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Further results on the smooth and nonsmooth solitons of the Novikov equation

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Abstract

This paper is concerned with the smooth and nonsmooth soliton solutions of the Novikov equation based on the bifurcation method of dynamical systems. Two interesting results are highlighted. First, the new Hamiltonian function is established in the case of \(\varphi ^2<c\) while \(\varphi ^2>c\) is discussed in Li (Int J Bifurcat Chaos 24(3):1450037 2014). Second, we prove that the corresponding traveling wave system of the Novikov equation exists new smooth and nonsmooth soliton solutions.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 11171115).

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Correspondence to Chaohong Pan.

Appendix

Appendix

We now present the progress for solving the integrations (3.5) and (3.9). To this end, we recall the integration

$$\begin{aligned} \int _{0}^{\xi }\hbox {d}\xi =\pm \sqrt{c}\int _{\psi _{3}}^{\psi } \frac{\hbox {d}\psi }{(1+\psi ^{2})(\psi -\psi _{1})\sqrt{\psi _{3}-\psi }}. \end{aligned}$$

Letting

$$\begin{aligned} \sqrt{\psi _{3}-\psi }=s, \end{aligned}$$

then we have

$$\begin{aligned} \psi =\psi _{3}-s^2. \end{aligned}$$

Further we get

$$\begin{aligned}&\int _{\psi _{3}}^{\psi }\frac{\hbox {d}\psi }{(1+\psi ^{2})(\psi -\psi _{1})\sqrt{\psi _{3}-\psi }}\nonumber \\&\quad =\int _{0}^{\sqrt{\psi _{3}-\psi }} \frac{-2\hbox {d}s}{\left[ 1+\left( \psi _{3}-s^2\right) ^2\right] \left( \psi _{3}-\psi _{1}-s^2\right) } \nonumber \\&\quad =\frac{-1}{\sqrt{\psi _{3}-\psi _{1}}} \nonumber \\&\qquad \times \int _{0}^{\sqrt{\psi _{3}-\psi }}\left[ \frac{\hbox {d}s}{\left[ 1+\left( \psi _{3}-s^2\right) ^2\right] \left( \sqrt{\psi _{3}-\psi _{1}}+s\right) }\right. \nonumber \\&\qquad \left. +\frac{\hbox {d}s}{\left[ 1+\left( \psi _{3}-s^2\right) ^2\right] \left( \sqrt{\psi _{3}-\psi _{1}}-s\right) }\right] . \end{aligned}$$

It is easy to check that

$$\begin{aligned}&\left( \psi _{3}-s^2\right) ^2+1\nonumber \\&\quad =\left( \psi _{3}-s^2+i\right) \left( \psi _{3}-s^2-i\right) \\&\quad =\left[ s^2+\left( \sqrt{\psi _{3}+i}+\sqrt{\psi _{3}-i}\right) s+\sqrt{\psi _{3}^2+1}\right] \nonumber \\&\qquad \left[ s^2-\left( \sqrt{\psi _{3}+i}+\sqrt{\psi _{3}-i}\right) s+\sqrt{\psi _{3}^2+1}\right] . \end{aligned}$$

We also have the following facts that

$$\begin{aligned}&\sqrt{\psi _{3}+i}=\left( \psi _{3}^2+1\right) ^{\frac{1}{4}}\,\hbox {e}^{\frac{\theta _{1}}{2}i},\\&\sqrt{\psi _{3}-i}=\left( \psi _{3}^2+1\right) ^{\frac{1}{4}}\,\hbox {e}^{-\frac{\theta _{1}}{2}i},\\&\sqrt{\psi _{3}+i}+\sqrt{\psi _{3}-i}=2\left( \psi _{3}^2+1\right) ^{\frac{1}{4}}\cos \theta _{1}, \end{aligned}$$

where \(\theta _{1}=\arctan \frac{1}{\psi _{3}}\). For simplicity, let

$$\begin{aligned} 2(\psi _{3}^2+1)^{\frac{1}{4}}\cos \theta _{1}= & {} \alpha _{1},\\ \sqrt{\psi _{3}^2+1}= & {} \beta _{1},\\ \sqrt{\psi _{3}-\psi _{1}}= & {} \gamma _{1}. \end{aligned}$$

