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Bidirectional solitons and interaction solutions for a new integrable fifth-order nonlinear equation with temporal and spatial dispersion

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Abstract

A new nonlinear integrable fifth-order equation with temporal and spatial dispersion is investigated, which can be used to describe shallow water waves moving in both directions. By performing the singularity manifold analysis, we demonstrate that this generalized model is integrable in the sense of Painlevé for one set of parametric choices. The simplified Hirota method is employed to construct the one-, two-, three-soliton solutions with non-typical phase shifts. Subsequently, an extended projective Riccati expansion method is presented and abundant travelling wave solutions are constructed uniformly. Furthermore, several new interaction solutions between periodic waves and kinky waves are also derived via a direct method. The rich interactions including overtaking collision, head-on collision and periodic-soliton collision are analyzed by some graphs.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11871328). The authors would like to sincerely and deeply thank the editor and the anonymous referees for their helpful comments and concrete constructive suggestions, which led to an improved version of this paper. The first author would like to thank professor Y.P. Liu for her useful and constructive discussions.

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  1. Department of Information Management, School of Management, Shanghai University, Shanghai, 200444, China

    Gui-Qiong Xu

  2. Department of Mathematics, Saint Xavier University, Chicago, IL, 60655, USA

    Abdul-Majid Wazwaz

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Correspondence to Gui-Qiong Xu.

