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Rogue waves, breather waves and solitary waves for a (3+1)-dimensional nonlinear evolution equation
Applied Mathematics Letters 97 (2019) 6–13
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Rogue waves, breather waves and solitary waves for a (3+1)-dimensional nonlinear evolution equation Jingjing Xie, Xiao Yang ∗ School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, PR China
article
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Article history: Received 22 March 2019 Received in revised form 5 May 2019 Accepted 5 May 2019 Available online 14 May 2019
abstract Based on the Hirota bilinear method, three types of exact solutions of a (3+1)dimensional nonlinear evolution equation are obtained, which contain rogue waves, breather waves and solitary waves. Moreover, the dynamical behaviors of all types of waves are discussed with graphic analysis. © 2019 Elsevier Ltd. All rights reserved.
Keywords: (3+1)-dimensional nonlinear evolution equation Bilinear form Rogue waves Breather waves Solitary waves
1. Introduction Lots of natural phenomena can be modeled by nonlinear evolution equations, exact solutions for such equations are helpful in the cognition and understanding of these phenomena. So finding exact solutions for nonlinear evolution equations is an indispensable item of related studies. Now, with the development of nonlinear theory, many effective methods to seek exact solutions have been proposed and developed, such as Hirota bilinear method [1], F-expansion method [2], tanh method [3,4], three-wave method [5], homogeneous balance method [6], exp-function method [7], Darboux and B¨acklund transformation method [8,9], inverse scattering method [10], the sine–cosine method [11] and so on [12,13]. Based on the methods, various types of exact solutions are provided, including soliton solutions, trigonometric function solutions and traveling waves. More recently, rogue waves, as a special type of exact solutions, have become the most attractive and important area in experiential observation and theoretical analysis. Rogue waves are also known as freak waves, monster episodic waves, killer waves, extreme waves and abnormal waves, which can be found in different fields as optical systems [14], superfluids [15], ocean [16], capillary flow [17], atmosphere [18], finance [19], etc. [20–23]. ∗
Corresponding author. E-mail address: [email protected] (X. Yang).
https://doi.org/10.1016/j.aml.2019.05.005 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
J. Xie and X. Yang / Applied Mathematics Letters 97 (2019) 6–13
7
In the present study, we focus on exact solutions for a (3+1)-dimensional nonlinear evolution equation 3uxz − 2uty − uxxxy + 2ux uy + 2uuxy + 2(ux ∂x−1 uy )x = 0,
(1.1)
which was first introduced in the discussion of some multidimensional nonlinear evolution equations [24]. Since its birth, Eq. (1.1) has attracted wide publicity [25–29]: N-soliton solution, resonant solution and complexion solution were obtained; positons, negatons, solitons and complexitons and their interaction solutions were given; N-soliton solution and its Wronskian form were constructed; bilinear representation, B¨ acklund transformation and exact solutions were investigated. All these interesting results have aroused our great concern, in the following, rogue waves, breather waves and solitary waves for the equation will be explored. The exploration is owing to Hirota bilinear method, based on which two bilinear forms of Eq. (1.1) are put forward. one leads to rogue waves and breather waves of the equation by using the homoclinic breather limit technique [30]. The other arrived at one-soliton, two-soliton and N-soliton solutions of the equation. The specific process can be seen in Sections 2 and 3, respectively. Besides, the dynamical behaviors of all types of waves are analyzed with graphic analysis. 2. Rogue waves and breather waves In this section, to construct rogue waves of Eq. (1.1), we give the related breather waves first. For this purpose, let us set ξ = x + y, which gives ∂ξ = ∂x = ∂y . Then Eq. (1.1) is rewritten as 3uξ,z − (2ut + uξξξ − 2uuξ )ξ + 2(uξ ∂ξ−1 uξ )ξ = 0.
(2.1)
By integrating it with respect to ξ, we have 3uz − 2ut − uξξξ + 4uuξ = 0.
(2.2)
Further, let η = ξ + t, then ∂η = ∂ξ = ∂t , thus Eq. (2.2) is turned into 3uz − 2uη − uηηη + 4uuη = 0.
(2.3)
Now, by considering that Eq. (1.1) has an equilibrium solution u0 , which is an arbitrary constant, we introduce the transformation u = u0 − 3(lnf )ηη . (2.4) Then by putting it into Eq. (2.3), we get − 9(lnf )ηη,z + 6(lnf )ηηη + 3(lnf )ηηηηη − 12(u0 − 3(lnf )ηη )(lnf )ηηη = 0.
(2.5)
Similarly, by integrating Eq. (2.5) with respect to η, we obtain − 3(lnf )η,z + 2(lnf )ηη + (lnf )ηηηη − 4u0 (lnf )ηη + 6((lnf )ηη )2 = 0,
(2.6)
which implies that Eq. (1.1) can be converted into the following bilinear form (Dη4 − 3Dη Dz + (2 − 4u0 )Dη2 )f · f = 0,
(2.7)
where Dx , Dy , Dz , Dt , Dη are Hirota’s bilinear operators [31] and defined by Dxn Dym Dzr Dts = (
∂ n ∂ ∂ m ∂ ∂ ∂ ∂ ∂ − ) ( − ) ( − ′ )r ( − ′ )s |x=x′ ,y=y′ ,z=z′ ,t=t′ , ∂x ∂x′ ∂y ∂y ′ ∂z ∂z ∂t ∂t
(2.8)
J. Xie and X. Yang / Applied Mathematics Letters 97 (2019) 6–13
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2 Dη4 f · f = 2(f f4η − 4fη f3η + 3f2η ),
Dη2 f · f = 2(f fηη − fη2 ),
Dη Dz f · f = 2(fη,z f − fz fη ).
