Exact Interaction Solutions of an Extended (2+1)-Dimensional Shallow Water Wave Equation

Transcription

1 Commun. Theor. Phys. 68 (017) Vol. 68, No., August 1, 017 Exact Interaction Solutions of an Extended (+1)-Dimensional Shallow Water Wave Equation Yun-Hu Wang ( 王云虎 ), 1, Hui Wang ( 王惠 ), 1, Hong-Sheng Zhang ( 张洪生 ), 3 and TEMUER Chaolu ( 特木尔朝鲁 ) 1 1 College of Art and Sciences, Shanghai Maritime University, Shanghai 01306, China Department of Mathematics, Shanghai University, Shanghai 00, China 3 College of Ocean Science and Engineering, Shanghai Maritime University, Shanghai 01306, China (Received Feruary 0, 017; revised manuscript received March 6, 017) Abstract Applying the consistent Riccati expansion method, the extended (+1)-dimensional shallow water wave equation is proved consistent Riccati solvable and the exact interaction solutions including soliton-cnoidal wave solutions, solitoff-typed solutions are obtained. With the help of the truncated Painlevé expansion, the corresponding nonlocal symmetry is also given, and furthermore, the nonlocal symmetry is localized by prolonging the related enlarged system. PACS numbers: 0.30.Ik, 05.5.Yv, 0.70.Wz DOI: / /68//165 Key words: soliton-cnoidal wave solutions, consistent Riccati expansion, nonlocal symmetry 1 Introduction The investigation in finding the exact solutions of nonlinear evolution equations (NLEEs) is important for the understanding of nonlinear phenomena in various fields of science, especially in physics. For example, the nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers are often modelled by the bell-shaped sech solutions and the kink-shaped tanh solutions. [1 Up to now, many powerful and systematic methods have been developed to construct explicit solitary wave solutions for the NLEEs, such as the Darboux transformation (DT), [ 3 the Bäcklund transformation (BT), [ and the function expansion method. [5 Recently, inspired from the nice results via nonlocal symmetry method, [6 7 Lou proposed the consistent Riccati expansion (CRE) method, [8 which is a lucid and systematic method to construct interaction solutions between different types of nonlinear excitations. Many interaction solutions for NLEEs, for instance, the Boussinesq Burgers equation, [9 Boiti Leon Pempinelli equation, [10 modified KP equation, [11 Sharma Tasso Olver equation, [1 breaking soliton equation, [13 Bogoyavlenskii coupled KdV system, [1 higher-order Broer Kaup system, [15 KdV-mKdV equation, [16 modified Bogoyavlenskii Schiff equation, [17 coupled mkdv BLMP system, [18 are discussed in detail. In this paper, we focus on the following extended (+1)-dimensional shallow water wave equation v t v x 1 x v y vv y + v y + αv y = 0, (1) which can be rewritten as [19 u xt u u y u xy u x + u xy + αu xy = 0, () through transformation v = u x. When α = 0, Eq. () can be reduced to the breaking soliton equation. [0 3 With the help of the Hereman s simplified method, Ref. [19 derived the multiple-soliton solutions of Eq. (). In Ref. [, the bilinear form, N-soliton solutions, Lax pair, bilinear BT, Darboux covariant Lax pair, and infinite conservation laws of Eq. () are presented by means of the binary Bell polynomial and Hirota bilinear method. To our knowledge, the soliton-cnoidal wave solutions, solitoff-typed solutions, nonlocal symmetry, and consistent Riccati solvability for Eq. () have not been reported. The purpose of this work