Therefore, we get

$$\begin{aligned}&\int _{0}^{\sqrt{\psi _{3}-\psi }}\left[ \frac{\hbox {d}s}{\left[ 1+\left( \psi _{3}-s^2\right) ^2\right] (\sqrt{\psi _{3}-\psi _{1}}+s)} \right. \nonumber \\&\quad \quad \left. +\frac{\hbox {d}s}{\left[ 1+\left( \psi _{3}-s^2\right) ^2\right] \left( \sqrt{\psi _{3}-\psi _{1}}-s\right) }\right] \\&\quad =I_{1}+I_{2}, \end{aligned}$$

where

$$\begin{aligned} I_{1}= & {} \int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{\left( s^2+\alpha _{1}s+\beta _{1}\right) \left( s^2-\alpha _{1}s+\beta _{1}\right) (\gamma _{1}+s)},\\ I_{2}= & {} \int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{\left( s^2+\alpha _{1}s+\beta _{1}\right) \left( s^2-\alpha _{1}s+\beta _{1}\right) (\gamma _{1}-s)}. \end{aligned}$$

Further, we have

$$\begin{aligned} I_{1}= & {} I_{11}+I_{12},\\ I_{2}= & {} I_{21}+I_{22}, \end{aligned}$$

where

$$\begin{aligned} I_{11}= & {} -\frac{1}{2\alpha _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{s\left( s^2+\alpha _{1}s+\beta _{1}\right) (s+\gamma _{1})},\\ I_{12}= & {} \frac{1}{2\alpha _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{s\left( s^2-\alpha _{1}s+\beta _{1}\right) (s+\gamma _{1})},\\ I_{21}= & {} -\frac{1}{2\alpha _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{s\left( s^2+\alpha _{1}s+\beta _{1}\right) (\gamma _{1}-s)},\\ I_{22}= & {} \frac{1}{2\alpha _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{s\left( s^2-\alpha _{1}s+\beta _{1}\right) (\gamma _{1}-s)}. \end{aligned}$$

It is easy to check that

$$\begin{aligned} I_{11}= & {} I_{111}+I_{112},\\ I_{12}= & {} I_{121}+I_{122},\\ I_{21}= & {} I_{211}+I_{212},\\ I_{22}= & {} I_{221}+I_{222}, \end{aligned}$$

where

$$\begin{aligned} I_{111}= & {} I_{211}=-\frac{1}{2\alpha _{1}\gamma _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{s\left( s^2+\alpha _{1}s+\beta _{1}\right) },\\ I_{112}= & {} \frac{1}{2\alpha _{1}\gamma _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{\left( s^2+\alpha _{1}s+\beta _{1}\right) (s+\gamma _{1})},\\ I_{121}= & {} I_{221}=\frac{1}{2\alpha _{1}\gamma _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{s\left( s^2-\alpha _{1}s+\beta _{1}\right) },\\ I_{122}= & {} -\frac{1}{2\alpha _{1}\gamma _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{\left( s^2-\alpha _{1}s+\beta _{1}\right) (s+\gamma _{1})},\\ I_{212}= & {} -\frac{1}{2\alpha _{1}\gamma _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{\left( s^2+\alpha _{1}s+\beta _{1}\right) (\gamma _{1}-s)},\\ I_{222}= & {} \frac{1}{2\alpha _{1}\gamma _{1}}\int _{0}^{\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}s}{\left( s^2-\alpha _{1}s+\beta _{1}\right) (\gamma _{1}-s)}. \end{aligned}$$

