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

Transcription

1 Pramana J. Phys :7 Indian Academy of Sciences Multiple periodic-soliton solutions of the 3 + -dimensional generalised shallow water equation YE-ZHOU LI and JIAN-GUO LIU,, School of Science, Beijing University of Posts and Telecommunications, Beijing 00876, China College of Computer, Jiangxi University of Traditional Chinese Medicine, Nanchang , Jiangxi, China Corresponding author @qq.com MS received 8 November 07; revised 9 December 07; accepted 7 December 07; published online 30 April 08 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 + -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. PACS Nos Interactions; periodic-soliton solutions; 3 + -dimensional generalised shallow water equation Wz; Yv; 0.30.Jr. Introduction Nonlinear partial differential equations play important roles in nonlinear science 8]. They are widely used to describe the phenomena in various fields such as optical fibre communications, ocean engineering, fluid dynamics, plasma physics, chemical physics, etc. The past work is mainly concerned with the solutions 9 3]. A variety of analytical and numerical methods have been proposed for studying soliton models, including inverse scattering method 4], Bäcklund transformation 5], Hirota direct method 6], homogeneous balance method 7 9], F-expansion method 0], similarity transformation ], etc. Shallow water equations, also called Saint-Venant equations in its unidimensional form after Adhémar Jean Claude Barré de Saint-Venant, are a category of hyperbolic partial differential equations or parabolic if viscous shear is considered describing the flow below a pressure surface in a fluid sometimes, but not necessarily, a free surface 4]. In this paper, we shall study the following 3 + -dimensional generalised shallow water equation 5,6]: u yt u xz 3u x u xy 3u y u xx + u xxxy = 0, where u = ux, y, z, t. Equation has applications in weather simulations, tidal waves, river and irrigation flows, tsunami prediction and so on, which was studied in different ways. By using the generalised tanh algorithm method with symbolic computation, Tian and Gao 7] obtained the solitontype solutions of eq.. Zayed 8] presented the travelling wave solutions of eq. bytheg /G- expansion method. Tang et al 9] derived the Grammian and Pfaffian derivative formulae, the Grammian and Pfaffian solutions of eq. by the Hirota bilinear form. In this work, we shall introduce new ansätz functions to study the 3 + -dimensional generalised shallow water equations and discuss their multiple periodicalsoliton solutions. This paper is organised as follows: In, weperform the extended variable-coefficient homogeneous balance method EvcHB on the 3 + -dimensional generalised shallow water equation and present a new auto-bäcklund transformation of eq.. In 3, the non-travelling wave soliton-type solutions of the 3+- dimensional generalised shallow water are investigated. In 4, the travelling wave multisoliton solutions are obtained by using the three-wave approach, which contain many singular periodic soliton solutions, periodic cross-kink waves solutions, two-solitary solutions and doubly periodic solitary solution. Finally, conclusions are given in 5.