For getting breather waves of Eq. (1.1), in light of the homoclinic breather limit method, we should seek the solution of Eq. (2.7) in the form f = e−p1 (η−az) + δ1 cos(p2 (η + bz)) + δ2 ep1 (η−az) ,
(2.9)
where p1 , p2 , δ1 , δ2 , a, b are real constants to be determined later. Substituting (2.9) into (2.7), we get an algebraic equation on ep1 (η−az) and e−p1 (η−az) . Then by equating the coefficients of all powers of eip1 (η−az) , sin(p2 (η + bz))eip1 (η−az) and cos(p2 (η + bz))ep1 (η−az) (i = −1, 0, 1) to zero, we obtain a set of algebraic equations for p1 , p2 , δ1 , δ2 , a, b, which is given as ⎧ ⎪ 12p21 aδ2 + 3δ12 p22 b + (2 − 4u0 )(4p21 δ2 − p22 δ12 ) + 4δ12 p42 + 16δ2 p41 = 0, ⎪ ⎪ ⎪ ⎪ 3 3 ⎪ ⎨ 3p1 p2 bδ1 − 3p1 p2 aδ1 − 2(2 − 4u0 )p1 p2 δ1 + 4p1 p2 δ1 − 4p1 p2 δ1 = 0, (2.10) 3p1 p2 aδ1 δ2 − 3p1 p2 bδ1 δ2 + 2(2 − 4u0 )p1 p2 δ1 δ2 + 4p31 p2 δ1 δ2 − 4p1 p32 δ1 δ2 = 0, ⎪ ⎪ ⎪ 2 2 2 2 4 4 2 2 ⎪ 3p2 bδ1 + 3p1 aδ1 + (2 − 4u0 )(p1 δ1 − p2 δ1 ) + p2 δ1 + p1 δ1 − 6δ1 p1 p2 = 0, ⎪ ⎪ ⎩ 3p21 aδ1 δ2 + 3p22 bδ1 δ2 + (2 − 4u0 )(p21 δ1 δ2 − p22 δ1 δ2 ) + p41 δ1 δ2 + p42 δ1 δ2 − 6p21 p22 δ1 δ2 = 0. Let p1 = p2 = p, then the above algebraic equations are solved by p2 = (
3a + 3b ), 4
b−a=
2(2 − 4u0 ) , 3
δ12 =
((−12a − 4(2 − 4u0 ) − 16p2 )δ2 ) , (3b − (2 − 4u0 ) + 4p2 )
(2.11)
where a, b, δ2 are real constants. Based on (2.11), solution (2.9) can be rewritten as f = e−p(η−az) + δ1 cos(p(η + bz)) + δ2 ep(η−az) , √ √ = 2 −δ2 cosh(p(η − az) + ln( −δ2 )) + δ1 cos(p(η + bz)),
(2.12)
√ √ 2 3a+3b 0 )−16p )δ2 with δ1 = ± (−12a−4(2−4u , p = ± 4 , a, b ∈ R. Substituting (2.11) into (2.12), we have the 3b−(2−4u0 )+4p2 following solutions according to the symbols + and −, respectively √ f1 (η, z) = 2 −δ2 cosh(p(η − (b − √ f2 (η, z) = 2 −δ2 cosh(p(η − (b −
√ 2 (2 − 4u0 ))z) + ln( −δ2 )) + g1 cos(p(η + bz)), 3 √ 2 (2 − 4u0 ))z) + ln( −δ2 )) − g1 cos(p(η + bz)), 3
(2.13) (2.14)
√ 2 0 )−16p )δ2 here g1 = (−12a−4(2−4u . By putting (2.13) and (2.14) into (2.4), solutions of (2.6) are gained and 3b−(2−4u0 )+4p2 expressed as √ √ 12δ2 p2 + 3g12 p2 − 12 −δ2 g1 p2 sinh(p(η − (b − 23 (2 − 4u0 ))z)) + ln( −δ2 ) sin(p(η + bz)) √ √ u1 = u0 + , (2.15) (2 −δ2 cosh(p(η − (b − 23 (2 − 4u0 ))z) + ln( −δ2 )) + g1 cos(p(η + bz)))2 √ √ 12δ2 p2 + 3g12 p2 + 12 −δ2 g1 p2 sinh(p(η − (b − 23 (2 − 4u0 ))z) + ln( −δ2 )) sin(p(η + bz)) √ √ u2 = u0 + . (2.16) (2 −δ2 cosh(p(η − (b − 23 (2 − 4u0 ))z) + ln( −δ2 )) − g1 cos(p(η + bz)))2 u1 , u2 are two homoclinic breather wave solutions, and when t → ±∞, they tend to a fixed point u0 . Since u1 and u2 are almost the same, in the rest of the section, only u2 will be taken into account. √ Let us begin with δ2 = −1, then we have ln( −δ2 ) = 0, thus u2 is simplified into u2 = u0 +
−12p2 + 3R2 p2 + 12Rp2 sinh(p(η − (b − 23 (2 − 4u0 ))z)) sin(p(η + bz)) , (2 cosh(p(η − (b − 23 (2 − 4u0 ))z)) − R cos(p(η + bz)))2
(2.17)
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Fig. 1. (Color online) Breather wave solution (2.17) for Eq. (1.1) by choosing suitable parameters as u0 = − 14 , p2 = 3, b = 3, a = 1, x = y = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
Fig. 2. (Color online) Breather wave solution (2.17) for Eq. (1.1) by choosing suitable parameters as u0 = 0, p2 = 41 , b = 56 , a = − 12 , x = y = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
with R = p = 0:
√
12a+4(2−4u0 )+16p2 . 3b−(2−4u0 )+4p2
In consideration of Eq. (2.17), along with the following Taylor expansions at
sin(p(η + bz)) = p(η + bz) + O(p2 ), sinh(p(η − az)) = p(η − az) + O(p2 ), 1 1 cos(p(η + bz)) = 1 − p2 (η + bz)2 + O(p3 ), cosh(p(η − az)) = 1 + p2 (η − az)2 + O(p3 ), 2 2 we obtain rogue wave solutions of Eq. (1.1) u = u0 +
24(x + y + t − (b − 23 (2 − 4u0 ))z)(x + y + t + bz) , ((x + y + t + bz)2 + (x + y + t − (b − 32 (2 − 4u0 ))z)2 )2
(2.18)
where b is an arbitrary real constant. Next, based on solutions (2.17) and (2.18), breather waves and rogue waves are presented in Figs. 1–2 and Figs. 3–4, respectively. From the figures, we can see that rogue wave is a special behavior of breather wave when choosing the same parameters.