A direct calculation yields

$$\begin{aligned}&I_{111}+I_{211}+I_{121}+I_{221}\\&\quad =-\frac{1}{\beta _{1}\gamma _{1}\sqrt{\delta _{1}}}\left[ -\arctan \frac{2\sqrt{\psi _{3}-\psi }+\alpha _{1}}{\sqrt{\delta _{1}}} +\arctan \frac{\alpha _{1}}{\sqrt{\delta _{1}}}\right. \nonumber \\&\qquad \left. +\frac{\sqrt{\delta _{1}}}{2\alpha _{1}}\ln \frac{\psi _{3}-\psi }{\psi _{3}-\psi +\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}}\right] \\&\qquad +\frac{1}{\beta _{1}\gamma _{1}\sqrt{\delta _{1}}}\left[ -\arctan \frac{2\sqrt{\psi _{3}-\psi }-\alpha _{1}}{\sqrt{\delta _{1}}}\right. \nonumber \\&\qquad +\arctan \frac{-\alpha _{1}}{\sqrt{\delta _{1}}}+\frac{\sqrt{\delta _{1}}}{2\alpha _{1}}\nonumber \\&\qquad \left. \times \ln \frac{\psi _{3}-\psi }{\psi _{3}-\psi -\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}}\right] \\&\quad =\frac{1}{\beta _{1}\gamma _{1}\sqrt{\delta _{1}}}\left[ \arctan \frac{2\alpha _{1}\sqrt{\delta _{1}}}{\delta _{1}-\alpha _{1}^2+4(\psi _{3}-\psi )}\right. \nonumber \\&\qquad +\arctan \frac{-2\alpha _{1}\sqrt{\delta _{1}}}{\delta _{1}-\alpha _{1}^2}+\frac{\sqrt{\delta _{1}}}{2\alpha _{1}}\nonumber \\&\qquad \left. \times \ln \frac{\psi _{3}-\psi +\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}}{\psi _{3}-\psi -\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}}\right] , \end{aligned}$$

where \(\delta _{1}=4\beta _{1}-\alpha _{1}^2=4\sqrt{\psi _{3}^{2}+1}\sin ^2\theta _{1}\). Letting \(s+\gamma _{1}=T\), we have

$$\begin{aligned} I_{112}= & {} \frac{1}{2\alpha _{1}\gamma _{1}} \nonumber \\&\int _{\gamma _{1}}^{\sqrt{\psi _{3}-\psi }+\gamma _{1}}\frac{\hbox {d}T}{\left[ T^2+(\alpha _{1}-2\gamma _{1})T+\gamma _{1}^2+\beta _{1}-\alpha _{1}\gamma _{1}\right] T},\\ I_{122}= & {} -\frac{1}{2\alpha _{1}\gamma _{1}} \nonumber \\&\int _{\gamma _{1}}^{\sqrt{\psi _{3}-\psi }+\gamma _{1}}\frac{\hbox {d}T}{\left[ T^2-(\alpha _{1}+2\gamma _{1})T+\gamma _{1}^2+\beta _{1}+\alpha _{1}\gamma _{1}\right] T},\\ I_{212}= & {} -\frac{1}{2\alpha _{1}\gamma _{1}} \nonumber \\&\int _{\gamma _{1}}^{\gamma _{1}-\sqrt{\psi _{3}-\psi }}\frac{-\hbox {d}T}{\left[ T^2-(\alpha _{1}+2\gamma _{1})T+\gamma _{1}^2+\beta _{1}+\alpha _{1}\gamma _{1}\right] T},\\ I_{222}= & {} \frac{1}{2\alpha _{1}\gamma _{1}} \nonumber \\&\int _{\gamma _{1}}^{\gamma _{1}-\sqrt{\psi _{3}-\psi }}\frac{-\hbox {d}T}{\left[ T^2+(\alpha _{1}-2\gamma _{1})T+\gamma _{1}^2+\beta _{1}-\alpha _{1}\gamma _{1}\right] T}. \end{aligned}$$