2 7 Page of Pramana J. Phys :7. Auto-Bäcklund transformation 6u 0y ξx 3 + 3ξ y ξσ ηξ xx ξx u 0x] In terms of the idea of EvcHB method 30], the solutions of eq. can be written as follows: ux, y, z, t=ψy, z, t x fξx, y, z, t, ηx, y, z, t] + u 0 x, y, z, t. In the following analysis, we shall stay with the following conditions: ψy, z, t =, η x x, y, z, t = η y x, y, z, t = 0 ηx, y, z, t = ηz, t. 3 Substituting and3 into eq., we obtain f ξξξξξ ξ y ξx f ξξξξξ xy ξx 3 η z f ξξη + f ξξξ ξ z ξ x + 6 f ξξξξ ξ y ξ xx ξ x + 6 f ξξξξ xxy ξ x + η t f ξξη + f ξξξ ξ t ξy ξ x + f ξξ ξ yt ξ x f ξξ ξ xz ξ x + f ξξξ ξ xy ξ xx ξ x + 4 f ξξξ ξ y ξ xxx ξ x + 4 f ξξ ξ xxxy ξ x + 3 f ξξξ ξ y ξ xx + u 0yt + f ξξ ξ y ξ xt u 0xz + η t f ξη + f ξξ ξ t ξ xy + f ξ ξ xyt η z f ξη + f ξξ ξ z ξxx f ξ ξ xxz + 6 f ξξ ξ xx ξ xxy + u 0xxxy + f ξξ ξ y ξ xxxx + f ξ ξ xxxxy 3 f ξξ ξx + u 0x + f ξ ξ xx fξξξ ξ y ξx + u 0xy ] + f ξξ ξ x ξ xy + ξ y ξ xx + f ξ ξ xxy + 4 f ξξ ξ xy ξ xxx 3 u 0y + f ξξ ξ y ξ x + f ξ ξ xy f ξξξ ξx f ξξξ xx ξ x + u 0xx + f ξ ξ xxx = 0. 4 Setting the coefficient of ξx 4 in 4 to zero yields an ordinary differentiable equation ODE for f ;thatis fξξξξξ 6 f ξξ f ξξξ ξy = 0, 5 which admits the solution f = lnξ + δη + ξσ η, 6 where δη and σηare differential functions. According to 6, we obtain f ξ = ξση] f ξξ, f ξξ = 3 f ξξξξ, f ξ f ξξ = ξση f ξξξ, f ξ f ξξξ = ξση f ξξξξ. 7 3 Substituting 5 and7 into4 yields a linear polynomial of f ξ, f ξξ,... Equating the coefficients of f ξ, f ξξ,...to zero, holds ξσ ηξx ξx ξ xy + ξ y ξ xx = 0, 8 ξ z ξx + ξ tξ y ξ x + 3ξση+ ] ξ y ξ xxx ξ x + 3 { 5ξση ] ξ xy ξ xx } +ξση+ ]ξ x ξ xxy ξx = 0, 9 3u 0xy ξx + ξ ytξ x ξ xz + 6u 0x ξ xy + 9u 0y ξ xx ξx + 4ξ xxxy ξ x ξ z ξ xx + ξ xy ξ t ξ xxx 3 ξσηξση 4] ξ xx ξ xxy + ξ xy ξ xxx + ξ y ξ xt 3u 0x ξ xx 3ξ x u 0xx + ξ xxxx = 0, 0 ξ xyt 3u 0xy ξ xx 3ξ xy u 0xx ξ xxz 3u 0x ξ xxy 3u 0y ξ xxx + ξ xxxxy = 0, σ ηη t ξ xy η z ξ xx = 0, u 0xxxy 3u 0x u 0xy 3u 0y u 0xx + u 0yt u 0xz = 0. 3 From and6, the new auto-bt for the dimensional generalised shallow water equation can be written as ux, y, z, t = lnξ + δη + ξση] x + u 0 x, y, z, t, 4 with ση, δη and ξ satisfying eqs 8 3. The meaning of auto-bt consisted of 8 4 isthatif u 0 x, y, z, t is a special solution of eq., then the expression 4 is another solution of eq.. 3. Non-travelling wave soliton-type solutions Now we use the new auto-bt consisted of eqs 8 4 to obtainexact solutionsof eq.. Starting from eq. 8, we obtain ση = 0. 5 Aiming at the non-travelling wave soliton-type solutions, we substitute u 0 = 0 and a trial solution ξx, y, z, t = + e mz,tx+ny,z,t 6 into eqs 9 3, where mz, t and ny, z, t are differentiable functions. Substituting 5 and6 into 9 3, we get ny, z, t = C z, t + m t z, t = C 3 zm z, my C 3 z, C z z, t = m3 + C t, 7 C 3 z where C z, t is an integrable function and C 3 z is an arbitrary differentiable function.

3 Pramana J. Phys :7 Page 3 of 7 Collecting all the above terms, we can derive the general solutions of eq. as follows: ux, y, z, t = mexm+ + e ymzm +C mt ymzm xm+ mt +C. 8 All parameters have been explained before. m = mz, t and n = ny, z, t satisfy constraint 7. Solution 8 contains more arbitrary parameters than the solution obtained before in refs 7 9]. 4. Travelling wave multisoliton solutions Aiming at the travelling wave multisoliton solutions, we now suppose that ση = 0, u 0 = v 0 and the real function ξx, y, z, t has the following ansätz: ξx, y, z, t = k e θ + e θ + k cosθ + k 3 coshθ 3 + k 4 sinhθ 4, 9 where θ i = α i x + β i y + γ i z + δ i t + σ i, i =,, 3, 4 and α i, β i, γ i, δ i are constants to be determined later, σ i and v 0 are arbitrary constants. Substituting eq. 9 into eqs 8 3 and equating all the coefficients of different powers of e θ,e θ,sinθ,cosθ,sinhθ 3, coshθ 3,sinhθ 4,coshθ 4 and constant term to zero, we obtain a set of algebraic equations for α i, β i, γ i, δ i, k i i =,, 3, 4. Solving the system with the aid of Mathematica, we obtain the following results: Case α = 0, δ = 0, β = 0, γ = 0, α 3 = 0, δ 3 = 0, β 4 = 0, γ 4 = 0, δ = α 3 + γ α, γ 3 = β 3γ, δ 4 = α 4γ α 3 4, 0 where, γ, β 3, α 4, α, k i and σ i i =,, 3, 4 are free real constants. Substituting these results into 9, we have ξx, y, z, t = k e y+zγ +σ + e y zγ σ + k cos xα + t α + γ α β ] + σ + k 3 cosh yβ 3 + zγ β 3 + σ 3 β + k 4 sinh xα 4 t α4 γ ] α 4 + σ 4. Therefore, we obtain the following periodic breather-type of kink three-soliton solutions for eq. as follows: { u = v 0 k 4 α 4 cosh xα 4 t α4 γ ] α 4 + σ 4 k α sin xα + t α + γ { / e y+zγ +σ k + e y zγ σ + k cos xα + t + k 3 cosh + k 4 sinh xα 4 t α + γ yβ 3 + zγ β 3 + σ 3 α4 γ α + σ ]} α + σ ] α 4 + σ 4 ]}. The expression is the periodic breather-type of kink three-soliton solutions of eq. which is a periodic wave in x, y and meanwhile is a two-soliton in x t and in y t, respectively. The evolution and mechanical feature of such waves are shown in figures 4. According to the dependent variable transformation in α = iα, σ = iσ, 3 where α and σ are real constants. Then, we obtain the cross-kink four-soliton solutions of eq. as follows: { u = k 4 α 4 cosh xα 4 t α4 γ ] α 4 + σ 4 β + k α sinh xα +t α γ ]} α +σ / {e y+zγ +σ k + e y zγ σ + k cosh xα t α γ ] α + σ + k 3 cosh yβ 3 + zγ β 3 + σ 3 β + k 4 sinh xα 4 t α4 γ ]} α 4 + σ 4 + v 0. 4 The physical structure of solution 4 is showed in figure 5. Case α = 0, δ = 0, β = 0, γ = 0, β 3 = 0, β 4 = 0, γ 3 = 0, γ 4 = 0, α 4 = ɛα 3,

4 7 Page 4 of Pramana J. Phys :7 γ = δ α 3, δ 3 = α 3 α α α 3 + δ, α δ 4 = ɛα 3 α 3 + α3 α δ, 5 α where, δ, α 3, α, k i and σ i i =,, 3, 4 are free real constants. Substituting these results into 9, we have ξx, y, z, t=e y+ zδ α3 α +σ k +e y zδ α3 α σ + k 3 cosh xα 3 + t α α 3 + δ ] α 3 + σ 3 α + k 4 sinh xɛα 3 tɛα3 + α 3 α ] δ α 3 + σ 4 α + k cosxα + tδ + σ, 6 where ε = ±. Therefore, we obtain the following doubly periodic breather-type of cross-kink two-soliton solutions for eq. as follows: { u = v 0 k α sinxα + tδ + σ + k 3 α 3 sinh xα 3 + t + ɛk 4 α 3 cosh α α 3 + δ ] α 3 + σ 3 α xɛα 3 tɛα3 + α 3 α δ α 3 α + σ 4 ] } /{e y+zγ +σ k +e y zγ σ + k cosxα + tδ + σ + k 3 cosh xα 3 + tα 3 α α 3 + δ ] + σ 3 α + k 4 sinh xɛα 3 tɛα 3α 3 + α 3 α ] } δ +σ 4. α 7 The expression 7 is the doubly periodic breather-type of cross-kink two-soliton solutions of eq. which are showed in figures 6 9. According to the dependent variable transformation in 7 α 3 = iα 3, σ 3 = iσ 3, σ 4 = iσ 4, k 4 = ik 4, 8 where α 3, σ 3, σ 4 and k 4 are real constants. Then, we obtain the cross-kink four-soliton solutions of eq. as follows: Figure. Plots of the periodic breather-type solution for α =, =, γ = 5, σ = σ = σ 3 =, σ 4 = 0.5, α 4 =, β 3 = 4, k =, k = 3, k 3 = 4, k 4 = 5, z = for a t =, b t = 0 and c t =. { u = v 0 k α sinxα + tδ + σ k 3 α 3 sin xα 3 + t α + α 3 + δ α 3 α

5 Pramana J. Phys :7 Page 5 of 7 Figure. Plots of the periodic breather-type solution for the same parameters as in figure, except that a y =, b y = 0 and c y =. ] + σ 3 ɛk 4 α 3 cos xɛα 3 + tɛ α3 + α 3 α ] } + δ α 3 + σ 4 / α { e y+zγ +σ k + e y zγ σ Figure 3. Plots of the periodic breather-type solution for the same parameters as in figure, except that a x =, b x = 0 and c x =. + k cosxα + tδ + σ + k 3 cos xα 3 +t +σ 3 ] k 4 sin xɛα 3 α + α 3 + δ α α 3 + tɛ α3 +α 3 α ] } + δ α 3 +σ 4. 9 α