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J. Xie and X. Yang / Applied Mathematics Letters 97 (2019) 6–13
Fig. 3. (Color online) Breather wave solution (2.18) for Eq. (1.1) by choosing suitable parameters as u0 = − 14 , p2 = 3, b = 3, a = 1, x = y = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
Fig. 4. (Color online) Breather wave solution (2.18) for Eq. (1.1) via choosing suitable parameters as u0 = 0, p2 = 14 , b = 65 , a = − 12 , x = y = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
3. Solitary waves In this section, for giving solitary waves, we select the transformation u = −3(ln f )xx , accordingly Eq. (1.1) is converted into (2Dy Dt + Dx3 Dy − 3Dx Dz )f · f = 0.
(3.1)
f (x, y, z, t) = 1 + f (1) ε + f (2) ε2 + f (3) ε3 + f (4) ε4 + · · ·.
(3.2)
Let us expand f in the form of ε
Then by substituting (3.2) into (3.1), and comparing the coefficients of εn (n = 1, 2, 3, . . .), we have a series of basic formula. And the first three are (1)
(1) (1) 2fyt + fxxxy − 3fxz = 0,
(3.3)
(2) (2) (2) 2(2fyt + fxxxy − 3fxz ) = −(2Dy Dt + Dx3 Dy − 3Dx Dz )f (1) · f (1) , (3) (3) (3) 2fyt + fxxxy − 3fxz = −(2Dy Dt + Dx3 Dy − 3Dx Dz )f (1) · f (2) .
(3.4) (3.5)
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Direct calculations imply that Eq. (3.3) has the following solution f (1) = exp η1 ,
η1 = κ1 x + µ1 y + ν1 z + ω1 t + τ1 ,
ω1 =
3 1 3 κ1 ν1 µ−1 1 − κ1 . 2 2
(3.6)
By taking solution f (1) into (3.4), we have (2)
(2) (2) 2fyt + fxxxy − 3fxz = 0.
(3.7)
When choosing a special solution of Eq. (3.7) as f (2) = 0, Eq. (3.5) gives (3)
(3) (3) 2fyt + fxxxy − 3fxz = 0,
(3.8)
which is similar to (3.7) in the form. By continuing the above procedure, we have f (3) = 0, f (4) = 0, f (5) = 0, f (6) = 0, f (7) = 0, · · ·. Then when ε = 1, by Eq. (3.2), we have f = 1 + exp(η1 ). Thus the one-soliton solution of Eq. (1.1) is given as follows u = −3(ln(1 + exp(η1 )))xx ,
(3.9)
with η1 = κ1 x + µ1 y + ν1 z + ω1 t + τ1 . Likewise, we know that Eq. (3.3) can be solved by the following f (1) f (1) = exp(η1 ) + exp(η2 ),
ηi = κi x + µi y + νi z + ωi t + τi (i = 1, 2),
ωi =
1 3 3 κi νi µ−1 i − κi . 2 2
(3.10)
By substituting (3.10) into (3.4), we have f (2) = exp(η1 + η2 + A12 ) (orf (2) = a12 exp(η1 + η2 )),
(3.11)
1 3 where ηi = κi x + µi y + νi z + ωi t + τi , ωi = 23 κi νi µ−1 i − 2 κi (i = 1, 2) and
exp(A12 ) = −
2(ω1 − ω2 )(µ1 − µ2 ) + (κ1 − κ2 )3 (µ1 − µ2 ) − 3(κ1 − κ2 )(ν1 − ν2 ) . 2(ω1 + ω2 )(µ1 + µ2 ) + (κ1 + κ2 )3 (µ1 + µ2 ) − 3(κ1 + κ2 )(ν1 + ν2 )
(3.12)
Then in the similar way of dealing with the one-soliton solution, let ε = 1, we get f (2) = 1 + exp(η1 ) + exp(η2 ) + exp(η1 + η2 + A12 ).
(3.13)
Thus the two-soliton solution of Eq. (1.1) is expressed in the following form u = −3(ln(1 + exp(η1 ) + exp(η2 ) + exp(η1 + η2 + A12 )))xx .
