Pramana – J. Phys. (2018) 90:71 © Indian Academy of Sciences https://doi.org/10.1007/s12043-018-1568-3

Multiple periodic-soliton solutions of the (3 + 1)-dimensional generalised shallow water equation

YE-ZHOU LI1 and JIAN-GUO LIU1,2 ,∗

1School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China 2College of Computer, Jiangxi University of Traditional Chinese Medicine, Nanchang 330004, Jiangxi, China ∗Corresponding author. E-mail: [email protected]

MS received 8 November 2017; revised 9 December 2017; accepted 27 December 2017; published online 30 April 2018

Abstract. Based on the extended variable-coefficient homogeneous balance method and two new ansätz functions, we construct auto-Bäcklund transformation and multiple periodic-soliton solutions of (3 + 1)-dimensional generalised shallow water equations. Completely new periodic-soliton solutions including periodic cross-kink wave, periodic two-solitary wave and breather type of two-solitary wave are obtained. In addition, cross-kink three-soliton and cross-kink four-soliton solutions are derived. Furthermore, propagation characteristics and interactions of the obtained solutions are discussed and illustrated in figures.

Keywords. Interactions; periodic-soliton solutions; (3 + 1)-dimensional generalised shallow water equation. PACS Nos 02.70.Wz; 05.45.Yv; 02.30.Jr

