department of Physics, Faculty of Mathematics and Natural Science, Universitas Andalas, Limau Manis, Padang, 25163, Indonesia

    Zulfi Abdullah & Ahmad Ripai

  2. Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Andalas, Limau Manis, Padang, 25163, Indonesia

    Mahdhivan Syafwan

  3. Department of Physics, Faculty of Mathematics and Natural Science, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, Indonesia

    Wahyu Hidayat

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  1. Zulfi Abdullah
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All authors contributed to the study conception and design. Material preparation, data collection and analysis performed by ZA, AR, MS, and WH. The first draft of the manuscript written by AR and all authors commented on previous versions of the manuscript. The final manuscript has been read and approved by all authors.

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Correspondence to Zulfi Abdullah.

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Appendix

Appendix

To prove the solution of \(q(s,\xi )\) in Eq. (25) as the traveling wave exact solution of the explicit-time nonlinear photoreactive dynamics equation in the low amplitude case, substitute Eq. (25) into Eq. (6). Thus, each of the terms can be written in the form

$$\begin{aligned}&T_{1}=ce^{ic(s-c\xi )}\sqrt{-\frac{-0.5c^{2}+\beta _{1}+\beta _{2}}{(-1+e^{-\tau })(\beta _{1}+2\beta _{2})}}~\nonumber \\&\quad \text {sech}\left[ \sqrt{2}~\sqrt{-0.5c^{2}+\beta _{1}+\beta _{2}}(s-c\xi ) \right] \nonumber \\&\quad \left( \sqrt{2}~c+2i\sqrt{-0.5c^{2}+\beta _{1}+\beta _{2}}\right. ~\nonumber \\&\quad \left. \text {tanh}\left[ \sqrt{2}~\sqrt{-0.5c^{2}+\beta _{1}+\beta _{2}}(s-c\xi ) \right] \right) , \end{aligned}$$
(S1)
$$\begin{aligned}&T_{2}=\frac{1}{2}~ce^{ic(s-c\xi )}\sqrt{-\frac{-0.5c^{2}+\beta _{1}+\beta _{2}}{(-1+e^{-\tau })(\beta _{1}+2\beta _{2})}}\nonumber \\&\quad ~\text {sech}\left[ \sqrt{2}~\sqrt{-0.5c^{2}+\beta _{1}+\beta _{2}}(s-c\xi ) \right] \nonumber \\&\quad \Biggl ( -1.41421 c^{2}+\left( 1.41421 c^{2} - 2.82843 \beta _{1} - 2.82843 \beta _{2} \right) ~\nonumber \\&\quad \text {sech}\left[ \sqrt{2}~\sqrt{-0.5c^{2}+\beta _{1}+\beta _{2}}(s-c\xi ) \right] ^{2}\nonumber \\&\quad -\left( 0+4i \right) c\sqrt{-0.5c^{2}+\beta _{1}+\beta _{2}}~\nonumber \\&\quad \text {tanh}\left[ \sqrt{2}~\sqrt{-0.5c^{2}+\beta _{1}+\beta _{2}} (s-c\xi )\right] \nonumber \\&\quad +\left( -1.41421 c^{2} + 2.82843 \beta _{1} + 2.82843 \beta _{2} \right) ~\nonumber \\&\quad \text {tanh}~\left[ \sqrt{2}~\sqrt{-0.5 c^{2}+\beta _{1}+\beta _{2}} \left( s-c\xi \right) \right] ^{2} \Biggl ), \end{aligned}$$
(S2)
$$\begin{aligned}&T_{3}=~-\sqrt{2}~e^{ic(s-c\xi )}\left( \beta _{1}+\beta _{2} \right) \nonumber \\&\quad \sqrt{-\frac{-0.5c^{2}+\beta _{1}+\beta _{2}}{(-1+e^{-\tau })(\beta _{1}+2\beta _{2})}}~\nonumber \\&\quad \text {sech}\left[ \sqrt{2}~\sqrt{-0.5c^{2}+\beta _{1}+\beta _{2}}(s-c\xi ) \right] , \end{aligned}$$
(S3)
$$\begin{aligned}&T_{4}=~-1.41421~e^{ic(s-c\xi )}\left( 1c^{2}-2\beta _{1}-2\beta _{2}\right) \nonumber \\&\quad \sqrt{-\frac{-0.5c^{2}+\beta _{1}+\beta _{2}}{(-1+e^{-\tau })(\beta _{1}+2\beta _{2})}}~\nonumber \\&\quad \text {sech}\left[ \sqrt{2}~\sqrt{-0.5c^{2}+\beta _{1}+\beta _{2}}(s-c\xi ) \right] ^{3}, \end{aligned}$$
(S4)

where \(T_{1}=iq_{\xi }\), \(T_{2}=1/2~q_{ss}\), \(T_{3}=-(\beta _{1}+\beta _{2})q\), and \(T_{4}=-\left( \beta _{1}+2\beta _{2} \right) \left( \exp \left[ -\tau \right] -1\right) \left| q \right| ^{2}q\). If we add up all the terms, then \(T_{1}+T_{2}+T_{3}+T_{4}=0\) (valid as an exact solution).

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Abdullah, Z., Ripai, A., Syafwan, M. et al. Traveling wave solutions for explicit-time nonlinear photorefractive dynamics equation. Nonlinear Dyn 111, 16515–16526 (2023). https://doi.org/10.1007/s11071-023-08610-8

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