(3.14)
Furthermore, the N-soliton solution of the (3+1)-dimensional equation (1.1) is presented as u = −3[ln(f )]xx , N ∑ f = Σρ=0,1 exp( ρi η i + j=1
∑
ρi ρj A12 ),
1≤i
(3.15) 3 1 3 κi νi µ−1 i − κi , 2 2 2(ωi − ωj )(µi − µj ) + (κi − κj )3 (µi − µj ) − 3(κi − κj )(νi − νj ) exp(Aij ) = − , 2(ωi + ωj )(µi + µj ) + (κi + κj )3 (µi + µj ) − 3(κi + κj )(νi + νj ) ∑ where κi , µi , νi , τi (i = 1, 2, . . . , N ) are arbitrary constants and ρ=0,1 is the sum of all possible combinations of ρi , ρj = 0, 1(i, j = 1, 2, . . . , N ). In the discussion that follows, we will show the propagation situations of the one-soliton wave solution (3.9) and two-soliton wave solution (3.14) by Figs. 5 and 6, respectively. ηi = κi x + µi y + νi z + ωi t + τi ,
ωi =
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Fig. 5. (Color online) One-soliton solution (3.9) for Eq. (1.1) via choosing suitable parameters as κ1 = 1, µ1 = 3, ν1 = 2, τ1 = 0, ω1 = 12 , y = z = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
Fig. 6. (Color online) Two-soliton solution (3.14) for Eq. (1.1) via choosing suitable parameters as κ1 = 1, µ1 = 3, ν1 = 2, τ1 = 0, ω1 = 21 , κ2 = −2, ν2 = −16, µ2 = 48, τ2 = 0, ω2 = 5, y = z = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
Acknowledgment This work was supported by National Natural Science Foundation of China (No. 11331008). References [1] R. Hirota, Direct Method in Soliton Theory, Cambridge University Press, New York, 2004. [2] A. Filiz, M. Ekici, A. Sonmezoglu, F-Expansion method and new exact solutions of the Sch¨ odinger-KdV equation, Sci. World J. (2014) 534063. [3] A.M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh-coth method, Appl. Math. Comput. 190 (2007) 633–640. [4] E.M.E. Zayed, H.M.A. Rahman, The extended tanh-method for finding traveling wave solutions of nonlinear evolution equations, Appl. Math. E-Notes 10 (2010) 235–245. [5] N. Liu, Y.S. Liu, New multi-soliton solution solutions of a (3+1)-dimensional nonlinear evolution equation, Comput. Math. Appl. 71 (2016) 1645–1654. [6] E.G. Fan, A note on the homogenous balance method, Phys. Lett. A 246 (1996) 403–406. [7] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals 30 (2006) 700–708. [8] X.M. Li, A.H. Chen, Darboux transformation and multi-soliton solutions of Boussinesq-Burgers equation, Phys. Lett. A 432 (2005) 413–420. [9] X.Y. Wen, N-fold darboux transformation and soliton solutions for Toda lattice equation, Rep. Math. Phys. 68 (2011) 211–223.
J. Xie and X. Yang / Applied Mathematics Letters 97 (2019) 6–13
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[10] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, 1991. [11] A.M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math. Comput. Modelling 40 (2004) 499–508. [12] C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett. 19 (1967) 1095–1097. [13] R. Hirota, S. Takeno(Ed.), Fundamental properties of the binary operators in soliton theory and their generalization, in: Dynamical Problem in Soliton Systems, in: Springer Series in Synergetiecs, 30, Springer, Berlin, 1985. [14] D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Optical rogue waves, Nature 450 (2007) 1054. [15] A.N. Ganshin, V.B. Efimov, G.V. Kolmakov, et al., Observation of an inverse energy Cascade in developed acoustic turbulence in superfluid helium, Phys. Rev. Lett. 101 (2008) 065303. [16] C. Kharif, E. Pelinovsky, A. Slunyaey, Rogue Waves in the Ocean, Observation, Theories and Modeling, Springer, New York, 2009. [17] M. Shats, H. Punzmann, H. Xia, Capillary rogue waves, Phys. Rev. Lett. 104 (2010) 104503. [18] L. Stenflo, M. Marklund, Rogue waves in the atmosphere, J. Plasma Phys. 76 (2010) 293–295. [19] Z.Y. Yan, Vector financial rogue waves, Phys. Lett. A 375 (2011) 4274. [20] D.H. Peregrine, Water waves, J. Aust. Math. Soc. Ser. B 25 (1983) 16. [21] X.B. Wang, S.F. Tian, H. Yan, T.T. zhang, On the solitary waves, breather waves and rogue waves to a generalized (3 + 1)-dimensional KadomtsevCPetviashvili equation, Comput. Math. Appl. 74 (2017) 556–563. [22] X.B. Wang, S.F. Tian, C.Y. Qin, T.T. zhang, Characteristics of the breather, rogue waves and solitary waves in a generalized (2+1)-dimensional Boussinesq equation, Europhys. Lett. 115 (2016) 10002. [23] X.B. Wang, S.F. Tian, C.Y. Qin, T.T. Zhang, Dynamics of the breathers, rogue waves and solitary waves in the (2+1)-dimensional Ito equation, Appl. Math. Lett. 68 (2017) 40–47. [24] X.G. Geng, Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations, J. Phys. A. Math. Gen. 36 (2003) 2289–2304. [25] Q.L. Zha, Z.B. Li, Darboux transformation and various solutions for a nonlinear evolution equation in (3+1)-dimensions, Modern Phys. Lett. B 22 (2008) 2945–2966. [26] Q.L. Zha, Z.B. Li, Positon, negaton, soliton and complexiton solutions to a four-dimensional evolution equation, Moderm Phys. Lett. B 23 (2009) 2971–2991. [27] X.G. Geng, Y.L. Ma, N-soliton solution and its wronskian form of a (3+1)-dimensional nonlinear evolution equation, Phys. Lett. A 369 (2007) 285–289. [28] J.P. Wu, A B¨ acklund transformation and explicit solutions of a (3+1)-dimensional soliton equation, 25, (12), 2008, 4192. [29] Q.L. Zha, Rogue waves and rational solutions of a (3+1)-dimensional nonlinear evolution equation, Phys. Lett. A 377 (2013) 3021–3026. [30] Z. Dai, S. Li, D. Li, A. Zhu, Periodic bifurcation and soliton deflexion for Kadomtsev–Petviashvili equation, Chin. Phys. Lett. 24 (2007) 1429–1432. [31] R. Hirota, Exact solution of the Korteweg–de Vries equation for multiple collisions, Phys. Rev. Lett. 27 (1971) 1192–1194.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Rogue waves, breather waves and solitary waves for a (3+1)-dimensional nonlinear evolution equation Jingjing Xie, Xiao Yang ∗ School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, PR China
article
info
Article history: Received 22 March 2019 Received in revised form 5 May 2019 Accepted 5 May 2019 Available online 14 May 2019
abstract Based on the Hirota bilinear method, three types of exact solutions of a (3+1)dimensional nonlinear evolution equation are obtained, which contain rogue waves, breather waves and solitary waves. Moreover, the dynamical behaviors of all types of waves are discussed with graphic analysis. © 2019 Elsevier Ltd. All rights reserved.