Therefore,

$$\begin{aligned}&I_{112}+I_{222}=\frac{1}{2\alpha _{1}\gamma _{1}}\nonumber \\&\quad \int _{\gamma _{1}-\sqrt{\psi _{3}-\psi }}^{\gamma _{1}+\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}T}{\left[ T^2+(\alpha _{1}-2\gamma _{1})T+\gamma _{1}^2+\beta _{1}-\alpha _{1}\gamma _{1}\right] T}\\&\quad =\frac{1}{4\alpha _{1}\gamma _{1}\left( \gamma _{1}^2+\beta _{1}-\alpha _{1}\gamma _{1}\right) }\nonumber \\&\qquad \times \ln \frac{\left( \gamma _{1}+\sqrt{\psi _{3}-\psi }\right) ^2\left[ \psi _{3}-\psi -\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}\right] }{\left( \gamma _{1}-\sqrt{\psi _{3}-\psi }\right) ^2\left[ \psi _{3}-\psi +\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}\right] }\\&\quad -\frac{\alpha _{1}-2\gamma _{1}}{2\alpha _{1}\gamma _{1}\left( \gamma _{1}^2+\beta _{1}-\alpha _{1}\gamma _{1}\right) \sqrt{\delta _{1}}}\nonumber \\&\qquad \times \bigg [\arctan \frac{1}{\sqrt{\delta _{1}}}\left( 2\sqrt{\psi _{3}-\psi }+\alpha _{1}\right) \nonumber \\&\qquad -\arctan \frac{1}{\sqrt{\delta _{1}}}\left( -2\sqrt{\psi _{3}-\psi }+\alpha _{1}\right) \bigg ]\\&\quad =\frac{1}{4\alpha _{1}\gamma _{1}\left( \gamma _{1}^2+\beta _{1}-\alpha _{1}\gamma _{1}\right) }\nonumber \\&\qquad \times \ln \frac{\left( \gamma _{1}+\sqrt{\psi _{3}-\psi }\right) ^2\left[ \psi _{3}-\psi -\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}\right] }{\left( \gamma _{1}-\sqrt{\psi _{3}-\psi }\right) ^2\left[ \psi _{3}-\psi +\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}\right] }\\&\quad -\frac{\alpha _{1}-2\gamma _{1}}{2\alpha _{1}\gamma _{1}(\gamma _{1}^2+\beta _{1}-\alpha _{1}\gamma _{1})\sqrt{\delta _{1}}}\nonumber \\&\qquad \times \arctan \frac{\sqrt{\psi _{3}-\psi }\sqrt{\delta _{1}}}{\beta _{1}-(\psi _{3}-\psi )}. \end{aligned}$$

and

$$\begin{aligned}&I_{122}+I_{212}=-\frac{1}{2\alpha _{1}\gamma _{1}}\nonumber \\&\qquad \int _{\gamma _{1}-\sqrt{\psi _{3}-\psi }}^{\gamma _{1}+\sqrt{\psi _{3}-\psi }}\frac{\hbox {d}T}{\left[ T^2-(\alpha _{1}+2\gamma _{1})T+\gamma _{1}^2+\beta _{1}+\alpha _{1}\gamma _{1}\right] T}\\&\quad =-\frac{1}{4\alpha _{1}\gamma _{1}\left( \gamma _{1}^2+\beta _{1}+A\gamma _{1}\right) }\nonumber \\&\qquad \times \ln \frac{\left( \gamma _{1}+\sqrt{\psi _{3}-\psi }\right) ^2\left[ \psi _{3}-\psi +\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}\right] }{\left( \gamma _{1}-\sqrt{\psi _{3}-\psi }\right) ^2\left[ \psi _{3}-\psi -\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}\right] }\\&\quad +\frac{\alpha _{1}+2\gamma _{1}}{2\alpha _{1}\gamma _{1}\left( \gamma _{1}^2+\beta _{1}+\alpha _{1}\gamma _{1}\right) \sqrt{\delta _{1}}}\nonumber \\&\qquad \times \bigg [\arctan \frac{1}{\sqrt{\delta _{1}}}\big (2\sqrt{\psi _{3}-\psi }-\alpha _{1}\big )\nonumber \\&\qquad -\arctan \frac{-1}{\sqrt{\delta _{1}}}\big (2\sqrt{\psi _{3}-\psi }+\alpha _{1}\big )\bigg ]\\&\quad =-\frac{1}{4\alpha _{1}\gamma _{1}\left( \gamma _{1}^2+\beta _{1}+\alpha _{1}\gamma _{1}\right) }\nonumber \\&\qquad \times \ln \frac{\left( \gamma _{1}+\sqrt{\psi _{3}-\psi }\right) ^2\left[ \psi _{3}-\psi +\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}\right] }{\left( \gamma _{1}-\sqrt{\psi _{3}-\psi }\right) ^2\left[ \psi _{3}-\psi -\alpha _{1}\sqrt{\psi _{3}-\psi }+\beta _{1}\right] }\\&\quad +\frac{\alpha _{1}+2\gamma _{1}}{2\alpha _{1}\gamma _{1}\left( \gamma _{1}^2+\beta _{1}+\alpha _{1}\gamma _{1}\right) \sqrt{\delta _{1}}}\nonumber \\&\qquad \times \arctan \frac{\sqrt{\psi _{3}-\psi }\sqrt{\delta _{1}}}{\beta _{1}-(\psi _{3}-\psi )}. \end{aligned}$$

