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Localized waves and mixed interaction solutions with dynamical analysis to the Gross–Pitaevskii equation in the Bose–Einstein condensate

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Abstract

We report a kind of breather, rogue wave, and mixed interaction structures on a variational background height in the Gross–Pitaevskii equation in the Bose–Einstein condensate by the generalized Darboux transformation method, and the effects of related parameters on rogue wave structures are discussed. Numerical simulation can discuss the dynamics and stability of these solutions. We numerically confirm that these are correct and can be reproduced from a deterministic initial profile. Results show that rogue waves and mixed interaction solutions can evolve with a small amplitude perturbation under the initial profile conditions, but breathers cannot. Therefore, these can be used to anticipate the feasibility of their experimental observation.

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Funding

This work has been supported by the National Natural Science Foundation of China (12075034, 11875008); Fundamental Research Funds for the Central Universities (2019XD-A09-3) ; the Open Research Fund of State Key Laboratory of Pulsed Power Laser Technology (No. SKL2018KF04).

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Authors and Affiliations

  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing, 100876, People’s Republic of China

    Haotian Wang & Wenjun Liu

  2. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, People’s Republic of China

    Qin Zhou

  3. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80257, Jeddah, 21589, Saudi Arabia

    Qin Zhou

  4. Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL, 35762-4900, USA

    Anjan Biswas

  5. Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Anjan Biswas

  6. Department of Applied Mathematics, National Research Nuclear University, 31 Kashirskoe Hwy, Moscow, 115409, Russian Federation

    Anjan Biswas

  7. Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa, 0204, Pretoria, South Africa

    Anjan Biswas

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  1. Haotian Wang
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Correspondence to Wenjun Liu.

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Appendix

Appendix

Second-order RW solution \(\psi \) in Fig. 1(a3)(b3) is expressed as \(q=-\frac{49152\ F}{G}\mathrm{e}^{2t-\frac{\mathrm{i}x^2}{2}+2\mathrm{i}\mathrm{e}^{4t}}\), where

$$\begin{aligned} \begin{aligned} F=\,&\Big [ {\frac{19325}{24}}+{\frac{3127\,\mathrm{i}}{16}}-\frac{1}{6}\,{x}^{6}+ \Big ({\frac{25}{4}}+\frac{\mathrm{i}}{2} \Big ) {x}^{4}\\&-\Big ( {\frac{1863}{16}}+{\frac{75\,\mathrm{i}}{4}} \Big ) {x}^{2} \Big ] {\mathrm{e}^{12\,t}}+ \Big [ -{\frac{1905653}{512}}\\&-{\frac{9775\,\mathrm{i}}{8}}\\&-\Big ( {\frac{627}{32}}+{\frac{25\,\mathrm{i}}{8}} \Big ) {x}^{4}+\Big ( {\frac{1925}{4}}+{\frac{1875\,\mathrm{i}}{16}} \Big ) {x}^{2} \Big ] {\mathrm{e}^{8\,t}}\\&-\frac{1}{2}\,x \Big [ -{\frac{315}{4}}-{\frac{75\,\mathrm{i}}{4}}-\frac{1}{2}\,{x}^{4}+\Big ( {\frac{25}{2}}+\mathrm{i}\Big ) {x}^{2} \Big ] {\mathrm{e}^{10\,t}}\\&+ \Big [ -{\frac{3115}{32}}-{\frac{125\,\mathrm{i}}{8}}-\frac{1}{2}\,{x}^{4}+\Big ( {\frac{25}{2}}+\mathrm{i}\Big ) {x}^{2} \Big ] {\mathrm{e}^{16\,t}}\\&+ \Big [ {\frac{9297875}{1024}}+{\frac{1950701\,\mathrm{i}}{512}}- \Big ( {\frac{385501}{512}}\\&+{\frac{3925\,\mathrm{i}}{16}}\Big ) {x}^{2} \Big ] {\mathrm{e}^{4\,t}}\\&+{\frac{25\,x}{8} \Big [ -{\frac{13}{8}}-{\frac{318\,\mathrm{i}}{25}}+\Big ( {\frac{206}{25}}+\mathrm{i} \Big ) {x}^{2} \Big ] {\mathrm{e}^{6\,t}}}\\&-\frac{1}{2}\,x \Big ( -{x}^{2}+{\frac{25}{2}}+\mathrm{i} \Big ) {\mathrm{e}^{14\,t}}+ \Big ( {\frac{25}{4}}+\frac{\mathrm{i}}{2}\\&-\frac{1}{2}\,{x}^{2}\Big ) {\mathrm{e}^{20\,t}}\\&-\Big ( {\frac{365837}{1024}}-{\frac{625\,\mathrm{i}}{128}} \Big ) x{\mathrm{e}^{2\,t}}-\frac{1}{6}\,{\mathrm{e}^{24\,t}}\\&-{\frac{224635711}{24576}}-{\frac{9674075\,\mathrm{i}}{2048}}+\frac{1}{4}\,{\mathrm{e}^{18\,t}}x, \end{aligned} \end{aligned}$$
$$\begin{aligned} G=\,&\big ( 4096\,{x}^{6}-153600\,{x}^{4}+2880000\,{x}^{2}\\&-20172800\big ) {\mathrm{e}^{12\,t}}\\&+ \big ( 484608\,{x}^{4}-12057600\,{x}^{2}+95075952 \big ) {\mathrm{e}^{8\,t}}\\&+ \big ( -6144\,{x}^{5}+153600\,{x}^{3}-976896\,x \big ) {\mathrm{e}^{10\,t}}\\&+ \big ( 12288\,{x}^{4}-307200\,{x}^{2}+2407680 \big ) {\mathrm{e}^{16\,t}}\\&+ \big ( 19226352\,{x}^{2}\\&-238192200 \big ) {\mathrm{e}^{4\,t}}+\big ( -635904\,{x}^{3}\\&+240000\,x\big ) {\mathrm{e}^{6\,t}}\\&+ \big ( -12288\,{x}^{3}+153600\,x \big ) {\mathrm{e}^{14\,t}}+ \big ( 12288\,{x}^{2}\\&-153600 \big ) {\mathrm{e}^{20\,t}}-6144\,x{\mathrm{e}^{18\,t}}+8534712\,x{\mathrm{e}^{2\,t}}\\&+4096\,{\mathrm{e}^{24\,t}}+248134411. \end{aligned}$$

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Wang, H., Zhou, Q., Biswas, A. et al. Localized waves and mixed interaction solutions with dynamical analysis to the Gross–Pitaevskii equation in the Bose–Einstein condensate. Nonlinear Dyn 106, 841–854 (2021). https://doi.org/10.1007/s11071-021-06851-z

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