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Various exact analytical solutions of a variable-coefficient Kadomtsev–Petviashvili equation

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Abstract

Under investigation is a generalized (3 + 1)-dimensional variable- coefficient Kadomtsev– Petviashvili equation in fluid mechanics. Various exact analytical solutions are obtained by Hirota’s bilinear method, such as lump-type, breather wave and kink-solitary wave solutions. We discuss the interaction between lump wave and solitary waves, and the interaction between lump wave and periodic wave. The physical structure and propagation characteristics of obtained solutions are shown by some 3D graphics.

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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

$$\begin{aligned} (1)\,\,\, \alpha _4(t)= & {} -\left[ \left( \alpha _1^2+\alpha _6^2\right) \alpha _1 h(t)+\alpha _3 \alpha _1^2 l(t)\right. \nonumber \\&+\,\alpha _3 \alpha _6^2 l(t)+\alpha _2^2 \alpha _1 m(t)\nonumber \\&-\,\alpha _7^2 \alpha _1 m(t)+2 \alpha _2 \alpha _6 \alpha _7 m(t)\nonumber \\&+\,\alpha _3^2 \alpha _1 n(t)-\alpha _8^2 \alpha _1 n(t)\nonumber \\&+\,2 \alpha _3 \alpha _6 \alpha _8 n(t)+\alpha _2 \nonumber \\&\left. \left( \alpha _1^2+\alpha _6^2\right) q(t)\right] /[\alpha _1^2+\alpha _6^2],\nonumber \\ \alpha _9(t)= & {} -\left[ \alpha _6 \left( \alpha _1^2+\alpha _6^2\right) h(t)\right. \nonumber \\&+\,\alpha _8 \left( \left( \alpha _1^2+\alpha _6^2\right) l(t)+2 \alpha _1 \alpha _3 n(t)\right) \nonumber \\&+\,\left( -\alpha _6 \alpha _2^2+2 \alpha _1 \alpha _7 \alpha _2+\alpha _6 \alpha _7^2\right) m(t)\nonumber \\&+\,\alpha _6 \left( \alpha _6 \alpha _7 q(t)-\alpha _3^2 n(t)\right) \nonumber \\&\left. +\,\alpha _6 \alpha _8^2 n(t)+\alpha _7 \alpha _1^2 q(t)\right] \bigg /\left[ \alpha _1^2+\alpha _6^2\right] ,\nonumber \\ \end{aligned}$$
(23)

with the constraints

$$\begin{aligned} \alpha _6\ne & {} 0, \alpha _1^2+\alpha _6^2\ne 0, 3 \left( \alpha _1^2+\alpha _6^2\right) {}^3 g(t)\nonumber \\&+\,\alpha _{11} \left[ \left( \alpha _2 \alpha _6-\alpha _1 \alpha _7\right) {}^2 m(t)\right. \nonumber \\&\left. +\left( \alpha _3 \alpha _6-\alpha _1 \alpha _8\right) {}^2 n(t)\right] =0.\nonumber \\ (2)\,\,\,\alpha _7= & {} -\frac{\alpha _1 \alpha _2}{\alpha _6},\nonumber \\ \alpha _8= & {} \frac{\alpha _3 \alpha _6}{\alpha _1},\nonumber \\ \alpha _4(t)= & {} -\alpha _1 h(t)-\alpha _3 l(t)+\frac{\alpha _1 \alpha _2^2 m(t)}{\alpha _6^2}\nonumber \\&-\,\frac{\alpha _3^2 n(t)}{\alpha _1}-\alpha _2 q(t),\nonumber \\ \alpha _9(t)= & {} \frac{\alpha _2 [\alpha _2 m(t)+\alpha _1 q(t)]}{\alpha _6}\nonumber \\&-\,\frac{\alpha _6 \left[ \alpha _1^2 h(t)+\alpha _3 \left( \alpha _1 l(t)+\alpha _3 n(t)\right) \right] }{\alpha _1^2}, \end{aligned}$$
(24)