1. Introduction flows, tsunami prediction and so on, which was studied in different ways. By using the Nonlinear partial differential equations play important generalised tanh algorithm method with symbolic com- roles in nonlinear science [1Ð8]. They are widely used putation, Tian and Gao [27] obtained the soliton- to describe the phenomena in various fields such as type solutions of eq. (1). Zayed [28] presented the optical fibre communications, ocean engineering, fluid travelling wave solutions of eq. (1)bythe(G/G)- dynamics, plasma physics, chemical physics, etc. The expansion method. Tang et al [29] derived the Gram- past work is mainly concerned with the solutions [9Ð mian and Pfaffian derivative formulae, the Grammian 13]. A variety of analytical and numerical methods have and Pfaffian solutions of eq. (1) by the Hirota bilinear been proposed for studying soliton models, including form. inverse scattering method [14], Bäcklund transforma- In this work, we shall introduce new ansätz functions tion [15], Hirota direct method [16], homogeneous to study the (3 + 1)-dimensional generalised shallow balance method [17Ð19], F-expansion method [20], water equations and discuss their multiple periodical- similarity transformation [21], etc. soliton solutions. Shallow water equations, also called Saint-Venant This paper is organised as follows: In ¤2,weper- equations in its unidimensional form after Adhémar form the extended variable-coefficient homogeneous Jean Claude Barré de Saint-Venant), are a category of balance method (EvcHB) on the (3 + 1)-dimensional hyperbolic partial differential equations (or parabolic if generalised shallow water equation and present a new viscous shear is considered) describing the flow below a auto-Bäcklund transformation of eq. (1). In ¤3,the pressure surface in a fluid (sometimes, but not necessar- non-travelling wave soliton-type solutions of the (3+1)- ily, a free surface) [22Ð24]. In this paper, we shall study dimensional generalised shallow water are investigated. the following (3 + 1)-dimensional generalised shallow In ¤4, the travelling wave multisoliton solutions are water equation [25,26]: obtained by using the three-wave approach, which con- tain many singular periodic soliton solutions, periodic u − u − 3u u − 3u u + u = 0, (1) yt xz x xy y xx xxxy cross-kink waves solutions, two-solitary solutions and where u = u(x, y, z, t). Equation (1) has applications doubly periodic solitary solution. Finally, conclusions in weather simulations, tidal waves, river and irrigation are given in ¤5. 71 Page 2 of 11 Pramana – J. Phys. (2018) 90:71   2. Auto-Bäcklund transformation −6u ξ 3 + 3ξ ξσ(η)ξ 2 − 2ξ 2u 0y x y xx x 0x + 3 [5ξσ(η)− 2] ξ ξ In terms of the idea of EvcHB method [30], the solutions xy xx +[ξσ(η)+ 2]ξ ξ ξ = 0, (9) of eq. (1) can be written as follows: x xxy x  − ξ 2 + ξ ξ − ξ + ξ + ξ ξ u(x, y, z, t)=ψ(y, z, t)∂xf[ξ(x, y, z, t), η(x, y, z, t)] 3u0xy x yt x 2 xz 6u0x xy 9u0y xx x + 4ξ ξ − ξ ξ + ξ (ξ − 2ξ ) + u0(x, y, z, t). (2) xxxy x z xx xy t xxx 3   In the following analysis, we shall stay with the follow- − ξσ(η)[ξσ(η)− 4] ξxxξxxy + ξxyξxxx ing conditions: 2 + ξy (ξxt − 3u0x ξxx − 3ξx u0xx + ξxxxx) = 0, (10) ψ(y, z, t) = 1,η(x, y, z, t) = η (x, y, z, t) = 0 x y ξxyt − 3u0xyξxx − 3ξxyu0xx − ξxxz ⇒ η(x, y, z, t) = η(z, t). (3) − 3u0x ξxxy − 3u0yξxxx + ξxxxxy = 0, (11)  Substituting (2)and(3) into eq. (1), we obtain σ (η)(η ξ − η ξ ) = 0, (12)   t xy z xx ξ ξ 4 + ξ ξ 3 − η + ξ ξ 2 u − 3u u − 3u u + u − u = 0. fξξξξξ y x 4 fξξξξ xy x z fξξη fξξξ z x 0xxxy 0x 0xy 0y 0xx 0yt 0xz 2 2 (13) + 6 fξξξξξ ξ ξ + 6 fξξξξ ξ  y xx x  xxy x + ηt fξξη + fξξξξt ξyξx + fξξξytξx From (2)and(6), the new auto-BT for the (3 + 1)- dimensional generalised shallow water equation can be − 2 fξξξxzξx + 12 fξξξξxyξxxξx written as + 4 fξξξξyξxxxξx + 4 fξξξxxxyξx + ξ ξ 2 + + ξ ξ − u(x, y, z, t) = [−2ln(ξ) + δ(η) + ξσ(η)] 3 fξξξ y xx u0yt fξξ y xt u0xz x + (η fξη + fξξξ )ξ + fξ ξ + u0(x, y, z, t), (14)  t t  xy xyt − η + ξ ξ − ξ + ξ ξ z fξη fξξ z xx fξ xxz 6 fξξ xx xxy with σ(η), δ(η) and ξ satisfying eqs (8)Ð(13). The + u + fξξξ ξ + fξ ξ meaning of auto-BT consisted of (8)Ð(14)isthatif 0xxxy y xxxx  xxxxy 2 2 u0(x, y, z, t) is a special solution of eq. (1), then the − 3 fξξξ + u + fξ ξ fξξξξ ξ + u x 0x xx  y x 0xy expression (14) is another solution of eq. (1). + fξξ(2ξ ξ + ξ ξ ) + fξ ξ x xy y xx xxy  + 4 fξξξ ξ − 3 u + fξξξ ξ + fξ ξ  xy xxx 0y y x xy × ξ 3 + ξ ξ + + ξ = . 3. Non-travelling wave soliton-type solutions fξξξ x 3 fξξ xx x u0xx fξ xxx 0 (4) ξ 4 Setting the coefficient of x in (4) to zero yields an ordi- Now we use the new auto-BT consisted of eqs (8)Ð(14) nary differentiable equation (ODE) for f ;thatis to obtain exact solutions of eq. (1). Starting from eq. (8),   we obtain fξξξξξ − 6 fξξ fξξξ ξy = 0, (5) which admits the solution σ(η) = 0. (15) f =−2ln(ξ) + δ(η) + ξσ(η), (6) Aiming at the non-travelling wave soliton-type solu- = where δ(η) and σ(η)are differential functions. Accord- tions, we substitute u0 0 and a trial solution ing to (6), we obtain ( , ) + ( , , ) ξ(x, y, z, t) = 1 + em z t x n y z t (16) − ξσ(η) 2 2 [2 ] 2 1 fξ = fξξ, fξξ = fξξξξ, into eqs (9)Ð(13), where m(z, t) and n(y, z, t) are dif- 2 3 2 − ξσ(η) ferentiable functions. Substituting (15)and(16)into fξ fξξ = fξξξ, (9)Ð(13), we get 2 2 − ξσ(η) my fξ fξξξ = fξξξξ. (7) n(y, z, t) = C1(z, t) + , 3 C3(z) Substituting (5)and(7)into(4) yields a linear poly- mt (z, t) = C3(z)mz, nomial of fξ , fξξ,.... Equating the coefficients of fξ , m3 + C C (z, t) = 1t , (17) fξξ,...to zero, holds 1z ( )   C3 z ξσ(η)ξ 2 ξ ξ + ξ ξ = , x x xy y xx 0 (8) where C1(z, t) is an integrable function and C3(z) is an − ξ ξ 2 + ξ ξ ξ + ξσ(η)+ ξ ξ ξ 2 z x 2 t y x [3 2] y xxx x arbitrary differentiable function. Pramana – J. Phys. (2018) 90:71 Page 3 of 11 71