6 7 Page 6 of Pramana J. Phys :7 Figure 4. Plots of the periodic breather-type solution for the same parameters as in figure, except that k3 = k4 = 0, and a t = 5, b t = 0 and c t = 5. The physical structure of solution 9 is shown in figure 0. Case 3 α = 0, δ = 0, β = 0, γ = 0, β3 = 0, β4 = 0, γ3 = 0, γ4 = 0, α3 = iα, β α3 δ α4 = i α, γ =, δ3 = iδ, δ4 = i δ, α 30 where β, δ, α, ki and σi i =,, 3, 4 are free real constants. Substituting these results into 9, we have Figure 5. Cross-kink four-soliton solutions 4 for the same parameters as in figure, except that α =, σ =, and a t = 5, b t = 0 and c t = 5. yβ zα3 δ β +σ yβ + zα3 δ β σ α α ξx, y, z, t = k e +e + k cosxα + tδ + σ + k3 cosxα + tδ iσ3 + ik4 sinx α + t δ iσ4, 3

7 Pramana J. Phys :7 Page 7 of 7 Figure 6. Doubly periodic breather-type solution 7 for α =, =, γ = 5, σ = σ = σ 3 =, σ 4 = 0.5, α 3 =, δ = 4, k =, k = 3, k 3 = 4 and k 4 = 5, ɛ =, z =, a t =, b t = 0 and c t =. Figure 7. Doubly periodic breather-type solution 7for the same parameters as in figure 6, except that ɛ =, and a t =, b t = 0 and c t =.

8 7 Page 8 of Pramana J. Phys :7 Figure 8. Doubly periodic breather-type solution 7for the same parameters as in figure 6, except that ɛ =, and a y =, b y = 0 and c y =. Figure 9. Doubly periodic breather-type solution 7for the same parameters as in figure 6, except that k = k = 0, and a t =, b t = 0 and c t =.

9 Pramana J. Phys :7 Page 9 of 7 Figure 0. Cross-kink four-soliton solution 9forα =, =, γ = 5, σ = σ = σ 3 =, σ 4 = 0.5, α 3 =, δ = 4, k =, k = 3, k 3 = 4 and k 4 = 5, ɛ =, z =, a t = 0, b t = 0 and c t = 0. Figure. Doubly periodic breather-type solution 3 for α =, =, γ = 5, σ = σ =, σ 3 = σ 4 = i, α 3 =, δ = 4, k =, k = 3, k 3 = 4 and k 4 = i, ɛ =, z =, a t = 0, b t = 0 and c t = 0.

10 7 Page 0 of Pramana J. Phys :7 iɛk 4 α cosxɛα + tɛδ iσ 4 } { / k e y zα3 δ α +σ + e y+ zα3 δ α σ + ik 4 sinxɛα + tɛδ iσ 4 + k cosxα + tδ + σ + k 3 cosxα + tδ iσ 3. 3 Expression 3 is another doubly periodic breathertype of cross-kink two-soliton solutions of eq. which are shown in figures and. } 5. Discussion and conclusion Figure. Doubly periodic breather-type solution 3 for the same parameters as in figure 9, except that σ 4 = 4i, k 4 = 5i, ɛ =, and a t = 0, b t = 0 and c t = 0. where ε =±. Therefore, we obtain another doubly periodic breather-type of cross-kink two-soliton solutions for eq. as follows: u 3 = v 0 + {k α sinxα + tδ + σ + k 3 α sinxα + tδ iσ 3 ] It is worth pointing out that singular periodic soliton solutions, periodic cross-kink waves solutions, twosolitary 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 9 is first used for solving the dimensional generalised shallow water equation. Compared with other literatures 5 9], ansätz 9 is simple and straightforward, and these obtained solutions contain more arbitrary parameters and richer physical significance. The dynamical behaviour of solitons in, 4, 7, 9and3 are shown in figures. Figures 9 illustrate the collisions between two parallel solitons, where the soliton of larger amplitude travels faster and moves across the soliton of smaller amplitude travelling slower. At the crossing instant, the larger amplitude is restrained, while the smaller amplitude is strengthened. In particular, figures c and8c give two solitons that almost merge into one single pulse at the moment t = andy =. 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 0. During the propagation, the interactions between solitons show periodical changes. The two solitons attract and repel each other periodically and form the bound solitons. The bound solitons exchange their energy periodically. By employing the extended variable-coefficient homogeneous balance method and extended three-wave type of ansätz approach 3 37], we obtained auto- Bäcklund transformation, non-travelling wave solitontype solutions and travelling wave multisoliton solutions of the 3 + -dimensional generalised shallow water equation. Furthermore, we observed and analysed the

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