with the constraints

$$\begin{aligned} \alpha _6\ne & {} 0, \alpha _1\ne 0,\nonumber \\&3 \left( \alpha _1^2+\alpha _6^2\right) g(t)\nonumber \\&+\,\frac{\alpha _{11} \alpha _2^2 m(t)}{\alpha _6^2}=0.\nonumber \\ (3)\,\,\,\alpha _7= & {} \frac{\alpha _2 \alpha _6}{\alpha _1}, \alpha _8{=} {-}\frac{\alpha _1 \alpha _3}{\alpha _6}{,}\nonumber \\ \alpha _4(t)= & {} {-}\alpha _1 h(t){-}\alpha _3 l(t){-}\frac{\alpha _2^2 m(t)}{\alpha _1}\nonumber \\&{+}\,\frac{\alpha _1 \alpha _3^2 n(t)}{\alpha _6^2}{-}\alpha _2 q(t){,}\nonumber \\ \alpha _9(t)= & {} \frac{\alpha _3 [\alpha _1 l(t){+}\alpha _3 n(t)]}{\alpha _6} \nonumber \\&{-}\,\frac{\alpha _6 \left[ \alpha _1^2 h(t)+\alpha _2 \left( \alpha _2 m(t)+\alpha _1 q(t)\right) \right] }{\alpha _1^2}, \end{aligned}$$
(25)

with the constraints

$$\begin{aligned} \alpha _6\ne & {} 0, \alpha _1\ne 0,\nonumber \\&3 \left( \alpha _1^2+\alpha _6^2\right) g(t)+\frac{\alpha _{11} \alpha _3^2 n(t)}{\alpha _6^2}=0.\nonumber \\ (4)\,\,\,\alpha _6= & {} 0{,} \alpha _9(t){=}{-}\frac{\alpha _8 \left( \alpha _1 l(t){+}2 \alpha _3 n(t)\right) {+}\alpha _7 \left( 2 \alpha _2 m(t){+}\alpha _1 q(t)\right) }{\alpha _1}{,}\nonumber \\ \alpha _4(t)= & {} {-}[\alpha _1^2 h(t){+}\alpha _3 \alpha _1 l(t){+}\left( \alpha _2^2{-}\alpha _7^2\right) m(t)\nonumber \\&{+}\,\alpha _3^2 n(t){-}\alpha _8^2 n(t){+}\alpha _2 \alpha _1 q(t)]/\alpha _1{,} \end{aligned}$$
(26)

with the constraints

$$\begin{aligned} \alpha _1\ne 0, 3 \alpha _1^4 g(t)+\alpha _{11} [\alpha _7^2 m(t)+\alpha _8^2 n(t)]=0. \end{aligned}$$

Substituting Eqs. (19), (20), (21) and (22) into Eqs. (3) and (4), respectively, the corresponding lump solution can be obtained. All solutions have been verified as correct by Mathematica.

