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Multiple rogue wave, breather wave and interaction solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation

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Abstract

Based on a direct variable transformation, we obtain multiple rogue wave solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation, including first-order, two-order and three-order rogue wave solutions. Their dynamic behaviors are shown by some 3D plots. Compared with Zha’s symbolic computation approach, we do not need to resort to Hirota bilinear form, and it can be used to deal with variable-coefficient integrable equations. Interaction solution between rogue wave and periodic wave is obtained by using the Hirota bilinear form. Abundant breather wave solutions are presented by a direct test function.

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Authors and Affiliations

  1. College of Computer, Jiangxi University of Traditional Chinese Medicine, Jiangxi, 330004, China

    Jian-Guo Liu

  2. Institute of artificial intelligence, Nanchang Institute of Science and Technology, Jiangxi, 330108, China

    Wen-Hui Zhu

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  1. Jian-Guo Liu
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Appendices

Appendix A

$$\begin{aligned} \vartheta _{32}= & {} \frac{11 \vartheta _{26}^5 \vartheta _{59}}{5832},\quad \vartheta _{31}=\frac{19 \vartheta _{26}^7 \vartheta _{59}^2}{3149280},\nonumber \\ \vartheta _{37}= & {} \frac{73 \vartheta _{26}^6 \vartheta _{59}}{244944},\quad \vartheta _{36}=\frac{107 \vartheta _{26}^8 \vartheta _{59}^2}{52907904},\nonumber \\ \vartheta _{55}= & {} -\frac{180}{\vartheta _{26}^2},\quad \vartheta _{38}=\frac{5 \vartheta _{26}^4}{432},\nonumber \\ \vartheta _{35}= & {} \frac{\vartheta _{26}^{10} \vartheta _{59}^3}{317447424}, \quad \vartheta _{43}=\frac{19 \vartheta _{26}^7 \vartheta _{59}}{979776}, \vartheta _{29}=\frac{5 \vartheta _{26}^2}{12}, \nonumber \\ \vartheta _{42}= & {} \frac{253 \vartheta _{26}^9 \vartheta _{59}^2}{793618560},\quad \vartheta _{41}=-\frac{\vartheta _{26}^{11} \vartheta _{59}^3}{28570268160},\nonumber \\ \vartheta _{52}= & {} \frac{11 \vartheta _{26}^6 \vartheta _{59}^3}{94478400}, \quad \vartheta _{33}=\frac{5 \vartheta _{26}^3}{54},\nonumber \\ \vartheta _{48}= & {} \frac{289 \vartheta _{26}^{10} \vartheta _{59}^2}{44442639360},\quad \vartheta _{40}=\frac{11 \vartheta _{26}^{13} \vartheta _{59}^4}{30855889612800},\nonumber \\ \vartheta _{61}= & {} \frac{7 \vartheta _{26}^9 \vartheta _{59}^3}{20407334400}, \quad \vartheta _{50}=\frac{\vartheta _{26}^6}{46656},\nonumber \\ \vartheta _{54}= & {} -\frac{19}{63} \vartheta _{26} \vartheta _{59},\quad \vartheta _{57}=-\frac{54}{\vartheta _{26}}, \nonumber \\ \vartheta _{49}= & {} \frac{29 \vartheta _{26}^8 \vartheta _{59}}{88179840},\quad \vartheta _{63}=\frac{\vartheta _{26}^5 \vartheta _{59}}{18144},\nonumber \\ \vartheta _{64}= & {} \frac{5 \vartheta _{26}^3}{216},\quad \vartheta _{47}=\frac{5707 \vartheta _{26}^{12} \vartheta _{59}^3}{53997806822400},\nonumber \\ \vartheta _{27}= & {} \frac{\vartheta _{26}^6 \vartheta _{59}^2}{699840},\quad \vartheta _{66}=-\frac{23 \vartheta _{26}^4 \vartheta _{59}}{13608},\nonumber \\ \vartheta _{25}= & {} \frac{7 \vartheta _{26}^3 \vartheta _{59}}{1620},\quad \vartheta _{56}=-\frac{1}{540} \vartheta _{26}^2 \vartheta _{59},\nonumber \\ \vartheta _{46}= & {} \frac{13381 \vartheta _{26}^{14} \vartheta _{59}^4}{23327052547276800}, \quad \vartheta _{67}=-\frac{5 \vartheta _{26}^2}{36},\nonumber \\ \vartheta _{53}= & {} -\frac{\vartheta _{26}^4 \vartheta _{59}^2}{58320},\quad \vartheta _{60}=\frac{1080}{\vartheta _{26}^3},\nonumber \\ \vartheta _{69}= & {} -\frac{3 \vartheta _{26}}{2},\quad \vartheta _{65}=-\frac{\vartheta _{26}^6 \vartheta _{59}^2}{2099520}, \nonumber \\ \vartheta _{28}= & {} \frac{23 \vartheta _{26}^4 \vartheta _{59}}{4536},\quad \vartheta _{30}=\frac{11 \vartheta _{26}^9 \vartheta _{59}^3}{5101833600},\nonumber \\ \vartheta _{39}= & {} -\nu ^2+\frac{279936 \nu ^2}{\vartheta _{26}^7}\quad +\,\frac{1331 \vartheta _{59}^5 \vartheta _{26}^{15}}{149959623518208000}, \nonumber \\ \vartheta _{34}= & {} -\frac{121 \vartheta _{26}^{12} \vartheta _{59}^4}{18513533767680},\quad \vartheta _{68}=\frac{13 \vartheta _{26}^3 \vartheta _{59}}{22680},\nonumber \\ \vartheta _{58}= & {} -\frac{19 \vartheta _{26}^3 \vartheta _{59}^2}{68040},\quad \vartheta _{44}=\frac{\vartheta _{26}^5}{1296},\nonumber \\ \vartheta _{45}= & {} \vartheta _1 \left( -\mu ^2-\nu ^2\right) \quad +\,\frac{\mu ^2 \vartheta _{26}}{6}+\frac{46656 \nu ^2}{\vartheta _{26}^6}\nonumber \\&+\,\frac{3509 \vartheta _{59}^5 \vartheta _{26}^{16}}{1259660837552947200}, \nonumber \\ \vartheta _{62}= & {} \frac{107 \vartheta _{26}^7 \vartheta _{59}^2}{617258880},\quad \vartheta _{51}=\vartheta _0 \left( -\mu ^2-\nu ^2\right) \nonumber \\&+\,\frac{\vartheta _{59} \left( 279936 \nu ^2+\mu ^2 \vartheta _{26}^7\right) }{7560 \vartheta _{26}^4}\nonumber \\&+\,\frac{14641 \vartheta _{59}^6 \vartheta _{26}^{18}}{20406505568357744640000},\nonumber \\ \varrho (t)= & {} -\frac{136080 \beta (t)}{\vartheta _{26}^4 \vartheta _{59}}. \end{aligned}$$
(28)