Keywords: (3+1)-dimensional nonlinear evolution equation Bilinear form Rogue waves Breather waves Solitary waves
1. Introduction Lots of natural phenomena can be modeled by nonlinear evolution equations, exact solutions for such equations are helpful in the cognition and understanding of these phenomena. So finding exact solutions for nonlinear evolution equations is an indispensable item of related studies. Now, with the development of nonlinear theory, many effective methods to seek exact solutions have been proposed and developed, such as Hirota bilinear method [1], F-expansion method [2], tanh method [3,4], three-wave method [5], homogeneous balance method [6], exp-function method [7], Darboux and B¨acklund transformation method [8,9], inverse scattering method [10], the sine–cosine method [11] and so on [12,13]. Based on the methods, various types of exact solutions are provided, including soliton solutions, trigonometric function solutions and traveling waves. More recently, rogue waves, as a special type of exact solutions, have become the most attractive and important area in experiential observation and theoretical analysis. Rogue waves are also known as freak waves, monster episodic waves, killer waves, extreme waves and abnormal waves, which can be found in different fields as optical systems [14], superfluids [15], ocean [16], capillary flow [17], atmosphere [18], finance [19], etc. [20–23]. ∗
Corresponding author. E-mail address: [email protected] (X. Yang).
https://doi.org/10.1016/j.aml.2019.05.005 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
J. Xie and X. Yang / Applied Mathematics Letters 97 (2019) 6–13
7
In the present study, we focus on exact solutions for a (3+1)-dimensional nonlinear evolution equation 3uxz − 2uty − uxxxy + 2ux uy + 2uuxy + 2(ux ∂x−1 uy )x = 0,
(1.1)
which was first introduced in the discussion of some multidimensional nonlinear evolution equations [24]. Since its birth, Eq. (1.1) has attracted wide publicity [25–29]: N-soliton solution, resonant solution and complexion solution were obtained; positons, negatons, solitons and complexitons and their interaction solutions were given; N-soliton solution and its Wronskian form were constructed; bilinear representation, B¨ acklund transformation and exact solutions were investigated. All these interesting results have aroused our great concern, in the following, rogue waves, breather waves and solitary waves for the equation will be explored. The exploration is owing to Hirota bilinear method, based on which two bilinear forms of Eq. (1.1) are put forward. one leads to rogue waves and breather waves of the equation by using the homoclinic breather limit technique [30]. The other arrived at one-soliton, two-soliton and N-soliton solutions of the equation. The specific process can be seen in Sections 2 and 3, respectively. Besides, the dynamical behaviors of all types of waves are analyzed with graphic analysis. 2. Rogue waves and breather waves In this section, to construct rogue waves of Eq. (1.1), we give the related breather waves first. For this purpose, let us set ξ = x + y, which gives ∂ξ = ∂x = ∂y . Then Eq. (1.1) is rewritten as 3uξ,z − (2ut + uξξξ − 2uuξ )ξ + 2(uξ ∂ξ−1 uξ )ξ = 0.
(2.1)
By integrating it with respect to ξ, we have 3uz − 2ut − uξξξ + 4uuξ = 0.
(2.2)
Further, let η = ξ + t, then ∂η = ∂ξ = ∂t , thus Eq. (2.2) is turned into 3uz − 2uη − uηηη + 4uuη = 0.
(2.3)
Now, by considering that Eq. (1.1) has an equilibrium solution u0 , which is an arbitrary constant, we introduce the transformation u = u0 − 3(lnf )ηη . (2.4) Then by putting it into Eq. (2.3), we get − 9(lnf )ηη,z + 6(lnf )ηηη + 3(lnf )ηηηηη − 12(u0 − 3(lnf )ηη )(lnf )ηηη = 0.
(2.5)
Similarly, by integrating Eq. (2.5) with respect to η, we obtain − 3(lnf )η,z + 2(lnf )ηη + (lnf )ηηηη − 4u0 (lnf )ηη + 6((lnf )ηη )2 = 0,
(2.6)
which implies that Eq. (1.1) can be converted into the following bilinear form (Dη4 − 3Dη Dz + (2 − 4u0 )Dη2 )f · f = 0,
(2.7)
where Dx , Dy , Dz , Dt , Dη are Hirota’s bilinear operators [31] and defined by Dxn Dym Dzr Dts = (
∂ n ∂ ∂ m ∂ ∂ ∂ ∂ ∂ − ) ( − ) ( − ′ )r ( − ′ )s |x=x′ ,y=y′ ,z=z′ ,t=t′ , ∂x ∂x′ ∂y ∂y ′ ∂z ∂z ∂t ∂t
(2.8)
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2 Dη4 f · f = 2(f f4η − 4fη f3η + 3f2η ),
Dη2 f · f = 2(f fηη − fη2 ),
Dη Dz f · f = 2(fη,z f − fz fη ).