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Appendices

Appendix A

$$\begin{aligned}&a_1 a_3^{2}\,(4\,c_{{1}}k_{{1}} k_2^{3}-4\,c_{{1}}k_1^{3}k_{{2}}-c_{{2}}k_1^{4}+6\,c_{{2}}k_1^{2}k_2^{2 }-c_{{2}}k_2^{4}\\&\quad +\,3c_1^{2}c_{{2}}-2c_{{1}}k_{{1}}k_{{2}}-c_2^{3}-c_{{2}}k_1^{2}+c_{{2}}k_2^{2})\,=\,0,\\&a_2 a_3^{2}\,(4\,c_{{1}}k_{{1}} k_2^{3}-4\,c_{{1}}k_1^{3}k_{{2}}-c_{{2}}k_1^{4}+6\,c_{{2}}k_1^{2}k_2^{2 }-c_{{2}}k_2^{4}\\&\quad +\,3c_1^{2}c_{{2}}-2c_{{1}}k_{{1}}k_{{2}}-c_2^{3}-c_{{2}}k_1^{2}+c_{{2}}k_2^{2})\,=\,0,\\&a_1 a_3^{2}\,( 6\,c_{{1}}k_1^{2}k_2^{2}-c_{{1}}k_1^{4}-c_{{1}}k_2^{4}+4\,c_{{2}}k_1^{3}k_{{2}}-4\,c_{{2 }}k_{{1}}k_2^{3}\\&\quad +\,c_1^{3}-3\,c_{{1}}c_2^{2}-c_{{1}}k_1^{2}+c_{{1}}k_2^{2}+2\,c_{{2}}k_{{1}}k_{{2}})\,=\,0,\\&a_2 a_3^{2}\,( 6\,c_{{1}}k_1^{2}k_2^{2}-c_{{1}}k_1^{4}-c_{{1}}k_2^{4}+4\,c_{{2}}k_1^{3}k_{{2}}-4\,c_{{2 }}k_{{1}}k_2^{3}\\&\quad +\,c_1^{3}-3\,c_{{1}}c_2^{2}-c_{{1}}k_1^{2}+c_{{1}}k_2^{2}+2\,c_{{2}}k_{{1}}k_{{2}})\,=\,0,\\&a_1^2 a_{{3}}\,(6\,c_{{1}}k_1^{2}k_2^{2} -c_{{1}}k_1^{4}-c_{{1}}k_2^{4}+4\,c_{{2}}k_1^{3}k_{{2}}-4\,c_{{2}}k_{{ 1}}k_2^{3}\\&\quad +\,c_1^{3}-3c_{{1}}c_2^{2}-c_{{1}}k_1^{2}+c_{{1}}k_2^{2}+2c_{{2}}k_{{1}}k_{{2}})\,=\,0,\\&a_2^2 a_{{3}}\,(6\,c_{{1}}k_1^{2}k_2^{2} -c_{{1}}k_1^{4}-c_{{1}}k_2^{4}+4\,c_{{2}}k_1^{3}k_{{2}}-4\,c_{{2}}k_{{ 1}}k_2^{3}\\&\quad +\,c_1^{3}-3c_{{1}}c_2^{2}-c_{{1}}k_1^{2}+c_{{1}}k_2^{2}+2c_{{2}}k_{{1}}k_{{2}})\,=\,0,\\&a_1^2 a_{{3}}(4\,c_{{1}}k_{{1}}k_2^{3}-4\,c_{{1}}k_1^{3}k_{{2}} -c_{{2}}k_1^{4}+6\,c_{{2}}k_1^{2}k_2^{2}-c_{{2}}k_2^{4}\\&\quad +\,3c_1^{2}c_{{2}}-2c_{{1}}k_{{1}}k_{{2}}-c_2^{3} -c_{{2}}k_1^{2}+c_{{2}}k_2^{2})\,=\,0,\\ \\&a_2^2 a_{{3}}(4\,c_{{1}}k_{{1}}k_2^{3}-4\,c_{{1}}k_1^{3}k_{{2}} -c_{{2}}k_1^{4}+6\,c_{{2}}k_1^{2}k_2^{2}-c_{{2}}k_2^{4}\\&\quad +\,3\,c_1^{2}c_{{2}}-2c_{{1}}k_{{1}}k_{{2}}-c_2^{3} -c_{{2}}k_1^{2}+c_{{2}}k_2^{2})\,=\,0,\\&a_3\,(4\,a_1a_2c_{{1}}k_{{1}}k_2^{3}-20\,a_{{1}}a_{{2}}c_{{1}}k_1^{3}k_{{2}} -7\,a_{{1}}a_{{2}}c_{{2}}k_1^{4}\\&\quad +\,2\,a_{{1}}a_{{2}}c_{{2}}k_1^{2}k_2^{2}+a_{{1}}a_{{2}}c_{{2}}k_2^{4} -4\,a_3^{2}c_{{2}}k_2^{4}-a_3^{2}c_2^{3}\\&\quad +\,9\,a_{{1}}a_{{2}}c_1^{2}c_{{2}} -6\,a_{{1}}a_{{2}}c_{{1}}k_{{1}}k_{{2}}+a_{{1}}a_{{2}}c_2^{3}\\&\quad -\,3a_{{1}}a_{{2}}c_{{2}}k_1^{2}-a_{{1}}a_{{2}}c_{{2}}k_2^{2}+a_3^{2}c_{{ 2}}k_2^{2})=0,\\&a_1\,(2a_3^{2}c_{{1}}k_1^{2}k_2^{2}\,-\,16a_{{1}}a_{{2}}c_{{1}}k_1^{4} \,-\,2a_3^2c_1k_2^4\\&\quad \,+\,2a_3^2c_2k_1^3k_2\,-\,6a_3^2c_2k_1k_2^3\,+\,4a_1a_2c_1^3\\&\quad \,-\,4a_1a_2c_1k_1^2\,-\,3a_3^2c_1c_2^2\,+\,a_3^2c_1k_2^2\\&\quad +\,2a_3^2c_2k_1k_2)\,=\,0,\\&a_2\,(2a_3^{2}c_{{1}}k_1^{2}k_2^{2}\,-\,16a_{{1}}a_{{2}}c_{{1}}k_1^{4} \,-\,2a_3^2c_1k_2^4\\&\quad \,+\,2a_3^2c_2k_1^3k_2\,-\,6a_3^2c_2k_1k_2^3\,+\,4a_1a_2c_1^3\\&\quad \,-\,4a_1a_2c_1k_1^2\,-\,3a_3^2c_1c_2^2\,+\,a_3^2c_1k_2^2\\&\quad \,+\,2a_3^2c_2k_1k_2)\,=\,0.\\ \end{aligned}$$

