Bäcklund transformation and soliton solutions in terms of the Wronskian for the Kadomtsev Petviashvili-based system in fluid dynamics

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1 Pramana J. Phys. (08) 90:45 Indian Academy of Sciences Bäcklund transformation and soliton solutions in terms of the Wronskian for the Kadomtsev Petviashvili-based system in fluid dynamics ZHOG DU, BO TIA, XI-YAG XIE, JU CHAI and XIAO-YU WU State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 00876, China Corresponding author. MS received 5 September 07; revised 6 October 07; accepted 7 October 07; published online 3 February 08 Abstract. In this paper, investigation is made on a Kadomtsev Petviashvili-based system, which can be seen in fluid dynamics, biology and plasma physics. Based on the Hirota method, bilinear form and Bäcklund transformation (BT) are derived. -soliton solutions in terms of the Wronskian are constructed, and it can be verified that the - soliton solutions in terms of the Wronskian satisfy the bilinear form and Bäcklund transformation. Through the -soliton solutions in terms of the Wronskian, we graphically obtain the kink-dark-like solitons and parallel solitons, which keep their shapes and velocities unchanged during the propagation. Keywords. Fluid dynamics; Kadomtsev Petviashvili-based system; Wronskian; Hirota method; soliton solutions; Bäcklund transformation. PACS os Yv; Fg; 0.30.Jr. Introduction onlinear evolution equations (LEEs) have been studied in fields such as the fluid dynamics, chemistry, biology, plasma physics, nonlinear optics, and condensed matter physics [ 6]. Efforts have been dedicated to find analytic solutions for the LEEs, including the soliton, periodic and rational solutions [ 3,7]. Methods such as the inverse scattering method [4], Painlevé analysis method [9], variable separation method [9], Bäcklund transformation (BT) [0], Hirota method [,] and Darboux transformation [3] have been developed to derive analytic solutions. Based on the bilinear forms for some LEEs, e.g., the Korteweg de Vries, Kadomtsev Petviashvili (KP) and breaking-soliton equations, the Wronskian technique has been developed to derive the -soliton solutions in terms of the Wronskian [4 ]. The KP equation [3,4], ( 4u t + 6uu x + u xxx ) x + 3u yy = 0, () has been used to describe the evolution of nonlinear, long waves of small amplitude with slow dependence on the transverse coordinate, where u is a real function with respect to two scaled space coordinates x, y and retarded time t, and the subscripts denote the partial derivatives. Through the inner parameter-dependent symmetry constraints H y = u xx uu x V x and V y = V xx (uv) x on eq. (), an integrable ( + )-dimensional KP-based system can be obtained [5 8], as follows: H t = 4[H xx + H 3 3HH x + 3H V x +3 y (VH) x ] x, (a) V t = 4(V xx + 3HV x + 3H V + 3V y V x ) x, (b) where H and V are the functions with respect to x, y and t.system() has also been proved to be an extension of the Whitham Broer Kaup system for the long waves by incorporating or mimicking the convective, dispersive and viscous effects [7]. Infinitely many Lax pairs for system () has been given in [5]. Kink soliton solutions, periodic soliton solutions and lattice soliton solutions for system () have been discussed in [6]. V-shaped and A-shaped soliton fusion and Y-shaped soliton fission phenomena for system () have been reported in [7]. Soliton-like solutions and periodic solutions for system () have been obtained in [8].