is to employ the truncated Painlevé expansion, consistent Riccati expansion (CRE) and symbolic computation to study the nonlocal symmetry, BT, consistent Riccati solvability and interaction solutions for Eq. (). In Sec., the CRE method is used to prove the CRE solvability of Eq. (). In Sec. 3, we will give some interaction solutions of Eq. (), including solitoncnoidal wave interaction solutions and solitoff-typed solutions. In Sec., the nonlocal symmetry related to truncated Painlevé expansion is studied. Section 5 will be our conclusions. Consistent Riccati Solvability As is well-known, investigation of integrability for NLEEs can be regarded as a pre-test and first step of its exact solvability. Among the methods employed to study the integrability of NLEEs, the Painlevé analysis not only Supported by the National Natural Science Foundation of China under Grant Nos , , , , and hwang@shmtu.edu.cn c 017 Chinese Physical Society and IOP Publishing Ltd

2 166 Communications in Theoretical Physics Vol. 68 can be used to verify the integrability, but also can be used to find exact solutions and other significant properties such as Lax pair and BT. [5 6 Recently, Lou defined the consistent Riccati solvability for NLEEs if it has a consistent Riccati expansion. [8 It has been shown that many more integrable systems are CRE solvable which indicates that the CRE solvable systems are integrable. [8 18 In the following, we will apply the CRE to Eq. (), and prove it is consistent Riccati solvable. For Eq. (), we give the possible CRE solution in the form of u = u 0 + u 1 R(w), (3) u 0, u 1, w are related to {t, x, y}, and function R(w) needs to satisfy the following Riccati equation R w = a 0 + a 1 R + a R, R R(w), () with R w indicates dr/dw, and a i (i = 0, 1, ) are arbitrary constants. By substituting the expansion (3) with Eq. () into Eq. () and vanishing different powers of R yields u 1 = a w x, u 0 = δ dx, w x w x δ = a 1 a 0 a, (5) and the function w only needs to satisfy C x + K x + SK x + KS x + αk x δw x w xy = 0, (6) C = w t, K = w y, S = w x 3 w w x w x w x wx (7) are usual Schwartzian variables. It is clear that if w is a solution of Eq. (6), then expression (3) is also a solution of Eq. (). Hereby, we have the CRE solvable theorem for Eq. (): Theorem 1 Equation () is CRE solvable and possesses the CRE solution u = δ w x wx dx + a w x R(w), (8) with w satisfies Eq. (6). It is easy to see that the expression (8) can also be regarded as the BT of Eq. (). As is well-known, the Riccati equation () has the following tanh function solution R(w) = 1 [ a 1 + ( 1 ) δ tanh δw, (9) a therefore, from the CRE solution (8), we can give the interaction solutions for Eq. () in the form as u = δ w x wx [ a 1 + ( 1 ) δ tanh δw w x. (10) Meanwhile, the solution of Eq. (1) can be also given as v = δ ( 1 ) w x tanh δw ( 1 ) δw tanh δw δ w x + 1 w x 1 w w x wx, (11) via the transformation v = u x. In the following section, starting from the expression (11) and w satisfies Eq. (6), we will study some interesting interaction solutions for Eq. (1). 