Bilinear form, soliton, breather, hybrid and periodic-wave solutions for a (3+1)-dimensional Korteweg–de Vries equation in a fluid

Fluids are studied in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3+1)-dimensional Korteweg–de Vries equation in a fluid. Bilinear form and N-soliton solutions are obtained, where N is a positive integer. Via the N-soliton solutions, we derive the higher-order breather solutions. We observe the interaction between the two perpendicular first-order breathers on the x-y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x-y$$\end{document} and x-z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x-z$$\end{document} planes and the interaction between the periodic line wave and the first-order breather on the y-z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y-z$$\end{document} plane, where x, y and z are the independent variables in the equation. We discuss the effects of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}, γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} and δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} on the amplitude of the second-order breather, where α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}, γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} and δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} are the constant coefficients in the equation: Amplitude of the second-order breather decreases as α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} increases; amplitude of the second-order breather increases as β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} increases; amplitude of the second-order breather keeps invariant as γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} or δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} increases. Via the N-soliton solutions, hybrid solutions comprising the breathers and solitons are derived. Based on the Riemann theta function, we obtain the periodic-wave solutions, and find that the periodic-wave solutions approach to the one-soliton solutions under a limiting condition.

However, to our knowledge, bilinear form, Nsoliton, breather, hybrid and periodic-wave solutions for Eq. (1) have not been considered, where N is a positive integer. In Sect. 2, bilinear form and N -soliton solutions for Eq. (1) will be studied. In Sect. 3, the higherorder breather solutions for Eq. (1) will be obtained. In Sect. 4, hybrid solutions comprising the solitons and breathers for Eq. (1) will be derived. In Sect. 5, periodic-wave solutions and their asymptotic behaviors for Eq. (1) will be investigated. Our conclusions will be given in Sect. 6.

Bilinear form and N-soliton solutions for Eq. (1)
Via the variable transformation [41] Eq. (1) can be converted into the following form: where f is a real differentiable function of x, y, z and t. Equation (6) can be integrated with respect to x, with the integral constant equal to zero, and the bilinear form for Eq. (1) is derived as where D x , D y , D z and D t are the bilinear differential operators defined by [29] D l with F being a differentiable function of x, y, z and t, G being a differentiable function of the formal variable x , y , z and t , while l, m, h and p being the non-negative integers.

The higher-order breather solutions for Eq. (1)
In this section, motivated by Ref.
[47], we construct the higher-order breather solutions for Eq. (1) with certain values of the parameters a ı 's, b ı 's, c ı 's and ξ 0 ı 's in N -Soliton Solutions (9) with N being an even integer. We assume that where a r 1 's, a r 2 's, b r 1 's, b r 2 's, c r 1 's, c r 2 's, ξ 0 r 1 's and ξ 0 r 2 's are all the real constants, * represents the complex conjugate, T is a positive integer, r = 1, 2, · · · , T and i 2 = −1. The T th-order breather solutions for Eq. (1) are given as where Taking T = 1 into Solutions (11), the first-order breather solutions can be derived as where The periods of the first-order breathers are 2π a 12 in the x direction, 2π a 11 b 12 +a 12 b 11 in the y direction and Fig. 1 The second-order breather via Solutions (11) a 11 c 12 +a 12 c 11 in the z direction. The locations of the first-order breathers are related to ξ 11 . When a 12 = b 12 = c 12 = ξ 0 12 = 0 in Solutions (12), the first-order breather solutions can be degenerated to the one-soliton solutions for Eq. (1).
When supposing T = 2 in Solutions (11), we can obtain the second-order breather solutions for Eq. (1). The second-order breathers depict the interactions between the two first-order breathers. As discussed on the first-order breathers, we can construct the second-order breathers comprising different first-order breathers. For example, the second-order breathers consisting of the two perpendicular first-order breathers can be constructed on the x − y plane under the condi- The same as Fig. 1  We observe the interaction between the two perpendicular first-order breathers on the x − y and x − z planes, as presented in Figs. 1a and b. In Fig. 1c, since a 11 = 0, a 11 b 11 − a 12 b 12 = 0 and a 11 c 11 − a 12 c 12 = 0, ξ 11 is not related to y and z. One of the first-order breathers is reduced into the periodic line wave. When t = − 1 3 , we observe the first-order breather, as shown in Fig. 1c 1 . As t goes by, we can find that the periodic line wave appears and interacts with the first-order breather, as shown in Fig. 1c 2 . When t = 1 3 , we find that the amplitude of the periodic line wave decreases and the amplitude of the first-order breather keeps unchanged, as shown in Fig. 1c 3 .
Furthermore, we investigate the influence of the coefficients in Eq. (1) on the amplitudes of the secondorder breathers. Comparing Fig. 2 with Fig. 1, when α increases, we find that the amplitude of the secondorder breather decreases. Comparing Fig. 3 with Fig. 1, when β increases, we observe that the amplitude of the second-order breather increases. Comparing