2 45 Page of 6 Pramana J. Phys. (08) 90:45 However, Bäcklund transformation (BT) in the bilinear form and soliton solutions in terms of the Wronskian for system () have not been derived. Motivated by the above, we shall derive the bilinear form, BT for system () using the Hirota method in. Using the Wronskian technique, -soliton solutions in terms of the Wronskian for system () will be constructed and verified in 3. Graphically, evolution of the one-soliton and parallel solitons will be presented. Finally, we shall work out our conclusions in 4.. Bilinear form and Bäcklund transformation Through the transformations W y = V x, P y = (VH) x [6], system () can be written as follows: H t + 4(H xx + H 3 3HH x + 3HW + 3P) x = 0, (3a) V t + 4(V xx + 3V x H + 3H V + 3VW) x = 0, (3b) W y V x = 0, P y (VH) x = 0. (3c) In order to obtain the bilinear form for eqs (3), we take the following dependent variable transformations: H = (log f ) x, (4a) V = (log f ) xy, (4b) W = (log f ) xx = H x, (4c) P = (log f ) x (log f ) xx = HH x, (4d) where f is a function of variables x, y and t. Substituting eqs (4) into eqs (3), we obtain f [(D x D t + D y D t + 4Dx 4 + 4D3 x D y) f f ] + f D x( f xx + f xy ) f x = 0, (5) and the bilinear form for eqs (3) is obtained as follows: (D x D t + D y D t + 4Dx 4 + 4D3 x D y) f f = 0, (6a) ( (D x + D y ) f = 0 x + ) f f = 0, (6b) where D x, D y and D t are the Hirota bilinear derivative operators [], defined as Dx m Dn y Dl t (α β) ( x ) ( m x ) n( t ) l t α(x, y, t)β(x, y, t ), x =x, y =y, t =t with x, y and t being the formal variables, m, n and l being the non-negative integers, while α is a function with respect to x, y and t, andβ is a function with respect to x, y and t. According to bilinear form (6), we shall derive the BT for eqs (3) between two different solutions H = (log f ) x, V = (log f ) xy, W = (log f ) xx and P = (log f ) x (log f ) xx, and H = (log f ) x, V = (log f ) xy, W = (log f ) xx and P = (log f ) x (log f ) xx for eqs (3). Setting P =[(D x D t + D y D t + 4Dx 4 + 4D3 x D y) f f ] f [(D x D t + D y D t + 4Dx 4 + 4D3 x D y) f f ] f, (7a) [( P = x + ) ] f f f [( x + ) ] f f f, (7b) with the following Hirota bilinear operator identities: (D x D t f f ) f (D x D t f f ) f = D x (D t f f ) ( f f ), (D y D t f f ) f (D y D t f f ) f = D y (D t f f ) ( f f ), D x (D y f f ) ( f f ) = D y (D x f f ) ( f f ), (Dx 4 f f ) f (Dx 4 f f ) f = D x (Dx 3 f f ) ( f f ) 6D x (Dx f f ) (D x f f ), (Dx 3 D y f f ) f (Dx 3 D y f f ) f = D y (D 3 x f f ) ( f f ) 6D x (D x D y f f ) (D x f f ), we may rewrite P and P as P = D x (D t f f ) ( f f ) + D y (D t f f ) ( f f ) + 8D x (Dx 3 f f ) ( f f ) 4 D x (Dx f f ) (D x f f ) + 8D y (Dx 3 f f ) ( f f ) 4D x (D x D y f f ) (D x f f ) = D x [(D t + 4Dx 3 + λd x) f f ] ( f f ) 4D x [(Dx λ) f f ] (D x f f ) + D y [(D t + 4Dx 3 + λd x) f f ] ( f f ) 4D x [(D x D y + λ) f f ] (D x f f ), (8a) P = f f ( f x f f f x + f y f f f y ) = [(D x + D y ) f f ] ( f f ). (8b) The BT for eqs (3) can be displayed as (D x + D y ) f f = 0, (9a) (D x D y μd x + λ) f f = 0, (9b)

3 Pramana J. Phys. (08) 90:45 Page 3 of 6 45 (Dx μd x λ) f f = 0, (9c) (D t + 4Dx 3 + λd x) f f = 0, (9d) where both μ and λ are the constants. In order to get the one-soliton solutions for eqs (3) from BT (9), a seed solution for eqs (3) can be taken as f =, i.e. H = 0, V = 0, W = 0andP = 0. Letting μ = 0andλ = k, and substituting the seed solution into BT (9), we obtain f = e ξ + e ξ, (0) where ξ = kx ky 6k 3 t + ξ 0, k and ξ 0 are all arbitrary constants. Substituting eq. (0) into eqs (4), we obtain one-soliton solutions for eqs (3), which can be expressed as H = k tanh ξ, (a) V = k sech ξ, (b) W = k sech ξ, (c) P = k 3 sech ξ tanh ξ. (d) 3. -Soliton solutions in terms of Wronskian Suppose that -soliton solutions in terms of Wronskian for eqs (3) can be defined by F = W (φ,φ,...,φ ) = φ φ () φ () φ φ () φ ().. φ φ () φ ( ) φ ( ). φ().... φ( ) =, () where φ i j = i φ j / x i (i =,,..., ), φ j = e ξ j + ( ) j+ e ξ j and ξ j = k j x k j y 6k 3 j t + ξ 0 j with ξ 0 j and k j ( j = 0,,...,) being the constants. It can be seen that φ j satisfies the following relations: φ j,y = φ j,x, (3a) φ j,xx = k j φ j, (3b) φ j,t = 6φ j,xxx. (3c) According to the properties of the derivative of Wronskian, the derivatives of F with respect to the variables x, y and t can be obtained as follows: F =, F x =,, F y =,, F t = 6( 4,,, 3,, + +, + ), F xx = 3,, +, +, F xy = F xx = ( 3,, +, + ), F xxx = 4,,, + 3,, + +, +, F xxy = F xxx = ( 4,,, + 3,, + +, + ), F xt = 6( 5, 3,,, 3, + +, + 3 ), F yt = F xt =6( 5, 3,,, 3,, + +, + 3 ), F xxxx = 5, 3,,, + 3 4,,, + + 3,, ,, + +, + 3, F xxxy = F xxxx = ( 5, 3,,, + 3 4,,, + + 3,, ,, + +, + 3 ), where K indicates that there are consecutive derivatives up to the order of K +. Substituting all the derivatives above F into bilinear form (6), we can obtain (D x D t + D y D t + 4D 4 x + 4D3 x D y)f F = { F t (F x + F y ) 4( 3F xy F xx 3F xx +3F x F xxy + F y F xxx + 4F x F xxx ) +F[F yt + F xt + 4(F xxxy + F xxxx )]} = { F t (F x F x ) 4[ 3F xx ( F xx + F xx ) 3F x F xxx F x F xxx + 4F x F xxx ] +F[ F xt + F xt + 4( F xxxx + F xxxx )]} = 0, (4a) ( x + ) F F = F(F x + F y ) = 0. (4b)

4 45 Page 4 of 6 Pramana J. Phys. (08) 90:45 Thus, we have verified that solutions () are solutions for bilinear form (6). Taking φ j = e ξ j + ( ) j+ e ξ j k F x F = k j F F x k j F x F and ξ j = k j x k j y 6k 3 j t + ξ 0 j with ξ 0 j and k j as some constants, we find that φ j satisfies relations (3). = 3,,τ (, + Hence, we can obtain the -soliton solutions for system () in terms of the Wronskian as follows: 3,, ) + ( 5, 3,,,τ H = (log ) x, (5a) 3, +,τ ), V = (log ) xy. (5b) In the following, we shall verify that solutions () satisfy BT (9). k F F x = k j F x F k j F F x The ( )-soliton solutions for bilinear form (6) =,τ (, + have the form of F 4,,, ) = W (φ,φ,...,φ,τ) φ φ () φ () φ ( ) 0, ( 3,,τ φ φ () φ () φ ( ) 0 4,,,τ ). = Besides, the Wronskian identities are listed as [0] φ φ () φ() φ( ) 0 φ φ () φ () φ ( ) α,...,α j, bα j,α j+,...,α =,τ, (6) where τ = (0, 0,...,0, ) T and T represents the = b j α,...,α, transpose of a matrix. Then, the derivatives of F are computed as follows: where α j s are -dimensional column vectors and bα j F =,τ, F x = 3,,τ, denotes (b α j,, b α j,,...,b α j, ) T. F y = F x = 3,,τ, Mab Mcd Mac Mbd + Mad Mbc F xy = 4,,,τ 3,,τ, F xx = = 4,,,τ + 3,,τ, M 0 abcd 0 Mabcd F xxx = 5, 3,,,τ where M is an ( ) matrix and a, b, c, d represent the -dimensional column vectors. + 4,,,τ + 3, +,τ, F t = 6( 5, 3,,,τ 4,,,τ + 3, +,τ ), k F F = F F F F k j k j =,τ ( 3,, +, + ) + ( 4,,,τ 3,,τ ), Taking μ = 0andλ = k, and substituting the derivatives of F and F into BT (9) with the Wronskian identities can yield (D x + D y )F F = (F x F F F x ) + (F y F F F y ) = 0, (7a) (D x D y μd x + λ)f F = F y F x F y F x + F F xy + FF xy + k F F 3 0 τ = = 0, (7b) 0 3 τ (Dx λ)f F = (Dx k )F F = F x F x + F F xx + FF xx k F F

5 Pramana J. Phys. (08) 90:45 Page 5 of 6 45 Figure. One soliton via solutions (5) with =. The parameters adopted here are k =.5, ξ 0 (b) t = 0; (c) t = 0.. = 0for(a) t = 0.; Figure. Parallel solitons via solutions (5) with =. The parameters adopted here are ξ 0 = 0, ξ 0 = 0. k =.4, k =.7 for(a) t = 0.; (b) t = 0; (c) t = τ = 0 3 τ = 0, (7c) (D t + 4Dx 3 + λd x)f F = (F x F xx F x F xx ) F [F t +4(3k F x + F xxx )] + F(F t + k F x + 4F xxx ) τ = τ τ τ = 0. (7d) Thus, we have verified that solutions () satisfy BT (9). Here, we present some figures to show the shapes and motion of the one soliton and two solitons for system (). Figures a c show the motion of a kink-dark-like soliton: H component shows the kink-shaped soliton while V component displays the soliton under the background which is like the dark soliton. It can be seen that the kink-dark-like soliton propagates stably, and keeps its width and amplitude invariant when propagating. Figures a c display the parallel propagation of twokink-dark-like solitons on the x y plane at different t s. We can see that the two-kink-dark-like solitons maintain their amplitudes and velocities. During the propagation,