3 Exact Interaction Solutions It is obvious that once the solutions of w satisfies Eq. (6) are obtained, then we can get the interaction solutions of Eq. (1). Here are some interesting examples. Example 1 Soliton Solution We first take the trivial w solution as w = κx + ιy + ωt + ξ 0, (1) then we obtain the single soliton solution for Eq. (1) as v = δ [ 1 κ tanh δ(κx + ιy + ωt + ξ0 ) δ κ, ω = κ ιδ ια. (13) Example Soliton-cnoidal Solutions In order to construct the soliton-cnoidal wave solutions, we take the form of function w as w = κ 1 x + ι 1 y + ω 1 t + G(θ), θ = κ x + ι y + ω t, (1) G 1 G 1 (θ) = G θ satisfies the following elliptic equation G 1θ = c 0 + c 1 G 1 + c G 1 + c 3 G c G 1, (15) with c i (i = 0,..., ) being constants. Substituting Eq. (1) with Eq. (15) into the w Eq. (6), we can obtain c 0 = δκ 1 3κ κ 1(c ι κ + αι + ω ) 3ι κ + κ 1(c 1 ι κ + αι 1 + ω 1 ) 3ι κ 3, c 3 = δκ 1 + c κ c 1κ 3κ 3κ 1 3κ + αι 1 1 3ι κ + ω 1 1 3ι κ ω α, c = δ. (16) 1 κ 1 κ ι 3κ 1 κ Then Eq. (1) has the following explicit solutions v = 1 [ (κ 1 + κ G 1 ) tanh 1 [ 1 (κ 1x + ι 1 y + ω 1 t + G) κ G 1θ tanh (κ 1x + ι 1 y + ω 1 t + G) κ G 1 + κ 1 κ 3 G κ 1κ G 1 κ G 1 G 1θθ + κ G 1θ + κ3 1κ G 1 κ 1 κ 3 G 1θθ + κ 1 (κ 1 + κ G 1 ), (17)

3 No. Communications in Theoretical Physics 167 with the relations (15) and (16). It is well known that the explicit solutions of Eq. (15) can be expressed by Jacobi elliptic functions, here, we just take a simple solution of Eq. (15) as G = 1 arctanh [sn (θ, n), (18) sn(θ, n) is the usual Jacobi elliptic sine function. Substituting expression (18) with Eq. (16) into Eq. (15) and vanishing all the different powers of sn, cn, and dn, we have c 1 = 0, c = n 1, a 0 = a 1 a 1 a, κ 1 = nκ, ω = n ι κ ι κ αι + αι 1 n + ω 1 n, (19) which leads Eq. (17) to the following soliton-cnoidal wave interaction solutions by choosing a 0 = 0, a 1 = a = 1, saying v = 16(S 1) κ 1 3CD(S 1)κ κ (S 1)(n S 1)κ 1κ 8CD(n S 1)κ 1 κ 3 + (n S 1) κ 8(CDκ S κ 1 + κ 1 ) T Sκ (n 1)(κ + κ 1 S κ 1 + CDκ 1 κ S n κ ) 1 (CDκ S κ 1 + κ 1 ) T + 16(CDκ S κ 1 + κ 1 ) [(7S n 8S n + S n 6S n + S 8n + 7)κ 8CD(s n S + n 1)κ 1 κ 3 (S 1)(n S 1)κ 1κ + 3CD(S 1)κ κ (S 1) κ 1, (0) [ 1 sn (θ, n) = S, cn (θ, n) = C, dn(θ, n) = D, T = tanh (κ 1x + ι 1 y + ω 1 t) + 1 arctanhsn(θ, n), and ω, κ 1 satisfy Eq. (19). From Fig. 1, we can find that a single soliton moving on a cnoidal wave background instead of on the plane continuous wave background. This kind of interactions can be used to display many interesting physical phenomena, such as the Fermionic quantum plasma. [7 Fig. 1 Plots of the single soliton on a cnoidal wave background expressed by Eq. (0) at t = 0 with the parameter selections {α, n, ι 1, ι, ω 1, ω, κ } = {1, 3, 1, 1, 1.5, 1.5, 1} of Eq. (1): (a) Two-dimensional image for v at t = 0; (b) (d) One-dimensional image for v at y = 0, x = 0 and its density plot, respectively. Example 3 Solitoff-Typed Solutions In order to obtain the solitoff-typed solutions, [8 9 we here define function w as P 1 P 1 (τ) = P τ satisfies the following Riccati equation w = κ 1 x + ι 1 y + ω 1 t + P (τ), τ = κ x + ι y + ω t, (1) P 1τ = P 1 P 1, () similar to the procedure as Example, one can obtain the following solitoff-typed solutions of Eq. (1) 1 [ κ v = ( e κx+ιy+ωt + 1) T + κ e κx+ιy+ωt T + κ ( e κx+ιy+ωt e κx+ιy+ωt 1), (3) [ κ T = tanh x ι 1y + αι 1 t + αι t + ω t 1 ln 1 ln( eκx+ιy+ωt + 1). ()