Appendix B

$$\begin{aligned} (1)\,\,\, \theta _3(t)= & {} \pm \varphi _1 \sqrt{\varphi _2^2+\varphi _6^2} \sqrt{\frac{\lambda _3^2}{\varphi _2^2 \left( \varphi _1^2+\varphi _{10}^2\right) }+\frac{4}{\varphi _1^2 \varphi _6^2-\varphi _2^2 \varphi _{10}^2}} \nonumber \\&\sqrt{\theta _1(t)}, \nonumber \\ \varphi _{11}= & {} \frac{\varphi _{10} \left( \varphi _1 \varphi _6 \left( \varphi _1^2+2 \varphi _5^2-\varphi _{10}^2\right) +\varphi _2 \varphi _5 \left( \varphi _{10}^2-\varphi _5^2\right) \right) }{\varphi _1 \varphi _5 \left( \varphi _1^2+\varphi _5^2\right) },\nonumber \\ \varphi _{12}= & {} \frac{\left( \varphi _1 \varphi _3+\varphi _5 \varphi _7\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _7-\varphi _3 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}},\nonumber \\ \varphi _5= & {} \frac{\varphi _1 \varphi _6}{\varphi _2}. \end{aligned}$$
(27)
$$\begin{aligned} (2)\,\,\, \theta _3(t)= & {} \pm \sqrt{\varphi _5^2+\varphi _{10}^2} \sqrt{\frac{\lambda _3^2 \varphi _5^2-\lambda _3^2 \varphi _{10}^2+8 \varphi _{10}^2}{2 \varphi _5^2 \varphi _{10}^2-2 \varphi _{10}^4}} \nonumber \\&\sqrt{\theta _1(t)}, \nonumber \\ \varphi _{11}= & {} \frac{\varphi _{10} \left( \varphi _1 \varphi _6 \left( \varphi _1^2+2 \varphi _5^2-\varphi _{10}^2\right) +\varphi _2 \varphi _5 \left( \varphi _{10}^2-\varphi _5^2\right) \right) }{\varphi _1 \varphi _5 \left( \varphi _1^2+\varphi _5^2\right) },\nonumber \\ \varphi _{12}= & {} \frac{\left( \varphi _1 \varphi _3+\varphi _5 \varphi _7\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _7-\varphi _3 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}}, \nonumber \\ \varphi _1= & {} \pm \varphi _{10}. \end{aligned}$$
(28)
$$\begin{aligned} (3)\,\,\, \theta _3(t)= & {} \pm \sqrt{\varphi _5^2+\varphi _{10}^2} \sqrt{\frac{\lambda _3^2 \varphi _5^2-\lambda _3^2 \varphi _{10}^2+8 \varphi _{10}^2}{2 \varphi _5^2 \varphi _{10}^2-2 \varphi _{10}^4}} \nonumber \\&\sqrt{\theta _1(t)}, \varphi _1=\pm \varphi _{10}, \nonumber \\ \varphi _{11}= & {} \frac{\varphi _{10} \left( \varphi _1 \varphi _6 \left( \varphi _1^2+2 \varphi _5^2-\varphi _{10}^2\right) +\varphi _2 \varphi _5 \left( \varphi _{10}^2-\varphi _5^2\right) \right) }{\varphi _1 \varphi _5 \left( \varphi _1^2+\varphi _5^2\right) },\nonumber \\ \varphi _{12}= & {} \frac{\varphi _{10} \left( \varphi _1 \varphi _7 \left( \varphi _1^2+2 \varphi _5^2-\varphi _{10}^2\right) +\varphi _3 \varphi _5 \left( \varphi _{10}^2-\varphi _5^2\right) \right) }{\varphi _1 \varphi _5 \left( \varphi _1^2+\varphi _5^2\right) }.\nonumber \\ \end{aligned}$$