For getting breather waves of Eq. (1.1), in light of the homoclinic breather limit method, we should seek the solution of Eq. (2.7) in the form f = e−p1 (η−az) + δ1 cos(p2 (η + bz)) + δ2 ep1 (η−az) ,
(2.9)
where p1 , p2 , δ1 , δ2 , a, b are real constants to be determined later. Substituting (2.9) into (2.7), we get an algebraic equation on ep1 (η−az) and e−p1 (η−az) . Then by equating the coefficients of all powers of eip1 (η−az) , sin(p2 (η + bz))eip1 (η−az) and cos(p2 (η + bz))ep1 (η−az) (i = −1, 0, 1) to zero, we obtain a set of algebraic equations for p1 , p2 , δ1 , δ2 , a, b, which is given as ⎧ ⎪ 12p21 aδ2 + 3δ12 p22 b + (2 − 4u0 )(4p21 δ2 − p22 δ12 ) + 4δ12 p42 + 16δ2 p41 = 0, ⎪ ⎪ ⎪ ⎪ 3 3 ⎪ ⎨ 3p1 p2 bδ1 − 3p1 p2 aδ1 − 2(2 − 4u0 )p1 p2 δ1 + 4p1 p2 δ1 − 4p1 p2 δ1 = 0, (2.10) 3p1 p2 aδ1 δ2 − 3p1 p2 bδ1 δ2 + 2(2 − 4u0 )p1 p2 δ1 δ2 + 4p31 p2 δ1 δ2 − 4p1 p32 δ1 δ2 = 0, ⎪ ⎪ ⎪ 2 2 2 2 4 4 2 2 ⎪ 3p2 bδ1 + 3p1 aδ1 + (2 − 4u0 )(p1 δ1 − p2 δ1 ) + p2 δ1 + p1 δ1 − 6δ1 p1 p2 = 0, ⎪ ⎪ ⎩ 3p21 aδ1 δ2 + 3p22 bδ1 δ2 + (2 − 4u0 )(p21 δ1 δ2 − p22 δ1 δ2 ) + p41 δ1 δ2 + p42 δ1 δ2 − 6p21 p22 δ1 δ2 = 0. Let p1 = p2 = p, then the above algebraic equations are solved by p2 = (
3a + 3b ), 4
b−a=
2(2 − 4u0 ) , 3
δ12 =
((−12a − 4(2 − 4u0 ) − 16p2 )δ2 ) , (3b − (2 − 4u0 ) + 4p2 )
(2.11)
where a, b, δ2 are real constants. Based on (2.11), solution (2.9) can be rewritten as f = e−p(η−az) + δ1 cos(p(η + bz)) + δ2 ep(η−az) , √ √ = 2 −δ2 cosh(p(η − az) + ln( −δ2 )) + δ1 cos(p(η + bz)),
(2.12)
√ √ 2 3a+3b 0 )−16p )δ2 with δ1 = ± (−12a−4(2−4u , p = ± 4 , a, b ∈ R. Substituting (2.11) into (2.12), we have the 3b−(2−4u0 )+4p2 following solutions according to the symbols + and −, respectively √ f1 (η, z) = 2 −δ2 cosh(p(η − (b − √ f2 (η, z) = 2 −δ2 cosh(p(η − (b −
√ 2 (2 − 4u0 ))z) + ln( −δ2 )) + g1 cos(p(η + bz)), 3 √ 2 (2 − 4u0 ))z) + ln( −δ2 )) − g1 cos(p(η + bz)), 3
(2.13) (2.14)
√ 2 0 )−16p )δ2 here g1 = (−12a−4(2−4u . By putting (2.13) and (2.14) into (2.4), solutions of (2.6) are gained and 3b−(2−4u0 )+4p2 expressed as √ √ 12δ2 p2 + 3g12 p2 − 12 −δ2 g1 p2 sinh(p(η − (b − 23 (2 − 4u0 ))z)) + ln( −δ2 ) sin(p(η + bz)) √ √ u1 = u0 + , (2.15) (2 −δ2 cosh(p(η − (b − 23 (2 − 4u0 ))z) + ln( −δ2 )) + g1 cos(p(η + bz)))2 √ √ 12δ2 p2 + 3g12 p2 + 12 −δ2 g1 p2 sinh(p(η − (b − 23 (2 − 4u0 ))z) + ln( −δ2 )) sin(p(η + bz)) √ √ u2 = u0 + . (2.16) (2 −δ2 cosh(p(η − (b − 23 (2 − 4u0 ))z) + ln( −δ2 )) − g1 cos(p(η + bz)))2 u1 , u2 are two homoclinic breather wave solutions, and when t → ±∞, they tend to a fixed point u0 . Since u1 and u2 are almost the same, in the rest of the section, only u2 will be taken into account. √ Let us begin with δ2 = −1, then we have ln( −δ2 ) = 0, thus u2 is simplified into u2 = u0 +
−12p2 + 3R2 p2 + 12Rp2 sinh(p(η − (b − 23 (2 − 4u0 ))z)) sin(p(η + bz)) , (2 cosh(p(η − (b − 23 (2 − 4u0 ))z)) − R cos(p(η + bz)))2
(2.17)
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Fig. 1. (Color online) Breather wave solution (2.17) for Eq. (1.1) by choosing suitable parameters as u0 = − 14 , p2 = 3, b = 3, a = 1, x = y = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
Fig. 2. (Color online) Breather wave solution (2.17) for Eq. (1.1) by choosing suitable parameters as u0 = 0, p2 = 41 , b = 56 , a = − 12 , x = y = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
with R = p = 0:
√
12a+4(2−4u0 )+16p2 . 3b−(2−4u0 )+4p2
In consideration of Eq. (2.17), along with the following Taylor expansions at
sin(p(η + bz)) = p(η + bz) + O(p2 ), sinh(p(η − az)) = p(η − az) + O(p2 ), 1 1 cos(p(η + bz)) = 1 − p2 (η + bz)2 + O(p3 ), cosh(p(η − az)) = 1 + p2 (η − az)2 + O(p3 ), 2 2 we obtain rogue wave solutions of Eq. (1.1) u = u0 +
24(x + y + t − (b − 23 (2 − 4u0 ))z)(x + y + t + bz) , ((x + y + t + bz)2 + (x + y + t − (b − 32 (2 − 4u0 ))z)2 )2
(2.18)
where b is an arbitrary real constant. Next, based on solutions (2.17) and (2.18), breather waves and rogue waves are presented in Figs. 1–2 and Figs. 3–4, respectively. From the figures, we can see that rogue wave is a special behavior of breather wave when choosing the same parameters.