4 168 Communications in Theoretical Physics Vol. 68 Fig. Plots of the solitoff-typed solutions expressed by Eq. (3) at t = 0 with the parameter selections {α, ι 1, ι, κ, ω } = {1,,,, } of Eq. (1): (a) Two-dimensional image for v at t = 0; (b) (d) One-dimensional image for v at y = 0, x = 0 and its density plot, respectively. Nonlocal Symmetry Related to Truncated Painlevé Expansion One knows that the symmetry method [30 31 is very effective for constructing explicit solutions of NLEEs. In Refs. [6 7, the nonlocal symmetries related to the BT and DT are used successfully to construct the interaction solutions among different types of nonlinear excitation, such as the soliton-cnoidal wave solutions. Recently, it is found that the Painlevé analysis can be also used to find nonlocal symmetries of NLEEs. [3 33 In this section, with the help of the truncated Painlevé expansion method, we will first give a non-auto BT of Eq. (), then write down the nonlocal symmetry related to the BT, and localize the nonlocal symmetry by introducing an enlarged system. According to the Painlevé analysis, [5 one can know that Eq. () exists the following truncated Painlevé expansion u = u 0 + u 1 f, (5) with u 0, u 1, f being the functions of (x, y, t). Substituting Eq. (5) into Eq. (), and vanishing all the coefficients of different powers of 1/f, we have u 0 = 1 f + 1 f x and the function f satisfies f dx, u 1 = f x, (6) f x C x + KS x + SK x + K x + αk x = 0. (7) Here, C, K, S are the Schwarzian variables shown in Eq. (7), which function w replaced with function f. It is known that the Schwarzian variables keep the Möbius invariance property f a + bf, (ad bc), (8) c + df which means that variable f possesses the point symmetry as k 0, k 1, k being constants. σ f = k 0 + k 1 f + k f, (9) Theorem If function f is a solution of system (7), then u = 1 f + 1 f dx f x (30) f x f is the solution of Eq. (). From the Theorem, on the one hand, one can construct various exact solutions of Eq. (), on the other hand, one can also obtain the nonlocal symmetry. As is known, under the invariant property f x u u + εσ u, (31) one knows that the symmetry equation for Eq. () reads σ u xt + σ u xy u x σ u y u xy σ u x u y σ u u σ u y + ασ u xy = 0. (3) It can be verified that Eq. () possesses a residual symmetry [3 σ u = f x, (33) with u and f satisfy Eqs. (7) and (30), respectively. By solving f from Eq. (30), one knows that the residual symmetry (33) of u is nonlocal, therefore, Eq. (33) is a nonlocal symmetry of Eq. (). To find out the symmetry group of the nonlocal symmetry (33), we need to extend the original system such that nonlocal symmetry becomes the local Lie point symmetries of an enlarged system. Therefore, we introduce a new dependent variable f 1 satisfies f 1 = f x. (3) Now one can easily verify that the nonlocal symmetry (33) of the original system () becomes Lie point symmetries of the enlarged system including Eqs. (), (30), and (3), namely σ u f 1 σ f 1 σ f = ff 1 f, (35) with σ f = f is obtained by choosing k 0 = k 1 = 0, k = 1 in Eq. (9).

5 No. Communications in Theoretical Physics 169 Based on the Lie s first theorem, the corresponding initial value problem of Eq. (35) reads dū(ϵ) = f 1 (ϵ), ū(0) = u, d f(ϵ) = f (ϵ), f(0) = f, d f 1 (ϵ) = f(ϵ) f 1 (ϵ), f1 (0) = f 1, (36) with arbitrary group parameter ϵ. By solving the initial value problem (36), we have the following finite transformation theorem: Theorem 3 If {u, f 1, f} is a solution of the prolonged system (), (30), and (3), then {ū(ϵ), f 1 (ϵ), f(ϵ)} are given by ū(ϵ) = u ϵf ϵf, f 1 f1 (ϵ) = (1 + ϵf), f f(ϵ) = 1 + ϵf.