Appendix B

In (56), the expression of \(\Lambda _1\) is listed as follows:

$$\begin{aligned} \Lambda _1= & {} [4c_2^2a_3^{2}(81k_1^{16}-900k_2^{2}k_1^{14}+4480k_2^{4}k_1^{12}\\&-13492k_2^{6}k_1^{10}+32730k_2^{8}k_1^{8}+24884k_2^{10}k_1^{6}\\&-8k_2^{6}k_1^{4}+2052k_2^{14}k_1^{2}-243k_2^{16}+324k_1^{14}\\&-2943k_2^{2}k_1^{12}+10922\,k_2^{4}k_1^{10}-18969k_2^{6}k_1^{8}\\&+81k_1^{8}+10399k_2^{10}k_1^{4}-4590k_2^{12}k_1^{2}+729k_2^{14}\\&+486k_1^{12}-3168k_2^{2}k_1^{10}+6315k_2^{4}k_1^{8}+54k_2^{8}k_1^{2}\\&+528k_2^{8}k_1^{4}+2484k_2^{10}k_1^{2}-729k_2^{12}+324k_1^{10}\\&-1107k_2^{2}k_1^{8}-226k_2^{4}k_1^{6}+1140k_2^{6}k_1^{6}+243k_2^{10}\\&-8120k_2^{12}k_1^{4}-1056k_2^{8}k_1^{6}+18k_1^{6}k_2^{2}+k_1^{4}k_2^{4}) \\&-4a_3^{2}k_1^{2}k_2^{2}(81k_1^{12}-738k_2^{2}k_1^{10}+2907k_2^{4}k_1^{8}\\&-8044k_2^{6}k_1^{6}-5881\,k_2^{8}k_1^{4}+1550\,k_2^{10}k_1^{2}\\&-243k_2^{12}+243k_1^{10}-1638k_2^{2}k_1^{8}+3730k_2^{4}k_1^{6}\\&+264\,k_2^{6}k_1^{4}-1789\,k_2^{8}k_1^{2}+486\,k_2^{10}+243\,k_1^{8}\\&-882\,k_1^{6}k_2^{2}-176\,k_1^{4}k_2^{4}-62\,k_1^{2}k_2^{6}-243\,k_2^{8}\\&+81k_1^{6}+18k_1^{4}k_2^{2}+k_1^{2}k_2^{4})(2k_1^{2}-2k_2^{2}+1)^{2}]\\&\quad /[16\,c_2^2\,a_{{2}}k_1^{2}( 2\,k_1^{2}-2\,k_2^{2}+1)( 81\,k_1^{12}+243k_1^{8}\\&+2907k_2^{4}k_1^{8}-8044k_2^{6}k_1^{6}-5881k_2^{8}k_1^{4}-243k_2^{12}\\&+1550\,k_2^{10}k_1^{2}+243\,k_1^{10}-1638\,k_2^{2}k_1^{8}+18k_1^{4}k_2^{2}\\ \\&+264k_2^{6}k_1^{4}-1789k_2^{8}k_1^{2}+486k_2^{10}-738\,k_2^{2}k_1^{10}\\&-882k_1^{6}k_2^{2}-176k_1^{4}k_2^{4}-62k_1^{2}k_2^{6}+3730k_2^{4}k_1^{6}\\&+81k_1^{6}-243 k_2^{8}+k_1^{2}k_2^{4})-16a_{{2}}k_1^{4}k_2^{2}(81k_1^{8}\\&-572k_1^{6}k_2^{2}+1958k_1^{4}k_2^{4}+1348k_1^{2}k_2^{6}-239k_2^{8}\\&+162k_1^{6}-654k_1^{4}k_2^{2}-66k_1^{2}k_2^{4}+238k_2^{6}\\&+81k_1^{4}+18k_1^{2}k_2^{2}+k_2^{4})(2k_1^{2}-2k_2^{2}+1)^{3}].\\ \end{aligned}$$

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Xu, GQ., Wazwaz, AM. Bidirectional solitons and interaction solutions for a new integrable fifth-order nonlinear equation with temporal and spatial dispersion. Nonlinear Dyn 101, 581–595 (2020). https://doi.org/10.1007/s11071-020-05740-1

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