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Analytical Study on the Generalized Fifth-Order Kaup–Kupershmidt Equation from the Shallow Water Wave

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Abstract

In this paper, the generalized fifth-order Kaup–Kupershmidt (KK) equation from the shallow water wave, have been investigated. With the help of bilinear method and auxiliary function, the multi-soliton solutions of the generalized fifth-order KK equation have been obtained, and those solutions have not been given before. From the analytical view, the interactions of the solitons have been presented. Through the discussion, we have found that the velocity of the soliton is related to the sign of the physical parameters \(b\) and \({{c}_{j}}\). Also, the (bright or dark-like) types of the soliton interaction are related to the physical parameters \(a\). Hence, the velocities and bright-dark types of the soliton interactions could be controlled by adjusting the physical parameters.

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Funding

This work has been supported by the National Natural Science Foundation of China under grant nos. 11426041 and 11605011.

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Authors and Affiliations

  1. Sports Business School, Beijing Sport University, no.48, 100084, Beijing, China

    Pan Wang, Jian-Rong Yang & Li Chen

  2. School of Management, Beijing Sport University, no.48, 100084, Beijing, China

    Sheng-Xin Li

  3. School of Information, Beijing Wuzi University, 101149, Beijing, China

    Feng-Hua Qi

Authors
  1. Pan Wang
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  2. Jian-Rong Yang
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  3. Li Chen
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  4. Sheng-Xin Li
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  5. Feng-Hua Qi
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Corresponding author

Correspondence to Pan Wang.

APPENDIX

APPENDIX

The related parameters for the three-soliton solution are as follows:

$$\begin{gathered} g = {{P}_{1}}{{e}^{{{{\theta }_{1}}}}} + {{P}_{2}}{{e}^{{{{\theta }_{2}}}}} + {{P}_{3}}{{e}^{{{{\theta }_{3}}}}} + {{P}_{7}}{{e}^{{{{\theta }_{1}} + {{\theta }_{2}}}}} + {{P}_{8}}{{e}^{{{{\theta }_{1}} + {{\theta }_{3}}}}} + {{P}_{9}}{{e}^{{{{\theta }_{2}} + {{\theta }_{3}}}}} + {{P}_{{10}}}{{e}^{{{{\theta }_{1}} + 2{{\theta }_{2}}}}} + {{P}_{{11}}}{{e}^{{2{{\theta }_{1}} + {{\theta }_{2}}}}} \\ + \;{{P}_{{12}}}{{e}^{{{{\theta }_{1}} + 2{{\theta }_{3}}}}} + {{P}_{{13}}}{{e}^{{2{{\theta }_{1}} + {{\theta }_{3}}}}} + {{P}_{{14}}}{{e}^{{{{\theta }_{2}} + 2{{\theta }_{3}}}}} + {{P}_{{15}}}{{e}^{{2{{\theta }_{2}} + {{\theta }_{3}}}}} + {{P}_{{16}}}{{e}^{{{{\theta }_{1}} + {{\theta }_{2}} + {{\theta }_{3}}}}} + {{P}_{{20}}}{{e}^{{{{\theta }_{1}} + {{\theta }_{2}} + 2{{\theta }_{3}}}}} \\ + \;{{P}_{{21}}}{{e}^{{{{\theta }_{1}} + 2{{\theta }_{2}} + {{\theta }_{3}}}}} + {{P}_{{22}}}{{e}^{{2{{\theta }_{1}} + {{\theta }_{2}} + {{\theta }_{3}}}}} + {{P}_{{23}}}{{e}^{{2{{\theta }_{1}} + 2{{\theta }_{2}} + {{\theta }_{3}}}}} + {{P}_{{24}}}{{e}^{{2{{\theta }_{1}} + {{\theta }_{2}} + 2{{\theta }_{3}}}}} + {{P}_{{25}}}{{e}^{{{{\theta }_{1}} + 2{{\theta }_{2}} + 2{{\theta }_{3}}}}}, \\ \end{gathered} $$
$${{P}_{7}} = \tfrac{{15a{{{({{c}_{1}} - {{c}_{2}})}}^{2}}(2c_{1}^{4} + 3c_{1}^{2}c_{2}^{2} + 2c_{2}^{4})}}{{(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2})}},\quad {{P}_{8}} = \tfrac{{15a{{{({{c}_{1}} - {{c}_{3}})}}^{2}}(2c_{1}^{4} + 3c_{1}^{2}c_{3}^{2} + 2c_{3}^{4})}}{{(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2})}},$$
$${{P}_{9}} = \tfrac{{15a{{{({{c}_{2}} - {{c}_{3}})}}^{2}}(2c_{2}^{4} + 3c_{2}^{2}c_{3}^{2} + 2c_{3}^{4})}}{{(c_{2}^{2} + {{c}_{2}}{{c}_{3}} + c_{3}^{2})}},\quad {{P}_{{10}}} = \tfrac{{15ac_{1}^{4}{{{({{c}_{1}} - {{c}_{2}})}}^{2}}(c_{1}^{2} - {{c}_{1}}{{c}_{2}} + c_{2}^{2})}}{{8{{{({{c}_{1}} + {{c}_{2}})}}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2})}},$$
$${{P}_{{11}}} = \tfrac{{15ac_{2}^{4}{{{({{c}_{1}} - {{c}_{2}})}}^{2}}(c_{1}^{2} - {{c}_{1}}{{c}_{2}} + c_{2}^{2})}}{{8{{{({{c}_{1}} + {{c}_{2}})}}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2})}},\quad {{P}_{{12}}} = \tfrac{{15ac_{1}^{4}{{{({{c}_{1}} - {{c}_{3}})}}^{2}}(c_{1}^{2} - {{c}_{1}}{{c}_{3}} + c_{3}^{2})}}{{8{{{({{c}_{1}} + {{c}_{3}})}}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2})}},$$