Collecting all the above terms, we can derive the Therefore, we obtain the following periodic general solutions of eq. (1) as follows: breather-type of kink three-soliton solutions for eq. (1) as follows: xm+ ymzm +C mt 1  ( , , , ) =−2me . γ u x y z t ym m (18) 2 1 xm+ z +C u1 = v0 − 2 k4α4 cosh xα4 − t α − α4 + σ4 1 + e mt 1 4 β  1 γ All parameters have been explained before. m = m(z, t) 2 1 − k2α2 sin xα2 + t α + α2 + σ2 and n = n(y, z, t) satisfy constraint (17). Solution (18) 2 β 1 contains more arbitrary parameters than the solution yβ1+zγ1+σ1 −yβ1−zγ1−σ1 obtained before in refs [27Ð29]. / e k1 + e  γ + α + α2 + 1 α + σ k2 cos x 2 t 2 2 2 β1 4. Travelling wave multisoliton solutions zγ β + k yβ + 1 3 + σ 3 cosh 3 β 3 Aiming at the travelling wave multisoliton solutions,  1 we now suppose that σ(η) = 0, u = v and the real γ 0 0 + α − α2 − 1 α + σ . function ξ(x, y, z, t) has the following ansätz: k4 sinh x 4 t 4 4 4 (22) β1