(29)
$$\begin{aligned} (4)\,\,\, \theta _3(t)= & {} \pm \sqrt{\varphi _5^2+\varphi _{10}^2} \sqrt{\frac{\lambda _3^2 \varphi _5^2-\lambda _3^2 \varphi _{10}^2+8 \varphi _{10}^2}{2 \varphi _5^2 \varphi _{10}^2-2 \varphi _{10}^4}} \nonumber \\&\sqrt{\theta _1(t)}, \varphi _1=\pm \varphi _{10}, \nonumber \\ \varphi _{11}= & {} \frac{\left( \varphi _1 \varphi _2+\varphi _5 \varphi _6\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _6-\varphi _2 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}},\nonumber \\ \varphi _{12}= & {} \frac{\varphi _{10} \left( \varphi _1 \varphi _7 \left( \varphi _1^2+2 \varphi _5^2-\varphi _{10}^2\right) +\varphi _3 \varphi _5 \left( \varphi _{10}^2-\varphi _5^2\right) \right) }{\varphi _1 \varphi _5 \left( \varphi _1^2+\varphi _5^2\right) }.\nonumber \\ \end{aligned}$$
(30)
$$\begin{aligned} (5)\,\,\, \theta _3(t)= & {} \pm \varphi _5 \sqrt{\varphi _3^2+\varphi _7^2} \nonumber \\&\sqrt{\frac{\lambda _3^2}{\varphi _3^2 \varphi _5^2 +\varphi _7^2 \varphi _{10}^2}+\frac{4}{\varphi _7^2 \left( \varphi _5^2-\varphi _{10}^2\right) }}\nonumber \\&\sqrt{\theta _1(t)}, \nonumber \\ \varphi _{11}= & {} \frac{\left( \varphi _1 \varphi _2+\varphi _5 \varphi _6\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _6-\varphi _2 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}},\nonumber \\ \varphi _1= & {} \frac{\varphi _3 \varphi _5}{\varphi _7}, \nonumber \\ \varphi _{12}= & {} \frac{\varphi _{10} \left( \varphi _1 \varphi _7 \left( \varphi _1^2{+}2 \varphi _5^2{-}\varphi _{10}^2\right) {+}\varphi _3 \varphi _5 \left( \varphi _{10}^2{-}\varphi _5^2\right) \right) }{\varphi _1 \varphi _5 \left( \varphi _1^2{+}\varphi _5^2\right) }{.}\nonumber \\ \end{aligned}$$
(31)
$$\begin{aligned} (6)\,\,\, \theta _3(t)= & {} \pm \sqrt{\varphi _1^2+\varphi _5^2} \nonumber \\&\sqrt{\frac{\lambda _3^2}{\varphi _1^2{+}\varphi _{10}^2}+\frac{2}{\varphi _5 \left( \varphi _5{+}\varphi _{10}\right) }{+}\frac{2}{\varphi _5^2{-}\varphi _5 \varphi _{10}}} \sqrt{\theta _1(t)}{,} \nonumber \\ \varphi _{12}= & {} \frac{\left( \varphi _1 \varphi _3{+}\varphi _5 \varphi _7\right) \varphi _{10}^2{+}\varphi _1 \varphi _5 \left( \varphi _1 \varphi _7{-}\varphi _3 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}}{,} m(t){=}0{.}\nonumber \\ \end{aligned}$$
(32)