Next, we present the progress of the integration

$$\begin{aligned} \int _{0}^{\xi }\hbox {d}\xi =\pm \sqrt{c}\int _{\psi _{4}}^{\psi } \frac{\hbox {d}\psi }{(1+\psi ^{2})(\psi _{2}-\psi )\sqrt{\psi _{4}-\psi }}. \end{aligned}$$

Letting

$$\begin{aligned} \sqrt{\psi _{4}-\psi }=s, \end{aligned}$$

i.e.

$$\begin{aligned} \psi =\psi _{4}-s^2, \end{aligned}$$

then we have

$$\begin{aligned}&\int _{\psi _{4}}^{\psi } \frac{d\psi }{(1+\psi ^{2})(\psi _{2}-\psi )\sqrt{\psi _{4}-\psi }}\nonumber \\&\quad =\int _{0}^{\sqrt{\psi _{4}-\psi }} \frac{-2ds}{\left[ 1+(\psi _{4}-s^2)^2\right] (\psi _{2}-\psi _{4}+s^2)}.\nonumber \\ \end{aligned}$$
(4.1)

It is easy to check that

$$\begin{aligned}&\left( \psi _{4}-s^2\right) ^2+1\\&\quad =\left( \psi _{4}-s^2+i\right) \left( \psi _{4}-s^2-i\right) \\&\quad =\left[ s^2+\left( \sqrt{\psi _{4}+i}+\sqrt{\psi _{4}-i}\right) s+\sqrt{\psi _{4}^2+1}\right] \nonumber \\&\qquad \left[ s^2-\left( \sqrt{\psi _{4}+i}+\sqrt{\psi _{4}-i}\right) s+\sqrt{\psi _{4}^2+1}\right] . \end{aligned}$$

It follows that

$$\begin{aligned}&\sqrt{\psi _{4}+i}=\left( \psi _{4}^2+1\right) ^{\frac{1}{4}}\,\hbox {e}^{\frac{\theta _{2}}{2}i},\\&\sqrt{\psi _{4}-i}=\left( \psi _{4}^2+1\right) ^{\frac{1}{4}}\,\hbox {e}^{-\frac{\theta _{2}}{2}i},\\&\sqrt{\psi _{4}+i}+\sqrt{\psi _{4}-i}=2(\psi _{4}^2+1)^{\frac{1}{4}}\cos \theta _{2}, \end{aligned}$$

where \(\theta _{2}=\arctan \frac{1}{\psi _{4}}\). For simplicity, let

$$\begin{aligned}&2\left( \psi _{4}^2+1\right) ^{\frac{1}{4}}\cos \theta _{2}=\alpha _{2},\\&\sqrt{\psi _{4}^2+1}=\beta _{2},\\&\psi _{2}-\psi _{4}=\gamma _{2}. \end{aligned}$$

Further we have

$$\begin{aligned} II= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }} \frac{-2\hbox {d}s}{\left[ 1+\left( \psi _{4}-s^2\right) ^2\right] \left( \psi _{2}-\psi _{4}+s^2\right) }\\= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{-2\hbox {d}s}{\left( s^2+\alpha _{2}s+\beta _{2}\right) \left( s^2-\alpha _{2}s+\beta _{2}\right) \left( s^2+\gamma _{2}\right) }\\= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{1}{\alpha _{2}s\left( s^2+\gamma _{2}\right) }\nonumber \\&\times \left[ \frac{1}{s^2+\alpha _{2}s+\beta _{2}}-\frac{1}{s^2-\alpha _{2}s+\beta _{2}}\right] \hbox {d}s\\= & {} II_{1}-II_{2}, \end{aligned}$$

where

$$\begin{aligned} II_{1}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }} \frac{\hbox {d}s}{\alpha _{2}s\left( s^2+\gamma _{2}\right) \left( s^2+\alpha _{2}s+\beta _{2}\right) }\nonumber \\= & {} II_{11}-II_{12},\\ II_{2}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{\alpha _{2}s\left( s^2+\gamma _{2}\right) \left( s^2-\alpha _{2}s+\beta _{2}\right) }\nonumber \\= & {} II_{22}-II_{21}, \end{aligned}$$