θ1 −θ1 ξ(x, y, z, t) = k1e + e + k2 cos(θ2) The expression (22) is the periodic breather-type of kink + k3 cosh(θ3) + k4 sinh(θ4) , (19) three-soliton solutions of eq. (1) which is a periodic wave in x, y and meanwhile is a two-soliton in x − t − where θi = αi x + βi y + γi z + δi t + σi , i = 1, 2, 3, 4 and in y t, respectively. The evolution and mechanical and αi , βi , γi , δi are constants to be determined later, feature of such waves are shown in figures 1Ð4. σi and v0 are arbitrary constants. Substituting eq. (19) According to the dependent variable transformation into eqs (8)Ð(13) and equating all the coefficients of in (22) θ −θ different powers of e 1 ,e 1 ,sin(θ2),cos(θ2),sinh(θ3), α = iα ,σ= iσ , (23) cosh(θ3),sinh(θ4),cosh(θ4) and constant term to zero, 2 21 2 21 we obtain a set of algebraic equations for αi , βi , γi , δi , where α21 and σ21 are real constants. Then, we obtain the ki (i = 1, 2, 3, 4). Solving the system with the aid of Mathematica, we obtain the following results: cross-kink four-soliton solutions of eq. (1) as follows:  Case 1 γ u = k α xα − t α2 − 1 α + σ 11 2 4 4 cosh 4 4 β 4 4 α = ,δ= ,β= ,γ= ,α= ,  1 1 0 1 0 2 0 2 0 3 0 γ γ α + k α xα +t α2 − 1 α +σ δ = ,β= ,γ= ,δ= α3 + 1 2 , 2 21 sinh 21 21 β 21 21 3 0 4 0 4 0 2 2 1 β1 yβ1+zγ1+σ1 −yβ1−zγ1−σ1 β γ α γ / −{e k1 + e γ = 3 1 ,δ= 4 1 − α3,  3 4 4 (20) γ β1 β1 + α − α2 − 1 α + σ k2 cosh x 21 t 21 21 21 β1 β γ β α α σ ( = , , , ) where 1, 1, 3, 4, 2, ki and i i 1 2 3 4 are zγ β + k yβ + 1 3 + σ free real constants. Substituting these results into (19), 3 cosh 3 β 3 we have  1 γ 2 1 β + γ +σ − β − γ −σ + k4 sinh xα4 − t α − α4 + σ4 + v0. ξ(x, y, z, t) = k ey 1 z 1 1 + e y 1 z 1 1 4 β 1  1 γ (24) + k xα + t α2 + 1 α 2 cos 2 2 β 2 1 The physical structure of solution (24) is showed in fig- zγ β ure 5. + σ + k yβ + 1 3 + σ 2 3 cosh 3 β 3  1 Case 2 γ + α − α2 − 1 α + σ . k4 sinh x 4 t 4 4 4 α = 0,δ= 0,β= 0,γ= 0,β= 0, β1 1 1 2 2 3 (21) β4 = 0,γ3 = 0,γ4 = 0,α4 = α3, 71 Page 4 of 11 Pramana – J. Phys. (2018) 90:71

β (δ − α3) δ γ = 1 2 2 ,δ= α −α2 − α2 + 2 , 1 α 3 3 2 3 α 2  2 α α3 + α2α − δ 3 2 3 2 2 δ4 =− , (25) α2 where β1, δ2, α3, α2, ki and σi (i = 1, 2, 3, 4) are free real constants. Substituting these results into (19), we have

(δ −α3)β (δ −α3)β z 2 2 1 z 2 2 1 yβ1+ α +σ1 −yβ1− α −σ1 ξ(x, y, z, t)=e 2 k +e 2  1 δ + k xα + t −α2 − α2 + 2 α + σ 3 cosh 3 2 3 α 3 3  2  (α3 + α2α − δ )α t 2 3 2 2 3 + k4 sinh x α3 − + σ4 α2

+ k2 cos(xα2 + tδ2 + σ2), (26) where ε =±1. Therefore, we obtain the following doubly periodic breather-type of cross-kink two-soliton solutions for eq. (1) as follows:

u2 = v0 − 2 − k2α2 sin(xα2 + tδ2 + σ2)  δ + k α xα + t −α2 − α2 + 2 α + σ 3 3 sinh 3 2 3 α 3 3  2 (α3 + α2α − δ )α t 2 3 2 2 3 + 2 k4α3 cosh x α3 − α2 

yβ1+zγ1+σ1 −yβ1−zγ1−σ1 + σ4 /{e k1 +e

+ k cos(xα + tδ + σ ) 2  2 2 2 δ + α + α −α2 − α2 + 2 + σ k3 cosh x 3 t 3 2 3 3 α2   α (α3 + α2α − δ ) t 3 2 3 2 2 + k4 sinh x α3 − +σ4 . α2 (27)

The expression (27) is the doubly periodic breather-type Figure 1. Plots of the periodic breather-type solution (22)for of cross-kink two-soliton solutions of eq. (1) which are α2 = 1, β1 = 2, γ1 =−5, σ1 = σ2 = σ3 =−2, σ4 = 0.5, α = β ======− showed in figures 6Ð9. 4 2, 3 4, k1 1, k2 3, k3 4, k4 5, z 2for =− = = According to the dependent variable transformation (a) t 2, (b) t 0 and (c) t 2. in (27)