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Fig. 3. (Color online) Breather wave solution (2.18) for Eq. (1.1) by choosing suitable parameters as u0 = − 14 , p2 = 3, b = 3, a = 1, x = y = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
Fig. 4. (Color online) Breather wave solution (2.18) for Eq. (1.1) via choosing suitable parameters as u0 = 0, p2 = 14 , b = 65 , a = − 12 , x = y = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
3. Solitary waves In this section, for giving solitary waves, we select the transformation u = −3(ln f )xx , accordingly Eq. (1.1) is converted into (2Dy Dt + Dx3 Dy − 3Dx Dz )f · f = 0.
(3.1)
f (x, y, z, t) = 1 + f (1) ε + f (2) ε2 + f (3) ε3 + f (4) ε4 + · · ·.
(3.2)
Let us expand f in the form of ε
Then by substituting (3.2) into (3.1), and comparing the coefficients of εn (n = 1, 2, 3, . . .), we have a series of basic formula. And the first three are (1)
(1) (1) 2fyt + fxxxy − 3fxz = 0,
(3.3)
(2) (2) (2) 2(2fyt + fxxxy − 3fxz ) = −(2Dy Dt + Dx3 Dy − 3Dx Dz )f (1) · f (1) , (3) (3) (3) 2fyt + fxxxy − 3fxz = −(2Dy Dt + Dx3 Dy − 3Dx Dz )f (1) · f (2) .
(3.4) (3.5)
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Direct calculations imply that Eq. (3.3) has the following solution f (1) = exp η1 ,
η1 = κ1 x + µ1 y + ν1 z + ω1 t + τ1 ,
ω1 =
3 1 3 κ1 ν1 µ−1 1 − κ1 . 2 2
(3.6)
By taking solution f (1) into (3.4), we have (2)
(2) (2) 2fyt + fxxxy − 3fxz = 0.
(3.7)
When choosing a special solution of Eq. (3.7) as f (2) = 0, Eq. (3.5) gives (3)
(3) (3) 2fyt + fxxxy − 3fxz = 0,
(3.8)
which is similar to (3.7) in the form. By continuing the above procedure, we have f (3) = 0, f (4) = 0, f (5) = 0, f (6) = 0, f (7) = 0, · · ·. Then when ε = 1, by Eq. (3.2), we have f = 1 + exp(η1 ). Thus the one-soliton solution of Eq. (1.1) is given as follows u = −3(ln(1 + exp(η1 )))xx ,
(3.9)
with η1 = κ1 x + µ1 y + ν1 z + ω1 t + τ1 . Likewise, we know that Eq. (3.3) can be solved by the following f (1) f (1) = exp(η1 ) + exp(η2 ),
ηi = κi x + µi y + νi z + ωi t + τi (i = 1, 2),
ωi =
1 3 3 κi νi µ−1 i − κi . 2 2
(3.10)
By substituting (3.10) into (3.4), we have f (2) = exp(η1 + η2 + A12 ) (orf (2) = a12 exp(η1 + η2 )),
(3.11)
1 3 where ηi = κi x + µi y + νi z + ωi t + τi , ωi = 23 κi νi µ−1 i − 2 κi (i = 1, 2) and
exp(A12 ) = −
2(ω1 − ω2 )(µ1 − µ2 ) + (κ1 − κ2 )3 (µ1 − µ2 ) − 3(κ1 − κ2 )(ν1 − ν2 ) . 2(ω1 + ω2 )(µ1 + µ2 ) + (κ1 + κ2 )3 (µ1 + µ2 ) − 3(κ1 + κ2 )(ν1 + ν2 )
(3.12)
Then in the similar way of dealing with the one-soliton solution, let ε = 1, we get f (2) = 1 + exp(η1 ) + exp(η2 ) + exp(η1 + η2 + A12 ).
(3.13)
Thus the two-soliton solution of Eq. (1.1) is expressed in the following form u = −3(ln(1 + exp(η1 ) + exp(η2 ) + exp(η1 + η2 + A12 )))xx .
(3.14)
Furthermore, the N-soliton solution of the (3+1)-dimensional equation (1.1) is presented as u = −3[ln(f )]xx , N ∑ f = Σρ=0,1 exp( ρi η i + j=1
∑
ρi ρj A12 ),
1≤i
(3.15) 3 1 3 κi νi µ−1 i − κi , 2 2 2(ωi − ωj )(µi − µj ) + (κi − κj )3 (µi − µj ) − 3(κi − κj )(νi − νj ) exp(Aij ) = − , 2(ωi + ωj )(µi + µj ) + (κi + κj )3 (µi + µj ) − 3(κi + κj )(νi + νj ) ∑ where κi , µi , νi , τi (i = 1, 2, . . . , N ) are arbitrary constants and ρ=0,1 is the sum of all possible combinations of ρi , ρj = 0, 1(i, j = 1, 2, . . . , N ). In the discussion that follows, we will show the propagation situations of the one-soliton wave solution (3.9) and two-soliton wave solution (3.14) by Figs. 5 and 6, respectively. ηi = κi x + µi y + νi z + ωi t + τi ,
ωi =
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Fig. 5. (Color online) One-soliton solution (3.9) for Eq. (1.1) via choosing suitable parameters as κ1 = 1, µ1 = 3, ν1 = 2, τ1 = 0, ω1 = 12 , y = z = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
Fig. 6. (Color online) Two-soliton solution (3.14) for Eq. (1.1) via choosing suitable parameters as κ1 = 1, µ1 = 3, ν1 = 2, τ1 = 0, ω1 = 21 , κ2 = −2, ν2 = −16, µ2 = 48, τ2 = 0, ω2 = 5, y = z = 0. (a) Perspective view of the wave. (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the t axis.