Appendix B

  1. (I)
    $$\begin{aligned} \theta _1= & {} 0,\varphi _8(t){=}[-\varphi _5 \left( 4 \varphi _9^3 \beta (t){+}2 \varphi _9 \gamma (t){+}\varphi _{12}(t)\right) +\,4 \varphi _9 \varphi _5^3 \beta (t)\nonumber \\&-\,2 \left( \varphi _6 \varphi _{10} \delta (t)+\varphi _7 \varphi _{11} \varrho (t)\right) ]/\varphi _9,\nonumber \\ \varphi _{12}(t)= & {} [-\varphi _9 \left[ \varphi _9^4 \beta (t){+}\varphi _9^2 \gamma (t){+}\left( \varphi _{10}^2{-}\varphi _6^2\right) \delta (t) {+}\left( \varphi _{11}^2{-}\varphi _7^2\right) \varrho (t)\right] \nonumber \\&+\,\varphi _5^2 \left( 2 \varphi _9^3 \beta (t)-\varphi _9 \gamma (t)\right) \nonumber \\&+\,3 \varphi _9 \varphi _5^4 \beta (t)-2 \varphi _5 [\varphi _6 \varphi _{10} \delta (t)\nonumber \\&+\varphi _7 \varphi _{11} \varrho (t)]]/(\varphi _5^2+\varphi _9^2),\nonumber \\ \beta (t)= & {} \frac{\left( \theta _2^2-\theta _3^2\right) \left[ \left( \varphi _6 \varphi _9-\varphi _5 \varphi _{10}\right) {}^2 \delta (t)+\left( \varphi _7 \varphi _9-\varphi _5 \varphi _{11}\right) {}^2 \varrho (t)\right] }{3 \left( \varphi _5^2+\varphi _9^2\right) {}^2 \left( \theta _2^2 \varphi _5^2+\theta _3^2 \varphi _9^2\right) }.\nonumber \\ \end{aligned}$$
    (29)
  2. (II)
    $$\begin{aligned} \theta _2= & {} 0, \varphi _4(t)=-[\varphi _1 \left[ 2 \varphi _9 \left[ 2 \left( \varphi _1^2+\varphi _9^2\right) \beta (t) +\gamma (t)\right] +\varphi _{12}(t)\right] \nonumber \\&\quad +2 \left( \varphi _2 \varphi _{10} \delta (t)+\varphi _3 \varphi _{11} \varrho (t)\right) ]/\varphi _9, \nonumber \\ \varphi _{12}(t)= & {} [\varphi _9 \left[ \varphi _9^4 \beta (t)+\varphi _9^2 \gamma (t)+\left( \varphi _2^2+\varphi _{10}^2\right) \delta (t)+\left( \varphi _3^2+\varphi _{11}^2\right) \varrho (t)\right] \nonumber \\&\quad +\varphi _1^2 \left( 2 \varphi _9^3 \beta (t)-\varphi _9 \gamma (t)\right) -3 \varphi _9 \varphi _1^4 \beta (t)\nonumber \\&\quad -2 \varphi _1 \left( \varphi _2 \varphi _{10} \delta (t)+\varphi _3 \varphi _{11} \varrho (t)\right) ] /(\varphi _1^2-\varphi _9^2),\nonumber \\ \beta (t)= & {} \frac{\left( \theta _1^2-\theta _3^2\right) \left[ \left( \varphi _2 \varphi _9-\varphi _1 \varphi _{10}\right) {}^2 \delta (t)+\left( \varphi _3 \varphi _9-\varphi _1 \varphi _{11}\right) {}^2 \varrho (t)\right] }{3 \left( \varphi _1^2-\varphi _9^2\right) {}^2 \left( \theta _1^2 \varphi _1^2-\theta _3^2 \varphi _9^2\right) }.\nonumber \\ \end{aligned}$$
    (30)
  3. (III)
    $$\begin{aligned} \varphi _1= & {} 0,\varphi _{11}=\frac{\varphi _7 \varphi _9}{\varphi _5}, \nonumber \\ \varphi _{10}= & {} \frac{\varphi _6 \varphi _9}{\varphi _5}, \varphi _4(t)=-\frac{2 \left[ \varphi _2 \varphi _6 \delta (t)+\varphi _3 \varphi _7 \varrho (t)\right] }{\varphi _5},\nonumber \\ \varphi _8(t)= & {} \frac{-\varphi _5 \left( 4 \varphi _9^3 \beta +2 \varphi _9 \gamma +\varphi _{12}\right) +4 \varphi _9 \varphi _5^3 \beta -2 \left( \varphi _6 \varphi _{10} \delta +\varphi _7 \varphi _{11} \varrho \right) }{\varphi _9},\nonumber \\ \varphi _{12}(t)= & {} \frac{\varphi _9^2 \left[ \left( \varphi _2^2-3 \varphi _6^2\right) \delta +\left( \varphi _3^2-3 \varphi _7^2\right) \varrho \right] -3 \varphi _5^2 \left( \varphi _9^2 \gamma +\varphi _2^2 \delta +\varphi _3^2 \varrho \right) }{3 \varphi _5^2 \varphi _9},\nonumber \\ \beta (t)= & {} -\frac{\varphi _2^2 \delta (t)+\varphi _3^2 \varrho (t)}{3 \varphi _5^2 \varphi _9^2}, \theta _1 = \epsilon \frac{\sqrt{\varphi _5^2+\varphi _9^2} \sqrt{\theta _2^2 \varphi _5^2+\theta _3^2 \varphi _9^2}}{\varphi _5 \varphi _9}, \end{aligned}$$
    (31)

    where \(\epsilon =\pm 1\).