α3 = iα31,σ3 = iσ31,σ4 = iσ41, k4 = ik41, (28) u21 = v0 − 2 − k2α2 sin(xα2 + tδ2 + σ2) where α31, σ31, σ41 and k41 are real constants. Then, we  δ obtain the cross-kink four-soliton solutions of eq. (1)as − α α + −α2 + α2 + 2 α k3 31 sin x 31 t 2 31 31 follows: α2 Pramana – J. Phys. (2018) 90:71 Page 5 of 11 71

Figure 2. Plots of the periodic breather-type solution (22)for Figure 3. Plots of the periodic breather-type solution (22)for the same parameters as in figure 1, except that (a) y =−2, the same parameters as in figure 2, except that (a) x =−2, (b) y = 0 and (c) y = 1. (b) x = 0 and (c) x = 2.

 + k cos(xα + tδ + σ ) 2  2 2 2 + σ − α α δ 31 2 k41 31 cos x 31 + k xα +t −α2 + α2 + 2 α 3 cos 31 2 31 α 31   2 t (−α3 + α2 α + δ )α + 2 31 2 2 31 + σ / +σ −k sin x α α 41 31 41 31 2  3 2 t (−α +α α2 + δ2)α31 yβ1+zγ1+σ1 −yβ1−zγ1−σ1 + 2 31 +σ . e k1 + e 41 (29) α2 71 Page 6 of 11 Pramana – J. Phys. (2018) 90:71

Figure 4. Plots of the periodic breather-type solution (22)for the same parameters as in figure 2, except that k3 = k4 = 0, and (a) t =−15, (b) t = 0 and (c) t = 15.

The physical structure of solution (29)isshownin figure 10. Figure 5. Cross-kink four-soliton solutions (24) for the same Case 3 parameters as in figure 1, except that α21 = 1, σ21 =−2, and (a) t =−5, (b) t = 0 and (c) t = 5. α1 = 0,δ1 = 0,β2 = 0,γ2 = 0,β3 = 0, β = 0,γ= 0,γ= 0,α= iα , 4 3 4 3 2 3 β α − δ (α3−δ )β (α3−δ )β 1 2 2 z 2 2 1 z 2 2 1 α =i α ,γ=− ,δ=iδ ,δ = i δ , yβ1− α +σ1 −yβ1+ α −σ1 4 2 1 3 2 4 2 ξ(x, y, z, t)=k 2 + 2 α2 1e e (30) + k2 cos(xα2 + tδ2 + σ2) + k (xα + tδ − iσ ) where β1, δ2, α2, ki and σi (i = 1, 2, 3, 4) are free real 3 cos 2 2 3 constants. Substituting these results into (19), we have + ik4 sin(x α2 + t δ2 − iσ4) , (31) Pramana – J. Phys. (2018) 90:71 Page 7 of 11 71

Figure 6. Doubly periodic breather-type solution (27)for Figure 7. α = 1, β = 2, γ =−5, σ = σ = σ =−2, σ = 0.5, Doubly periodic breather-type solution (27)forthe 2 1 1 1 2 3 4 =− a α = 2, δ = 4, k = 1, k = 3, k = 4 and k = 5, = 1, same parameters as in figure 6, except that 1, and ( ) 3 2 1 2 3 4 t =− b t = c t = z =−1, (a) t =−2, (b) t = 0 and (c) t = 2. 2, ( ) 0 and ( ) 2. 71 Page 8 of 11 Pramana – J. Phys. (2018) 90:71

Figure 8. Doubly periodic breather-type solution (27)forthe Figure 9. Doubly periodic breather-type solution (27)forthe same parameters as in figure 6, except that =−1, and (a) same parameters as in figure 6, except that k1 = k2 = 0, and y =−2, (b) y = 0 and (c) y = 2. (a) t =−2, (b) t = 0 and (c) t = 2. Pramana – J. Phys. (2018) 90:71 Page 9 of 11 71