In the above solutions, \(\varphi _4(t)\), g(t), \(\theta _2(t)\), \(\varphi _8(t)\) and \(\varphi _{13}(t)\) are the same as those of Eq. (18).

$$\begin{aligned} (7)\,\,\theta _2(t)= & {} 0, \theta _3(t)=\lambda _4 \sqrt{\theta _1(t)}, \nonumber \\ g(t)= & {} \frac{\left( \lambda _4^2{-}4\right) [\left( \varphi _2 \varphi _{10}{-}\varphi _1 \varphi _{11}\right) {}^2 m(t){+}\left( \varphi _3 \varphi _{10}{-}\varphi _1 \varphi _{12}\right) {}^2 n(t)]}{3 \left( \varphi _1^2{+}\varphi _{10}^2\right) {}^2 \left( \lambda _4^2 \varphi _{10}^2{+}4 \varphi _1^2\right) }{,}\nonumber \\ \varphi _4(t)= & {} \left[ \theta _1(t) \left[ -4 g(t) \left( \lambda _4^2 \varphi _{10}^4+4 \varphi _1^4\right) \right. \right. \nonumber \\&+\,h(t) \left( \lambda _4^2 \varphi _{10}^2-4 \varphi _1^2\right) +\lambda _4^2 \varphi _{10} \varphi _{12} l(t)\nonumber \\&-\,4 \varphi _1 \varphi _3 l(t)+\lambda _4^2 \varphi _{11}^2 m(t)-4 \varphi _2^2 m(t)\nonumber \\&+\,\lambda _4^2 \varphi _{12}^2 n(t)-4 \varphi _3^2 n(t)\nonumber \\&\left. +\lambda _4^2 \varphi _{10} \varphi _{11} q(t)-4 \varphi _1 \varphi _2 q(t)+\lambda _4^2 \varphi _{10} \varphi _{13}(t)\right] \nonumber \\&\left. -2 \varphi _1 \theta _1'(t)\right] /[4 \varphi _1 \theta _1(t)],\nonumber \\ \varphi _{13}(t)= & {} -\left[ -4 \lambda _4^2 \varphi _{10}^5 g(t)-16 \varphi _1^2 \varphi _{10}^3 g(t)\right. \nonumber \\&+\,h(t) \left( \lambda _4^2 \varphi _{10}^3+4 \varphi _1^2 \varphi _{10}\right) \nonumber \\&+\,\varphi _{12} \left( \lambda _4^2 \varphi _{10}^2 l(t)+4 \varphi _1^2 l(t)+8 \varphi _3 \varphi _1 n(t)\right) \nonumber \\&+\,(\lambda _4^2 \varphi _{10} \varphi _{11}^2-4 \varphi _{10} \varphi _2^2\nonumber \\&+\,8 \varphi _1 \varphi _{11} \varphi _2) m(t)+\lambda _4^2 \varphi _{12}^2 \varphi _{10} n(t)\nonumber \\&-\,4 \varphi _3^2 \varphi _{10} n(t)+\,\lambda _4^2 \varphi _{11} \varphi _{10}^2 q(t)\nonumber \\&\left. +4 \varphi _1^2 \varphi _{11} q(t)\right] /(\lambda _4^2 \varphi _{10}^2+4 \varphi _1^2), \end{aligned}$$
(33)

where \(\lambda _4\) is integral constant.