and

$$\begin{aligned} II_{11}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{\alpha _{2}s(\alpha _{2}s+\beta _{2}-\gamma _{2})\left( s^2+\gamma _{2}\right) }\nonumber \\= & {} II_{111}-II_{112},\\ II_{12}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }} \frac{\hbox {d}s}{\alpha _{2}s(\alpha _{2}s+\beta _{2}-\gamma _{2})\left( s^2+\alpha _{2}s+\beta _{2}\right) }\nonumber \\= & {} II_{121}-II_{122},\\ II_{21}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }} \frac{\hbox {d}s}{\alpha _{2}s(\alpha _{2}s+\gamma _{2}-\beta _{2})\left( s^2+\gamma _{2}\right) }\nonumber \\= & {} II_{211}-II_{212},\\ II_{22}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }} \frac{\hbox {d}s}{\alpha _{2}s(\alpha _{2}s+\gamma _{2}-\beta _{2})\left( s^2-\alpha _{2}s+\beta _{2}\right) }\nonumber \\= & {} II_{221}-II_{222}. \end{aligned}$$

It is easy to check that

$$\begin{aligned} II_{111}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{(\gamma _{2}-\beta _{2})(\alpha _{2}s+\beta _{2}-\gamma _{2})\left( s^2+\gamma _{2}\right) }\\= & {} \frac{1}{\alpha _{2}(\gamma _{2}-\beta _{2})}\nonumber \\&\int _{\omega }^{\sqrt{\psi _{4}-\psi }+\omega }\frac{\hbox {d}T}{T\left( T^2-2\omega T+\omega ^2+\gamma _{2}\right) },\\ II_{112}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{(\gamma _{2}-\beta _{2})\alpha _{2}s\left( s^2+\gamma _{2}\right) },\\ II_{121}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{(\gamma _{2}-\beta _{2})(\alpha _{2}s+\beta _{2}-\gamma _{2})\left( s^2+\alpha _{2}s+\beta _{2}\right) }\\= & {} \frac{1}{\alpha _{2}(\gamma _{2}-\beta _{2})}\nonumber \\&\int _{\omega }^{\sqrt{\psi _{4}-\psi }+\omega }\frac{\hbox {d}T}{T\left[ T^2+(\alpha _{2}-2\omega )T+\omega ^2-\alpha _{2}\omega +\beta _{2}\right] },\\ II_{122}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{(\gamma _{2}-\beta _{2})\alpha _{2}s\left( s^2+\alpha _{2}s+\beta _{2}\right) },\\ II_{211}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{(\gamma _{2}-\beta _{2})\alpha _{2}s\left( s^2+\gamma _{2}\right) },\\ II_{212}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{(\gamma _{2}-\beta _{2})(\alpha _{2}s+\gamma _{2}-\beta _{2})\left( s^2+\gamma _{2}\right) }\\= & {} \frac{1}{\alpha _{2}(\gamma _{2}-\beta _{2})}\int _{-\omega }^{\sqrt{\psi _{4}-\psi }-\omega } \frac{\hbox {d}T}{T\left( T^2+2\omega T+\omega ^2+\gamma _{2}\right) },\\ II_{221}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{(\gamma _{2}-\beta _{2})\alpha _{2}s\left( s^2-\alpha _{2}s+\beta _{2}\right) },\\ II_{222}= & {} \int _{0}^{\sqrt{\psi _{4}-\psi }}\frac{\hbox {d}s}{(\gamma _{2}-\beta _{2})(\alpha _{2}s+\gamma _{2}-\beta _{2})\left( s^2-\alpha _{2}s+\beta _{2}\right) }\\= & {} \frac{1}{\alpha _{2}(\gamma _{2}-\beta _{2})}\\&\times \int _{-\omega }^{\sqrt{\psi _{4}-\psi }-\omega }\frac{\hbox {d}T}{T\left[ T^2-(\alpha _{2}-2\omega )T+\omega ^2+\alpha _{2}\omega +\beta _{2}\right] },\\ \end{aligned}$$

where

$$\begin{aligned} \omega =\frac{\beta _{2}-\gamma _{2}}{\alpha _{2}}. \end{aligned}$$