  4. (IV)
    $$\begin{aligned} \varphi _5= & {} 0,\varphi _{11}=\frac{\varphi _3 \varphi _9}{\varphi _1},\nonumber \\ \varphi _{10}= & {} \frac{\varphi _2 \varphi _9}{\varphi _1}, \varphi _8(t)=-\frac{2 \left[ \varphi _2 \varphi _6 \delta (t)+\varphi _3 \varphi _7 \varrho (t)\right] }{\varphi _1},\nonumber \\ \varphi _4(t)= & {} \frac{\varphi _1^2 \left( -3 \varphi _9^2 \gamma +\varphi _6^2 \delta +\varphi _7^2 \varrho \right) +3 \varphi _9^2 \left[ \left( \varphi _6^2-\varphi _2^2\right) \delta +\left( \varphi _7^2-\varphi _3^2\right) \varrho \right] }{3 \varphi _1 \varphi _9^2},\nonumber \\ \varphi _{12}(t)= & {} \frac{3 \varphi _1^2 \left( -\varphi _9^2 \gamma +\varphi _6^2 \delta +\varphi _7^2 \varrho \right) +\varphi _9^2 \left[ \left( \varphi _6^2-3 \varphi _2^2\right) \delta +\left( \varphi _7^2-3 \varphi _3^2\right) \varrho \right] }{3 \varphi _1^2 \varphi _9},\nonumber \\ \beta (t)= & {} -\frac{\varphi _6^2 \delta (t)+\varphi _7^2 \varrho (t)}{3 \varphi _1^2 \varphi _9^2}, \theta _2 = \epsilon \frac{\sqrt{\left( \varphi _9^2-\varphi _1^2\right) \left( \theta _1^2 \varphi _1^2-\theta _3^2 \varphi _9^2\right) }}{\varphi _1 \varphi _9}.\nonumber \\ \end{aligned}$$
    (32)
  5. (V)
    $$\begin{aligned} \varphi _7= & {} \frac{\varphi _3 \varphi _5}{\varphi _1},\nonumber \\ \varphi _{11}= & {} \frac{\varphi _3 \varphi _9}{\varphi _1}, \nonumber \\ \beta (t)= & {} -\frac{\left( \varphi _2 \varphi _5-\varphi _1 \varphi _6\right) {}^2 \delta (t)}{3 \left( \varphi _1^2+\varphi _5^2\right) {}^2 \varphi _9^2},\nonumber \\ \varphi _{10}= & {} \frac{-\varphi _5 \varphi _6 \varphi _1^2+\varphi _2 \left( \varphi _5^2+\varphi _9^2\right) \varphi _1+\varphi _5 \varphi _6 \varphi _9^2}{\left( \varphi _1^2+\varphi _5^2\right) \varphi _9},\nonumber \\ \varphi _4(t)= & {} -\frac{\varphi _1 \left[ 2 \varphi _9 \left[ 2 \left( \varphi _1^2+\varphi _9^2\right) \beta +\gamma \right] +\varphi _{12}\right] +2 \left( \varphi _2 \varphi _{10} \delta +\varphi _3 \varphi _{11} \varrho \right) }{\varphi _9},\nonumber \\ \varphi _8(t)= & {} \frac{-\varphi _5 \left( 4 \varphi _9^3 \beta +2 \varphi _9 \gamma +\varphi _{12}\right) +4 \varphi _9 \varphi _5^3 \beta -2 \left( \varphi _6 \varphi _{10} \delta +\varphi _7 \varphi _{11} \varrho \right) }{\varphi _9},\nonumber \\ \varphi _{12}= & {} [-\varphi _9 \left[ \varphi _9^4 \beta +\varphi _9^2 \gamma +\left( \varphi _{10}^2-\varphi _6^2\right) \delta +\left( \varphi _{11}^2-\varphi _7^2\right) \varrho \right] +3 \varphi _9 \varphi _5^4 \beta \nonumber \\&+ \varphi _5^2 \left( 2 \varphi _9^3 \beta -\varphi _9 \gamma \right) -2 \varphi _5 \left( \varphi _6 \varphi _{10} \delta +\varphi _7 \varphi _{11} \varrho \right) ]/(\varphi _5^2+\varphi _9^2),\nonumber \\ \theta _1= & {} \epsilon \sqrt{\varphi _5^2+\varphi _9^2} \sqrt{\frac{\theta _3^2 \left( \varphi _1^2-\varphi _9^2\right) -\theta _2^2 \left( \varphi _1^2+\varphi _5^2\right) }{\left( \varphi _1^2+\varphi _5^2\right) \left( \varphi _1^2-\varphi _9^2\right) }}. \end{aligned}$$