$$\begin{aligned} (8)\,\,\, \theta _2(t)= & {} \lambda _3 \sqrt{\theta _1(t)}, \theta _3(t)=0, \nonumber \\ g(t)= & {} \frac{\left( \lambda _3^2{-}4\right) [\left( \varphi _2 \varphi _5{-}\varphi _1 \varphi _6\right) {}^2 m(t){+}\left( \varphi _3 \varphi _5{-}\varphi _1 \varphi _7\right) {}^2 n(t)]}{3 \left( \varphi _1^2{+}\varphi _5^2\right) {}^2 \left( \lambda _3^2 \varphi _5^2{+}4 \varphi _1^2\right) }{,}\nonumber \\ \varphi _4(t)= & {} -\left[ 2 \theta _1(t) \left[ g(t) \left[ \left( \lambda _3^2-16\right) \varphi _1^4\right. \right. \right. \nonumber \\&\left. -6 \lambda _3^2 \varphi _5^2 \varphi _1^2-3 \lambda _3^2 \varphi _5^4\right] \nonumber \\&+\,\left( \lambda _3^2-4\right) \varphi _1^2 h(t)+\left( \lambda _3^2-4\right) \nonumber \\&\left[ \varphi _1 \varphi _3 l(t)+\varphi _2^2 m(t)+\varphi _3^2 n(t)\right. \nonumber \\&\left. \left. +\varphi _1 \varphi _2 q(t)\right] \right] +\left( \lambda _3^2-4\right) \nonumber \\&\left. \varphi _1 \theta _1'(t)\right] /\left[ 2 \left( \lambda _3^2-4\right) \varphi _1 \theta _1(t)\right] ,\nonumber \\ \varphi _8(t)= & {} \left[ 4 g(t) \left[ \left( \lambda _3^2-1\right) \varphi _5^4+3 \varphi _1^4+6 \varphi _5^2 \varphi _1^2\right] \right. \nonumber \\&-\,\left( \lambda _3^2-4\right) \left[ \varphi _5^2 h(t)+\varphi _7 \varphi _5 l(t)+\varphi _6^2 m(t)\right. \nonumber \\&\left. \left. +\varphi _7^2 n(t)+\varphi _6 \varphi _5 q(t)\right] \right] /\left[ \left( \lambda _3^2-4\right) \varphi _5\right] . \end{aligned}$$
(34)
$$\begin{aligned} (9)\,\,\, \theta _1(t)= & {} 0, \theta _2(t)=\lambda _5, \theta _3(t)=\lambda _6, \varphi _5=\pm \varphi _{10} \nonumber \\ \varphi _{11}= & {} \frac{\varphi _{10} \left( \left( \varphi _5 \varphi _6-\varphi _1 \varphi _2\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _2 \varphi _5+\varphi _1 \varphi _6\right) \right) }{2 \varphi _1^2 \varphi _5^2+\left( \varphi _5^2-\varphi _1^2\right) \varphi _{10}^2},\nonumber \\ \varphi _{12}= & {} \frac{\varphi _{10} \left( \left( \varphi _5 \varphi _7-\varphi _1 \varphi _3\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _3 \varphi _5+\varphi _1 \varphi _7\right) \right) }{2 \varphi _1^2 \varphi _5^2+\left( \varphi _5^2-\varphi _1^2\right) \varphi _{10}^2},\nonumber \\ g(t)= & {} \frac{\left( \varphi _2 \varphi _{10}-\varphi _1 \varphi _{11}\right) {}^2 m(t)+\left( \varphi _3 \varphi _{10}-\varphi _1 \varphi _{12}\right) {}^2 n(t)}{3 \varphi _5^2 \left( \varphi _1^2+\varphi _{10}^2\right) {}^2},\nonumber \\ \varphi _8(t)= & {} -\left[ -\left( \varphi _1^2+\varphi _{10}^2\right) \left( \varphi _5^4+3 \right. \right. \nonumber \\&\left. \left( \varphi _1^2+2 \varphi _5^2\right) \varphi _{10}^2\right) g(t)+\varphi _5^2 (\varphi _1^2\nonumber \\&+\,\varphi _{10}^2) h(t)+\varphi _5 \varphi _7 \varphi _1^2 l(t)+\varphi _5 \varphi _7 \varphi _{10}^2 l(t)\nonumber \\&+\,\varphi _6^2 \varphi _1^2 m(t)+\varphi _{11}^2 \varphi _1^2 m(t)\nonumber \\&-\,2 \varphi _2 \varphi _{10} \varphi _{11} \varphi _1 m(t)+\varphi _2^2 \varphi _{10}^2 m(t)\nonumber \\&+\,\varphi _6^2 \varphi _{10}^2 m(t)+\varphi _7^2 \varphi _1^2 n(t)\nonumber \\&+\,\varphi _{12}^2 \varphi _1^2 n(t)-2 \varphi _3 \varphi _{10} \varphi _{12} \varphi _1 n(t)+\varphi _3^2 \varphi _{10}^2 n(t)\nonumber \\&+\,\varphi _7^2 \varphi _{10}^2 n(t)\nonumber \\&\left. +\varphi _5 \varphi _6 \varphi _1^2 q(t)+\varphi _5 \varphi _6 \varphi _{10}^2 q(t)\right] \bigg /\left[ \varphi _5 \left( \varphi _1^2+\varphi _{10}^2\right) \right] ,\nonumber \\ \varphi _4(t)= & {} -\left[ \left( \varphi _1^2+\varphi _5^2\right) \left( \varphi _1^2-\varphi _5^2-6 \varphi _{10}^2\right) g(t)\right. \nonumber \\&+\,\left( \varphi _1^2+\varphi _5^2\right) h(t)\nonumber \\&+\varphi _1 \varphi _3 l(t)+\varphi _5 \varphi _7 l(t)+\varphi _2^2 m(t)\nonumber \\&+\,\varphi _6^2 m(t)+\varphi _3^2 n(t)\nonumber \\&+\,\varphi _7^2 n(t)+\varphi _1 \varphi _2 q(t)+\varphi _5 \varphi _6 q(t)\nonumber \\&\left. +\varphi _5 \varphi _8(t)\right] /\varphi _1,\nonumber \\ \varphi _{13}(t)= & {} -\left[ -\left( \varphi _5^4+6 \varphi _{10}^2 \varphi _5^2+\varphi _{10}^4\right) g(t)\right. \nonumber \\&+\,\left( \varphi _5^2+\varphi _{10}^2\right) h(t)+\varphi _5 \varphi _7 l(t)\nonumber \\&+\,\varphi _{10} \varphi _{12} l(t)+\varphi _6^2 m(t)+\varphi _{11}^2 m(t)\nonumber \\&+\,\varphi _7^2 n(t)+\varphi _{12}^2 n(t)\nonumber \\&+\,\varphi _5 \varphi _6 q(t)+\varphi _{10} \varphi _{11} q(t)\nonumber \\&\left. +\varphi _5 \varphi _8(t)\right] /\varphi _{10}. \end{aligned}$$
(35)