$${{P}_{{13}}} = \tfrac{{15ac_{3}^{4}{{{({{c}_{1}} - {{c}_{3}})}}^{2}}(c_{1}^{2} - {{c}_{1}}{{c}_{3}} + c_{3}^{2})}}{{8{{{({{c}_{1}} + {{c}_{3}})}}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2})}},\quad {{P}_{{14}}} = \tfrac{{15ac_{2}^{4}{{{({{c}_{2}} - {{c}_{3}})}}^{2}}(c_{2}^{2} - {{c}_{2}}{{c}_{3}} + c_{3}^{2})}}{{8{{{({{c}_{2}} + {{c}_{3}})}}^{2}}(c_{2}^{2} + {{c}_{2}}{{c}_{3}} + c_{3}^{2})}},$$
$$\begin{gathered} {{P}_{{16}}} = \{ 15a[2c_{2}^{4}c_{3}^{4}{{(c_{2}^{2} - c_{3}^{2})}^{2}}(2c_{2}^{4} + 3c_{2}^{2}c_{3}^{2} + 2c_{3}^{4}) + c_{1}^{{12}}(4c_{2}^{4} - 2c_{2}^{2}c_{3}^{2} + 4c_{3}^{4}) - c_{1}^{8}{{(c_{2}^{2} - c_{3}^{2})}^{2}}(4c_{2}^{4} + 5c_{2}^{2}c_{3}^{2} + 4c_{3}^{4}) \\ - \;c_{1}^{8}{{(c_{2}^{2} - c_{3}^{2})}^{2}}(4c_{2}^{4} + 5c_{2}^{2}c_{3}^{2} + 4c_{3}^{4}) - c_{1}^{{10}}(2c_{2}^{6} + c_{2}^{4}c_{3}^{2} + c_{2}^{2}c_{3}^{4} + 2c_{3}^{6}) - c_{1}^{2}c_{2}^{2}c_{3}^{2}{{(c_{2}^{2} - c_{3}^{2})}^{2}}(2c_{2}^{6} + 5c_{2}^{4}c_{3}^{2} + 5c_{2}^{2}c_{3}^{4} + 2c_{3}^{6}) \\ \end{gathered} $$
$$\begin{gathered} - \;c_{1}^{6}(2c_{2}^{{10}} - 3c_{2}^{8}c_{3}^{2} + 4c_{2}^{6}c_{3}^{4} + 4c_{2}^{4}c_{3}^{6} - 3c_{2}^{2}c_{3}^{8} + 2c_{3}^{{10}}) + c_{1}^{4}(4c_{2}^{{12}} - c_{2}^{{10}}c_{3}^{2} + 2c_{2}^{8}c_{3}^{4} - 4c_{2}^{6}c_{3}^{6} + 2c_{2}^{4}c_{3}^{8} - c_{2}^{2}c_{3}^{{10}} + 4c_{3}^{{12}})]\} \\ \times \;{{[2{{({{c}_{1}} + {{c}_{2}})}^{2}}{{({{c}_{1}} + {{c}_{3}})}^{2}}{{({{c}_{2}} + {{c}_{3}})}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2})(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2})(c_{2}^{2} + {{c}_{2}}{{c}_{3}} + c_{3}^{2})]}^{{ - 1}}}, \\ \end{gathered} $$
$$\begin{gathered} {{P}_{{20}}} = 15a{{({{c}_{1}} - {{c}_{2}})}^{2}}{{({{c}_{1}} - {{c}_{3}})}^{2}}{{({{c}_{2}} - {{c}_{3}})}^{2}}(2c_{1}^{4} + 3c_{1}^{2}c_{2}^{2} + 2c_{2}^{4})(c_{1}^{2} - {{c}_{1}}{{c}_{3}} + c_{3}^{2})(c_{2}^{2} - {{c}_{2}}{{c}_{3}} + c_{3}^{2}) \\ \times \;{{[16{{({{c}_{1}} + {{c}_{3}})}^{2}}{{({{c}_{2}} + {{c}_{3}})}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2})(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2})(c_{2}^{2} + {{c}_{2}}{{c}_{3}} + c_{3}^{2})]}^{{ - 1}}}, \\ \end{gathered} $$
$$\begin{gathered} {{P}_{{21}}} = 15a{{({{c}_{1}} - {{c}_{2}})}^{2}}{{({{c}_{1}} - {{c}_{3}})}^{2}}{{({{c}_{2}} - {{c}_{3}})}^{2}}(2c_{1}^{4} + 3c_{1}^{2}c_{3}^{2} + 2c_{3}^{4})(c_{1}^{2} - {{c}_{1}}{{c}_{2}} + c_{2}^{2})(c_{2}^{2} - {{c}_{2}}{{c}_{3}} + c_{3}^{2}) \\ \times \;{{[16{{({{c}_{1}} + {{c}_{2}})}^{2}}{{({{c}_{2}} + {{c}_{3}})}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2})(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2})(c_{2}^{2} + {{c}_{2}}{{c}_{3}} + c_{3}^{2})]}^{{ - 1}}}, \\ \end{gathered} $$