Hybrid solutions comprising the solitons and breathers for Eq. (1)
Hybrid solutions consisting of the solitons and breathers for Eq. (1) are derived with the following conditions in Solutions (9): where and are the positive integers, a s1 's, b s1 's, c s1 's and ξ 0 s1 's are the real constants. Fig. 4 The same as Fig. 1 except that γ = 10 For example, the hybrid solutions comprising the first-order breather and one kink-type soliton are obtained from Solutions (9) with the parameters as Under the condition a 11 a 31 = a 11 b 11 −a 12 b 12 we observe that the first-order breather is parallel to the one kink-type soliton, as shown in Fig. 7. When t = 0, the first-order breather interacts with the kink-type soliton. After the interaction, the first-order breather and the kink-type soliton keep their velocities and shapes unchanged, meaning that the interaction is elastic. When N = 4, = 1 and = 2 via Conditions (13), the interaction among the first-order breather and two kink-type solitons is shown in Fig. 8.
According to the above discussion, we find that Periodic-Wave Solutions (22) approach to the onesoliton solutions with the limiting condition → 0.

Conclusions
Fluids have been studied in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we have studied a (3+1)-dimensional KdV equation in a fluid, i.e., Eq. (1). Via Transformation (5) and the bilinear method, Bilinear Form (7) and N -Soliton Solutions (9) for Eq. (1) have been obtained, where N is a positive integer. Based on N -Soliton Solutions (9), we have obtained the T th-order breather solutions for Eq. (1), i.e., Solutions (11), where T is a positive integer. Hybrid solutions comprising the solitons and breathers for Eq. (1) have been constructed via N -Soliton Solutions (9) under Conditions (13). Via the Riemann theta function, Periodic-Wave Solutions (22) for Eq. (1) have been obtained. Via Solutions (11), we have observed the interaction between the two perpendicular first-order breathers on the x − y and x − z planes and the interaction between the periodic line wave and the first-order breather on the y − z plane, as shown in Fig. 1. Furthermore, we have discussed the effects of α, β, γ and δ on the amplitude of the second-order breather, where α, β, γ and δ are the constant coefficients in Eq. (1): Amplitude of the second-order breather decreases as α increases, as shown in Fig. 2; amplitude of the second-order breather increases as β increases, as shown in Fig. 3; amplitude of the second-order breather keeps invariant as γ or δ increases, as shown in Figs. 4 and 5. We have observed the interaction between the two parallel firstorder breathers, as shown in Fig. 6. We have seen that the first-order breather interacts with one kink-type soliton and the interaction is elastic, as shown in Fig. 7. Interaction among the first-order breather and two kinktype solitons has been shown in Fig. 8. With the discussion of the asymptotic behaviors of Periodic-Wave Solutions (22), we have found that under a certain limit process, Periodic-Wave Solutions (22) approach to the one-soliton solutions.