A direct calculation yields

$$\begin{aligned} II_{112}-II_{211}= & {} 0,\\ II_{122}-II_{221}= & {} \frac{1}{2\alpha _{2}\beta _{2}(\gamma _{2}-\beta _{2})}\nonumber \\&\times \ln \left| \frac{\beta _{2}-\alpha _{2}\sqrt{\psi _{4}-\psi }+\psi _{4}-\psi }{\beta _{2}+\alpha _{2}\sqrt{\psi _{4}-\psi }+\psi _{4}-\psi }\right| \\&-\frac{1}{\beta _{2}\sqrt{\delta _{2}}(\gamma _{2}-\beta _{2})}\nonumber \\&\times \left( \arctan \left[ \frac{2\sqrt{\psi _{4}-\psi }+\alpha _{2}}{\sqrt{\delta _{2}}}\right] \right. \nonumber \\&\left. -\arctan \left[ \frac{2\sqrt{\psi _{4}-\psi }-\alpha _{2}}{\sqrt{\delta _{2}}}\right] \right) , \end{aligned}$$
$$\begin{aligned} II_{111}= & {} \frac{1}{\alpha _{2}(\gamma _{2}-\beta _{2})(\omega ^2+\gamma _{2})}\nonumber \\&\times \left[ \ln \frac{\sqrt{\psi _{4}-\psi }+\omega }{\psi _{4}-\psi +\gamma _{2}}-\frac{1}{2}\right. \nonumber \\&\left. \times \ln \frac{\omega ^2}{\gamma _{2}}+\frac{\omega }{\sqrt{\gamma _{2}}}\arctan \frac{\sqrt{\psi _{4}-\psi }}{\sqrt{\gamma _{2}}}\right] ,\\ II_{212}= & {} \frac{1}{\alpha _{2}(\gamma _{2}-\beta _{2})(\omega ^2+\gamma _{2})}\nonumber \\&\times \left[ \frac{1}{2}\ln \frac{(\sqrt{\psi _{4}-\psi }-\omega )^2}{\psi _{4}-\psi +\gamma _{2}}-\frac{1}{2}\right. \nonumber \\&\left. \times \ln \frac{\omega ^2}{\gamma _{2}}-\frac{\omega }{\sqrt{\gamma _{2}}}\arctan \frac{\sqrt{\psi _{4}-\psi }}{\sqrt{\gamma _{2}}}\right] , \end{aligned}$$
$$\begin{aligned} II_{121}= & {} \frac{1}{\alpha _{2}(\gamma _{2}-\beta _{2})(\omega ^2-\alpha _{2}\omega +\beta _{2})}\nonumber \\&\times \bigg [\ln \frac{\sqrt{\psi _{4}-\psi }+\omega }{\psi _{4}-\psi -\alpha _{2} \sqrt{\psi _{4}-\psi }+\beta _{2}}\\&-\frac{1}{2}\ln \frac{\omega ^2}{\beta _{2}}- \frac{\alpha _{2}-2\omega }{\sqrt{\delta _{2}}}\nonumber \\&\times \arctan \frac{2\sqrt{\psi _{4}-\psi }+\alpha _{2}}{\sqrt{\delta _{2}}}+\frac{\alpha _{2}-2\omega }{\sqrt{\delta _{2}}}\nonumber \\&\times \arctan \frac{\alpha _{2}}{\sqrt{\delta _{2}}}\bigg ],\\ II_{222}= & {} \frac{1}{\alpha _{2}(\gamma _{2}-\beta _{2})(\omega ^2-\alpha _{2}\omega +\beta _{2})}\nonumber \\&\times \bigg [\frac{1}{2}\ln \frac{(\sqrt{\psi _{4}-\psi }-\omega )^2}{\psi _{4}-\psi +\alpha _{2}\sqrt{\psi _{4}-\psi }+\beta _{2}}\\&-\frac{1}{2}\ln \frac{\omega ^2}{\beta _{2}} +\frac{\alpha _{2}-2\omega }{\sqrt{\delta _{2}}}\nonumber \\&\times \arctan \frac{2\sqrt{\psi _{4}-\psi }-\alpha _{2}}{\sqrt{\delta _{2}}}\nonumber \\&+\frac{\alpha _{2}-2\omega }{\sqrt{\delta _{2}}}\arctan \frac{\alpha _{2}}{\sqrt{\delta _{2}}}\bigg ]. \end{aligned}$$

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Pan, C., Li, S. Further results on the smooth and nonsmooth solitons of the Novikov equation. Nonlinear Dyn 86, 779–788 (2016). https://doi.org/10.1007/s11071-016-2921-z

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