    (33)
  6. (VI)
    $$\begin{aligned} \varphi _6= & {} \frac{\varphi _2 \varphi _5}{\varphi _1}, \varphi _{10}=\frac{\varphi _2 \varphi _9}{\varphi _1}, \beta (t)=-\frac{\left( \varphi _3 \varphi _5-\varphi _1 \varphi _7\right) {}^2 \varrho (t)}{3 \left( \varphi _1^2+\varphi _5^2\right) {}^2 \varphi _9^2},\nonumber \\ \varphi _{11}= & {} \frac{-\varphi _5 \varphi _7 \varphi _1^2+\varphi _3 \left( \varphi _5^2+\varphi _9^2\right) \varphi _1+\varphi _5 \varphi _7 \varphi _9^2}{\left( \varphi _1^2+\varphi _5^2\right) \varphi _9},\nonumber \\ \varphi _4(t)= & {} -\frac{\varphi _1 \left[ 2 \varphi _9 \left[ 2 \left( \varphi _1^2+\varphi _9^2\right) \beta +\gamma \right] +\varphi _{12}\right] +2 \left( \varphi _2 \varphi _{10} \delta +\varphi _3 \varphi _{11} \varrho \right) }{\varphi _9},\nonumber \\ \varphi _8(t)= & {} \frac{-\varphi _5 \left( 4 \varphi _9^3 \beta +2 \varphi _9 \gamma +\varphi _{12}\right) +4 \varphi _9 \varphi _5^3 \beta -2 \left( \varphi _6 \varphi _{10} \delta +\varphi _7 \varphi _{11} \varrho \right) }{\varphi _9},\nonumber \\ \varphi _{12}= & {} [-\varphi _9 \left[ \varphi _9^4 \beta +\varphi _9^2 \gamma +\left( \varphi _{10}^2-\varphi _6^2\right) \delta +\left( \varphi _{11}^2-\varphi _7^2\right) \varrho \right] \nonumber \\&+3 \varphi _9 \varphi _5^4 \beta \nonumber \\&+ \varphi _5^2 \left( 2 \varphi _9^3 \beta -\varphi _9 \gamma \right) -2 \varphi _5 \left( \varphi _6 \varphi _{10} \delta +\varphi _7 \varphi _{11} \varrho \right) ]/(\varphi _5^2+\varphi _9^2),\nonumber \\ \theta _1= & {} \epsilon \sqrt{\varphi _5^2+\varphi _9^2} \sqrt{\frac{\theta _3^2 \left( \varphi _1^2-\varphi _9^2\right) -\theta _2^2 \left( \varphi _1^2+\varphi _5^2\right) }{\left( \varphi _1^2+\varphi _5^2\right) \left( \varphi _1^2-\varphi _9^2\right) }}. \end{aligned}$$
    (34)

Substituting Eqs. (29)–(34) into Eq. (19), respectively, we can obtain six different sets of breather wave solutions.

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Liu, JG., Zhu, WH. Multiple rogue wave, breather wave and interaction solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation. Nonlinear Dyn 103, 1841–1850 (2021). https://doi.org/10.1007/s11071-020-06186-1

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  • DOI: https://doi.org/10.1007/s11071-020-06186-1

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