Acknowledgment This work was supported by National Natural Science Foundation of China (No. 11331008). References [1] R. Hirota, Direct Method in Soliton Theory, Cambridge University Press, New York, 2004. [2] A. Filiz, M. Ekici, A. Sonmezoglu, F-Expansion method and new exact solutions of the Sch¨ odinger-KdV equation, Sci. World J. (2014) 534063. [3] A.M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh-coth method, Appl. Math. Comput. 190 (2007) 633–640. [4] E.M.E. Zayed, H.M.A. Rahman, The extended tanh-method for finding traveling wave solutions of nonlinear evolution equations, Appl. Math. E-Notes 10 (2010) 235–245. [5] N. Liu, Y.S. Liu, New multi-soliton solution solutions of a (3+1)-dimensional nonlinear evolution equation, Comput. Math. Appl. 71 (2016) 1645–1654. [6] E.G. Fan, A note on the homogenous balance method, Phys. Lett. A 246 (1996) 403–406. [7] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals 30 (2006) 700–708. [8] X.M. Li, A.H. Chen, Darboux transformation and multi-soliton solutions of Boussinesq-Burgers equation, Phys. Lett. A 432 (2005) 413–420. [9] X.Y. Wen, N-fold darboux transformation and soliton solutions for Toda lattice equation, Rep. Math. Phys. 68 (2011) 211–223.
J. Xie and X. Yang / Applied Mathematics Letters 97 (2019) 6–13
13
[10] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, 1991. [11] A.M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math. Comput. Modelling 40 (2004) 499–508. [12] C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett. 19 (1967) 1095–1097. [13] R. Hirota, S. Takeno(Ed.), Fundamental properties of the binary operators in soliton theory and their generalization, in: Dynamical Problem in Soliton Systems, in: Springer Series in Synergetiecs, 30, Springer, Berlin, 1985. [14] D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Optical rogue waves, Nature 450 (2007) 1054. [15] A.N. Ganshin, V.B. Efimov, G.V. Kolmakov, et al., Observation of an inverse energy Cascade in developed acoustic turbulence in superfluid helium, Phys. Rev. Lett. 101 (2008) 065303. [16] C. Kharif, E. Pelinovsky, A. Slunyaey, Rogue Waves in the Ocean, Observation, Theories and Modeling, Springer, New York, 2009. [17] M. Shats, H. Punzmann, H. Xia, Capillary rogue waves, Phys. Rev. Lett. 104 (2010) 104503. [18] L. Stenflo, M. Marklund, Rogue waves in the atmosphere, J. Plasma Phys. 76 (2010) 293–295. [19] Z.Y. Yan, Vector financial rogue waves, Phys. Lett. A 375 (2011) 4274. [20] D.H. Peregrine, Water waves, J. Aust. Math. Soc. Ser. B 25 (1983) 16. [21] X.B. Wang, S.F. Tian, H. Yan, T.T. zhang, On the solitary waves, breather waves and rogue waves to a generalized (3 + 1)-dimensional KadomtsevCPetviashvili equation, Comput. Math. Appl. 74 (2017) 556–563. [22] X.B. Wang, S.F. Tian, C.Y. Qin, T.T. zhang, Characteristics of the breather, rogue waves and solitary waves in a generalized (2+1)-dimensional Boussinesq equation, Europhys. Lett. 115 (2016) 10002. [23] X.B. Wang, S.F. Tian, C.Y. Qin, T.T. Zhang, Dynamics of the breathers, rogue waves and solitary waves in the (2+1)-dimensional Ito equation, Appl. Math. Lett. 68 (2017) 40–47. [24] X.G. Geng, Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations, J. Phys. A. Math. Gen. 36 (2003) 2289–2304. [25] Q.L. Zha, Z.B. Li, Darboux transformation and various solutions for a nonlinear evolution equation in (3+1)-dimensions, Modern Phys. Lett. B 22 (2008) 2945–2966. [26] Q.L. Zha, Z.B. Li, Positon, negaton, soliton and complexiton solutions to a four-dimensional evolution equation, Moderm Phys. Lett. B 23 (2009) 2971–2991. [27] X.G. Geng, Y.L. Ma, N-soliton solution and its wronskian form of a (3+1)-dimensional nonlinear evolution equation, Phys. Lett. A 369 (2007) 285–289. [28] J.P. Wu, A B¨ acklund transformation and explicit solutions of a (3+1)-dimensional soliton equation, 25, (12), 2008, 4192. [29] Q.L. Zha, Rogue waves and rational solutions of a (3+1)-dimensional nonlinear evolution equation, Phys. Lett. A 377 (2013) 3021–3026. [30] Z. Dai, S. Li, D. Li, A. Zhu, Periodic bifurcation and soliton deflexion for Kadomtsev–Petviashvili equation, Chin. Phys. Lett. 24 (2007) 1429–1432. [31] R. Hirota, Exact solution of the Korteweg–de Vries equation for multiple collisions, Phys. Rev. Lett. 27 (1971) 1192–1194.