Figure 10. Cross-kink four-soliton solution (29)forα2 = 1, Figure 11. Doubly periodic breather-type solution (32)for β1 =−2, γ1 = 5, σ1 = σ2 = σ31 =−2, σ41 = 0.5, α31 = 2, α2 = 1, β1 = 2, γ1 =−5, σ1 = σ2 =−2, σ3 = σ4 = i, δ2 = 4, k1 =−1, k2 =−3, k3 =−4 and k41 =−5, = 1, α3 = 2, δ2 = 4, k1 = 1, k2 = 3, k3 = 4 and k4 = i, = 1, z =−1, (a) t =−10, (b) t = 0 and (c) t = 10. z =−1, (a) t =−10, (b) t = 0 and (c) t = 10. 71 Page 10 of 11 Pramana – J. Phys. (2018) 90:71

− 2i k α cos(x α + t δ − iσ )}  4 2 2 2 4 (α3−δ )β (α3−δ )β z 2 2 1 z 2 2 1 yβ1− α +σ1 −yβ1+ α −σ1 / k1e 2 + e 2 

+ ik4 sin(x α2 + t δ2 − iσ4)

+ k2 cos(xα2 + tδ2 + σ2) + k3 cos(xα2 + tδ2 − iσ3) . (32) Expression (32) is another doubly periodic breather- type of cross-kink two-soliton solutions of eq. (1)which are shown in figures 11 and 12.

5. Discussion and conclusion

It is worth pointing out that singular periodic soliton solutions, periodic cross-kink waves solutions, two- solitary solutions and doubly periodic solitary solutions can be viewed as special cases of the obtained results when the free parameters are properly adjusted/tuned. Ansätz (19) is first used for solving the (3 + 1)- dimensional generalised shallow water equation. Com- pared with other literatures [25Ð29], ansätz (19)is simple and straightforward, and these obtained solutions contain more arbitrary parameters and richer physical significance. The dynamical behaviour of solitons in (22), (24), (27), (29)and(32) are shown in figures 1Ð12. Figures 1Ð 9 illustrate the collisions between two parallel solitons, where the soliton of larger amplitude travels faster and moves across the soliton of smaller amplitude (travel- ling slower). At the crossing instant, the larger amplitude is restrained, while the smaller amplitude is strength- ened. In particular, figures 1cand8c give two solitons that almost merge into one single pulse at the moment t = 2andy = 2. After the strikes (crossing), the two solitons keep travelling separately, with their original amplitudes, widths and velocities. If a soliton and a singular breather travel with the same velocity, they can form a bound state, as presented in figures 10Ð12. During the propagation, the interactions Figure 12. Doubly periodic breather-type solution (32)for between solitons show periodical changes. The two soli- the same parameters as in figure 9, except that σ = 4i, 4 tons attract and repel each other periodically and form k4 =−5i, =−1, and (a) t =−10, (b) t = 0 and (c) t = 10. the bound solitons. The bound solitons exchange their energy periodically. By employing the extended variable-coefficient where ε =±1. Therefore, we obtain another doubly homogeneous balance method and extended three-wave periodic breather-type of cross-kink two-soliton solu- type of ansätz approach [31Ð37], we obtained auto- tions for eq. (1) as follows: Bäcklund transformation, non-travelling wave soliton- type solutions and travelling wave multisoliton solutions = v + { α ( α + δ + σ ) u3 0 2k2 2 [sin x 2 t 2 2 of the (3 + 1)-dimensional generalised shallow water + k3α2 sin(xα2 + tδ2 − iσ3)] equation. Furthermore, we observed and analysed the Pramana – J. Phys. (2018) 90:71 Page 11 of 11 71 evolution process of interaction with the time, including P A Clarkson (Cambridge University Press, London, the degeneracy of soliton, periodic bifurcation and soli- 1990) ton deflection of two-wave, fission and fusion of breather [15] J G Liu, Y Z Li and G M Wei, Chin. Phys. Lett. 23, 1670 two-wave, and so on. 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