$$\begin{gathered} {{P}_{{22}}} = 15a{{({{c}_{1}} - {{c}_{2}})}^{2}}{{({{c}_{1}} - {{c}_{3}})}^{2}}{{({{c}_{2}} - {{c}_{3}})}^{2}}(2c_{2}^{4} + 3c_{2}^{2}c_{3}^{2} + 2c_{3}^{4})(c_{1}^{2} - {{c}_{1}}{{c}_{2}} + c_{2}^{2})(c_{1}^{2} - {{c}_{1}}{{c}_{3}} + c_{3}^{2}) \\ \times \;{{[16{{({{c}_{1}} + {{c}_{2}})}^{2}}{{({{c}_{1}} + {{c}_{3}})}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2})(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2}) \times (c_{2}^{2} + {{c}_{2}}{{c}_{3}} + c_{3}^{2})]}^{{ - 1}}}, \\ \end{gathered} $$
$$\begin{gathered} {{P}_{{23}}} = 15ac_{3}^{4}{{({{c}_{1}} - {{c}_{2}})}^{4}}{{({{c}_{1}} - {{c}_{3}})}^{2}}{{({{c}_{2}} - {{c}_{3}})}^{2}}{{(c_{1}^{2} - {{c}_{1}}{{c}_{2}} + c_{2}^{2})}^{2}}(c_{1}^{2} - {{c}_{1}}{{c}_{3}} + c_{3}^{2})(c_{2}^{2} - {{c}_{2}}{{c}_{3}} + c_{3}^{2}) \\ \times \;{{[128{{({{c}_{1}} + {{c}_{2}})}^{4}}{{({{c}_{1}} + {{c}_{3}})}^{2}}{{({{c}_{2}} + {{c}_{3}})}^{2}}{{(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2})}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2})(c_{2}^{2} + {{c}_{2}}{{c}_{3}} + c_{3}^{2})]}^{{ - 1}}}, \\ \end{gathered} $$
$$\begin{gathered} {{P}_{{24}}} = 15ac_{2}^{4}{{({{c}_{1}} - {{c}_{3}})}^{4}}{{({{c}_{1}} - {{c}_{2}})}^{2}}{{({{c}_{2}} - {{c}_{3}})}^{2}}(c_{1}^{2} - {{c}_{1}}{{c}_{2}} + c_{2}^{2}){{(c_{1}^{2} - {{c}_{1}}{{c}_{3}} + c_{3}^{2})}^{2}}(c_{2}^{2} - {{c}_{2}}{{c}_{3}} + c_{3}^{2}) \\ \times \;{{[128{{({{c}_{1}} + {{c}_{3}})}^{4}}{{({{c}_{1}} + {{c}_{2}})}^{2}}{{({{c}_{2}} + {{c}_{3}})}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2}){{(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2})}^{2}}(c_{2}^{2} + {{c}_{2}}{{c}_{3}} + c_{3}^{2})]}^{{ - 1}}}, \\ \end{gathered} $$
$$\begin{gathered} {{P}_{{25}}} = 15ac_{1}^{4}{{({{c}_{2}} - {{c}_{3}})}^{4}}{{({{c}_{1}} - {{c}_{2}})}^{2}}{{({{c}_{1}} - {{c}_{3}})}^{2}}(c_{1}^{2} - {{c}_{1}}{{c}_{2}} + c_{2}^{2})(c_{1}^{2} - {{c}_{1}}{{c}_{3}} + c_{3}^{2}){{(c_{2}^{2} - {{c}_{2}}{{c}_{3}} + c_{3}^{2})}^{2}} \\ \times \;{{[128{{({{c}_{2}} + {{c}_{3}})}^{4}}{{({{c}_{1}} + {{c}_{2}})}^{2}}{{({{c}_{1}} + {{c}_{3}})}^{2}}(c_{1}^{2} + {{c}_{1}}{{c}_{2}} + c_{2}^{2})(c_{1}^{2} + {{c}_{1}}{{c}_{3}} + c_{3}^{2}){{(c_{2}^{2} + {{c}_{2}}{{c}_{3}} + c_{3}^{2})}^{2}}]}^{{ - 1}}}. \\ \end{gathered} $$

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Wang, P., Yang, JR., Chen, L. et al. Analytical Study on the Generalized Fifth-Order Kaup–Kupershmidt Equation from the Shallow Water Wave. Comput. Math. and Math. Phys. 60, 1480–1487 (2020). https://doi.org/10.1134/S0965542520090158

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