6 45 Page 6 of 6 Pramana J. Phys. (08) 90:45 there is no energy exchange between the two solitons. In addition to the parallel solitons, the KPII equation has a variety of the soliton interactions, e.g. elastic interaction and inelastic interaction [30 3]. umbers of the incoming and outgoing solitons of the elastic interaction are the same, when compared with the inelastic interaction between two solitons for eq. () [30,3]. 4. Conclusions Investigation has been made on KP-based system () in fluid dynamics. Using the Hirota method and symbolic computation, we have deduced bilinear form (6) and BT (9) for system (), and constructed -soliton solutions (5) in terms of the Wronskian for system () from BT (9). Using the Wronskian technique, it has been verified that solutions (5) satisfy bilinear form (6)and BT (9). Figures a c have displayed the kink-dark-like soliton for system (), and the soliton keeps its shape unchanged during the propagation. Figures a c have illustrated the parallel propagation of two-kink-darklike solitons on the x y plane at different t s, during which the two solitons maintain their amplitudes and velocities unchanged. Acknowledgements This work has been supported by the ational atural Science Foundation of China under Grant os 7707, 703 and 47050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 07ZZ05) and by the Fundamental Research Funds for the Central Universities of China under Grant o. 0BUP- TYB0. References [] R K Bullough and P J Caudrey, Solitons (Springer, Berlin, 980) [] M P Barnett, J F Capitani, J Von Zur Gathen and J Gerhard, Int. J. Quant. Chem. 00, 80 (004) [3] G Das and J Sarma, Phys. Plasmas 6, 4394 (999) [4] M J Ablowitz and P A Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, Cambridge, 99) [5] M Iwao and R Hirota, J. Phys. Soc. Jpn 66, 577 (997) [6] M Wadati, J. Phys. Soc. Jpn 3, 68 (97) [7] T Xu, J Li, H Q Zhang, Y X Zhang, Z Z Yao and B Tian, Phys. Lett. A 369, 458 (007) M Li, T Xu and D X Meng, J. Phys. Soc. Jpn 85(), 400 (06) [8] F Caruello and M Tabor, Physica D 39, 77 (989) [9] X Y Tang and S Y Lou, J. Math. Phys. 44, 4000 (003) [0] M Wadati, J. Phys. Soc. Jpn 38, 673 (975) [] R Hirota, Direct method in soliton theory (Springer, Berlin, 980) [] S Zhang, C Tian and W Y Qian, Pramana J. Phys. 86, 59 (06) [3] V B Matveev and M A Salle, Darboux transformation and soliton (Springer, Berlin,99) [4] M Li and T Xu, Phys. Rev. E 9, 0330 (05) [5] T Xu, M Li and L Li, EPL 09, (05) [6] T Xu and Y Zhang, onlinear Dyn. 73, 485 (03) [7] M Li, T Xu, L Wang and F H Qi, Appl. Math. Lett. 60, 8 (06) [8] T Xu, H J Li, H J Zhang, M Li and S Lan, Appl. Math. Lett. 63, 88 (07) [9] T Xu, C J Liu, F H Qi, C X Li and D X Meng, J. onlin. Math. Phys. 4, 6 (07) [0] J J immo and C Freeman, Phys. Lett. A 95, 4 (983) [] C Freeman and J J immo, Phys. Lett. A 95, (983) [] B Qin, B Tian, L C Liu, X H Meng and W J Liu, Commun. Theor. Phys. 54, 059 (00) [3] B Kadomtsev and V Petviashvili, Sov. Phys. Dokl. 5, 539 (970) [4] Y akamura, H Bailung, K E Lonngren, Phys. Plasmas 6, 3466 (999) [5] S Y Lou and X B Hu, J. Math. Phys. 38, 640 (997) [6] C L Bai and H Zhao, Phys. Scr. 73, 49 (006) [7] C Q Dai and C Y Liu, onlinear Anal. 7, 7 (0) [8] J Q Mei, D S Li and H Q Zhang, Chaos Solitons Fractals, 669 (004) [9] F Galas, J. Phys. A 5, 98 (995) [30] G Biondini and S Chakravarty, J. Math. Phys. 47, (006) [3] S Chakravarty and Y Kodama, J. Phys. A 4, 7509 (008) [3] M S Khatun, M F Hoque and M A Rahman, Pramana J. Phys. 88, 86 (07)