where \(\lambda _5\) and \(\lambda _6\) are integral constants.

$$\begin{aligned} (10)\,\,\, \theta _1(t)= & {} 0, \theta _2(t)=\lambda _5, \theta _3(t)=\lambda _6, \varphi _5=\pm \varphi _{10}, \nonumber \\ \varphi _{11}= & {} \frac{\left( \varphi _1 \varphi _2+\varphi _5 \varphi _6\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _6-\varphi _2 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}},\nonumber \\ \varphi _{12}= & {} \frac{\varphi _{10} \left( \left( \varphi _5 \varphi _7-\varphi _1 \varphi _3\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _3 \varphi _5+\varphi _1 \varphi _7\right) \right) }{2 \varphi _1^2 \varphi _5^2+\left( \varphi _5^2-\varphi _1^2\right) \varphi _{10}^2}. \end{aligned}$$
(36)
$$\begin{aligned} (11)\,\,\, \theta _1(t)= & {} 0, \theta _2(t)=\lambda _5, \theta _3(t)\nonumber \\= & {} \pm \varphi _1 \sqrt{\frac{\varphi _3^2+\varphi _7^2}{\varphi _1^2 \varphi _3^2+\varphi _{10}^2 \varphi _3^2}} \lambda _5, \varphi _5=\frac{\varphi _1 \varphi _7}{\varphi _3}, \nonumber \\ \varphi _{11}= & {} \frac{\left( \varphi _1 \varphi _2+\varphi _5 \varphi _6\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _6-\varphi _2 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}},\nonumber \\ \varphi _{12}= & {} \frac{\varphi _{10} \left( \left( \varphi _5 \varphi _7-\varphi _1 \varphi _3\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _3 \varphi _5+\varphi _1 \varphi _7\right) \right) }{2 \varphi _1^2 \varphi _5^2+\left( \varphi _5^2-\varphi _1^2\right) \varphi _{10}^2}. \end{aligned}$$
(37)
$$\begin{aligned} (12)\,\,\, \theta _1(t)= & {} 0, \theta _2(t)=\lambda _5, \theta _3(t)=\sqrt{\frac{\varphi _1^2+\varphi _5^2}{\varphi _1^2+\varphi _{10}^2}} \lambda _5, \nonumber \\ \varphi _{11}= & {} \frac{\left( \varphi _1 \varphi _2+\varphi _5 \varphi _6\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _6-\varphi _2 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}},\nonumber \\ \varphi _{12}= & {} \frac{\left( \varphi _1 \varphi _3+\varphi _5 \varphi _7\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _7-\varphi _3 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}}. \end{aligned}$$
(38)
$$\begin{aligned} (13)\,\,\, \theta _1(t)= & {} 0, \theta _2(t)=\lambda _5, \theta _3(t)\nonumber \\= & {} \sqrt{\frac{\varphi _1^2+\varphi _5^2}{\varphi _1^2+\varphi _{10}^2}} \lambda _5, \varphi _5=\pm \varphi _{10}, \nonumber \\ \varphi _{11}= & {} \frac{\left( \varphi _1 \varphi _2+\varphi _5 \varphi _6\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _6-\varphi _2 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}},\nonumber \\ \varphi _{12}= & {} \frac{\left( \varphi _1 \varphi _3+\varphi _5 \varphi _7\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _7-\varphi _3 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}}. \end{aligned}$$
(39)
$$\begin{aligned} (14)\,\,\, \theta _1(t)= & {} 0, \theta _2(t)=\lambda _5, \theta _3(t)\nonumber \\= & {} \varphi _1 \sqrt{\frac{\varphi _2^2+\varphi _6^2}{\varphi _2^2 \left( \varphi _1^2+\varphi _{10}^2\right) }} \lambda _5, \varphi _5=\frac{\varphi _1 \varphi _6}{\varphi _2}, \nonumber \\ \varphi _{11}= & {} \frac{\varphi _{10} \left( \left( \varphi _5 \varphi _6-\varphi _1 \varphi _2\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _2 \varphi _5+\varphi _1 \varphi _6\right) \right) }{2 \varphi _1^2 \varphi _5^2+\left( \varphi _5^2-\varphi _1^2\right) \varphi _{10}^2},\nonumber \\ \varphi _{12}= & {} \frac{\left( \varphi _1 \varphi _3+\varphi _5 \varphi _7\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _7-\varphi _3 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}}. \end{aligned}$$
(40)
$$\begin{aligned} (15)\,\,\, \theta _1(t)= & {} 0, \theta _2(t)=\lambda _5, \theta _3(t)\nonumber \\= & {} \varphi _1 \sqrt{\frac{\varphi _2^2+\varphi _6^2}{\varphi _2^2 \left( \varphi _1^2+\varphi _{10}^2\right) }} \lambda _5, \varphi _5=\pm \varphi _{10}, \nonumber \\ \varphi _{11}= & {} \frac{\varphi _{10} \left( \left( \varphi _5 \varphi _6-\varphi _1 \varphi _2\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _2 \varphi _5+\varphi _1 \varphi _6\right) \right) }{2 \varphi _1^2 \varphi _5^2+\left( \varphi _5^2-\varphi _1^2\right) \varphi _{10}^2},\nonumber \\ \varphi _{12}= & {} \frac{\left( \varphi _1 \varphi _3+\varphi _5 \varphi _7\right) \varphi _{10}^2+\varphi _1 \varphi _5 \left( \varphi _1 \varphi _7-\varphi _3 \varphi _5\right) }{\left( \varphi _1^2+\varphi _5^2\right) \varphi _{10}}. \end{aligned}$$
(41)

In Eqs. (36)–(41), \(\varphi _4(t)\), g(t), \(\theta _2(t)\), \(\varphi _8(t)\) and \(\varphi _{13}(t)\) are the same as those of Eq. (35). All solutions have been verified as correct by Mathematica.

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Liu, JG., Zhu, WH. Various exact analytical solutions of a variable-coefficient Kadomtsev–Petviashvili equation. Nonlinear Dyn 100, 2739–2751 (2020). https://doi.org/10.1007/s11071-020-05629-z

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