(37) Theorem 3 provides a way to obtain new solution of Eq. () from the old ones. 5 Conclusions In this paper, we investigate the extended (+1)- dimensional shallow water wave equation by means of CRE, truncated Painlevé expansion and symbolic computation. On the one hand, by using the CRE method, Eq. () is proved CRE solvable. It is interesting that the CRE solvable can be regarded as the pre-test for finding the exact interaction solutions of NLEEs. And on this basis we obtain the soliton-cnoidal wave interaction solutions (0) and solitoff-typed solutions (3) for Eq. (1). On the other hand, with the help of truncated Painlevé expansion (5) of Eq. (), a non-auto BT (30) is obtained, and from which, we present the nonlocal symmetry (33) of Eq. (). Furthermore, the nonlocal symmetry (33) is localized by prolonging the original system () to the enlarged system (), (30), and (3). The finite transformation (37) of the nonlocal system are also obtained by solving the standard Lie s initial value problem (36). It has been shown that Eq. () is CRE solvable and possesses nonlocal symmetry (33), therefore, some other interesting problems such as the nonlocal symmetry reduction, rational solutions, and so on [3 35 are worthy of further study, which will be discussed in our future work. References [1 Z. H. Yang, Commun. Theor. Phys. 6 (006) 807. [ V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons, Springer, Berlin (1991). [3 C. H. Gu, H. S. Hu, and Z. X. Zhou, Darboux Transformations in Soliton Theory and Its Geometric Applications, Shanghai Science and Technology Press, Shanghai (1999). [ C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, Academic Press, New York (198). [5 E. G. Fan, Phys. Lett. A 77 (000) 1. [6 S. Y. Lou, X. R. Hu, and Y. Chen, J. Phys. A. Math. Theor. 5 (01) [7 X. R. Hu, S. Y. Lou, and Y. Chen, Phys. Rev. E 85 (01) [8 S. Y. Lou, arxiv: v[nlin, (013). [9 Y. H. Wang, Appl. Math. Lett. 38 (01) 100. [10 Y. H. Wang and H. Wang, Phys. Scr. 89 (01) [11 B. Ren, Phys. Scr. 90 (015) [1 H. Pu and M. Jia, Commun. Theor. Phys. 6 (015) 63. [13 W. G. Cheng, B. Li, and Y. Chen, Commun. Theor. Phys. 63 (015) 59. [1 X. R. Hu and Y. Q. Li, Appl. Math. Lett. 51 (016) 0. [15 X. P. Xin and X. Q. Liu, Commun. Theor. Phys. 66 (016) 79. [16 H. C. Hu, M. Y. Tan, and X. Hu, J. Association Arab Univ. Basic Appl. Sci. 1 (016) 6. [17 L. L. Huang and Y. Chen, Chin. Phys. B 5 (016) [18 J. C. Chen, Z. Y. Ma, and Y. H. Hu, Chin. Phys. Lett. 3 (017) [19 A. M. Wazwaz, Stud. Math. Sci. 1 (010) 1. [0 F. Calogero and A. Degnsperis, Nuovo. Cimento. B 3 (1976) 01. [1 F. Calogero and A. Degnsperis, Nuovo. Cimento. B 39 (1977) 1. [ Z. Y. Yan and H. Q. Zhang, Comput. Math. Appl. (00) 139. [3 H. H. Hao, D. J. Zhang, J. B. Zhang, and Y. Q. Yao, Commun. Theor. Phys. 53 (010) 30. [ Y. H. Wang and Y. Chen, Chin. Phys. B (013) [5 J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. (1983) 5. [6 J. Weiss, J. Math. Phys. (1983) 105. [7 A. J. Keane, A. Mushtaq, and M. S. Wheatland, Phys. Rev. E 83 (011) [8 S. Y. Lou and X. Y. Tang, Methods of Nonlinear Mathematical Physics, Science Press, Beijing (006) (in Chinese). [9 S. H. Ma and J. P. Fang, Commun. Theor. Phys. 5 (009) 61. [30 P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York (1993). [31 G. W. Bluman and K. Kumei, Symmetries and Differential Equations, Springer, Berlin (1989). [3 S. Y. Lou, arxiv.org/abs/ v1, (013). [33 X. R. Hu and Y. Chen, Z. Naturforsch. 70 (015) 79. [3 H. W. Yang, X. R. Wang, and B. S. Yin, Nonlinear Dyn. 76 (01) 175. [35 Y. Zhang, H. H. Dong, X. E. Zhang, and H. W. Yang, Comput. Math. Appl. 73 (017) 6.