Painlevé Analysis and Darboux Transformation for a Variable-Coefficient Boussinesq System in Fluid Dynamics with Symbolic Computation

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1 Commun. Theor. Phys. (Beijing China) 53 (2010) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53 No. 5 May Painlevé Analysis and Darboux Transformation for a Variable-Coefficient Boussinesq System in Fluid Dynamics with Symbolic Computation LI Hong-Zhe (Ó ) 1 TIAN Bo ( þ) 123 LI Li-Li (ÓÛ ) 1 and ZHANG Hai-Qiang ( Ö) 1 1 School of Science P.O. Box 122 Beijing University of Posts and Telecommunications Beijing China 2 State Key Laboratory of Software Development Environment Beijing University of Aeronautics and Astronautics Beijing China 3 Key Laboratory of Optical Communication and Lightwave Technologies Ministry of Education P.O. Box 128 Beijing University of Posts and Telecommunications Beijing China (Received February ; revised manuscript received March ) Abstract The new soliton solutions for the variable-coefficient Boussinesq system whose applications are seen in fluid dynamics are studied in this paper with symbolic computation. First the Painlevé analysis is used to investigate its integrability properties. For the identified case we give the Lax pair of the system is found and then the Darboux transformation is constructed. At last some new soliton solutions are presented via the Darboux method. Those solutions might be of some value in fluid dynamics. PACS numbers: Ge Yv Key words: variable-coefficient Boussinesq system Lax pair Darboux transformation soliton solutions symbolic computation 1 Introduction The Boussinesq system u tt + αu xx + γ u xxxx + β(u 2 ) xx = 0 (1) has been first introduced in Ref. [1]. It has been found that Eq. (1) and its variants arise in several physical applications including the long waves in shallow water [1] one-dimensional nonlinear lattice waves [2] vibrations in a nonlinear string [3] and ion sound waves in a plasma. [4] The nonlinear partial differential equations (NPDEs) are of current importance in many fields [5 7] and Eq. (1) and its variants are a class of the NPDEs. In recent years there has been much interest in the variants of the Boussinesq system. [8] In view of the work before including the ones in Ref. [8] we find that many variants of the Boussinesq system are the reduced form of the following variable-coefficient Boussinesq system [9] u t + α 1 (t)v x + β 1 (t)uu x + γ 1 (t)u xx = 0 v t + α 2 (t)uv x + β 2 (t)vu x γ 2 (t)v xx + p(t)u xxx = 0 (2) where α i (t) β i (t) γ i (t) (i = 1 2) and p (t) are all smooth functions of time t. In fact when we let α 1 (t) = β 1 (t) = α 2 (t) = β 2 (t)=1 γ 1 (t) = γ 2 (t) = µ and p(t) = ν system (2) will reduce to the Whitham Broer Kaup (WBK) equations [10] u t + uu x + v x + µu xx = 0 v t + (uv) x + νu xxx µv xx = 0 (3) which are also the generalized form of the classical long-wave equations describing the shallow water wave with diffusion [11] and the Boussinesq Burgers (BB) equations. [12] Moreover system (2) also includes the Kupershmidt equations u t + uu x + v x + δu xx = 0 v t + (uv) x δv xx = 0 (4) with some analytic solutions reported in Ref. [13] and the following equations u t + uu x + v x = 0 v t + (uv) x + u xxx = 0 (5) with the periodic-wave and solitary/traveling-wave solutions reported in Ref. [14]. In view of the variants of Boussinesq system as mentioned above we have become interested in system (2) in this paper. With symbolic computation [5 7] we will first perform the Painlevé analysis to system (2) and get the constraints on the coefficients in Sec. 2. Under the constraint conditions given in Sec. 2 we will derive the Lax pair associated with system (2) in Sec. 3. Then from this spectral problem the Darboux transformation (DT) of system (2) will be constructed in Sec. 4. Applications of the obtained DT to deriving some new soliton solutions and discussions on Supported by the National Natural Science Foundation of China under Grant No by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. BUAA-SKLSDE-09KF-04 Beijing University of Aeronautics and Astronautics by the National Basic Research Program of China (973 Program) under Grant No. 2005CB and by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos and Chinese Ministry of Education Corresponding author tian.bupt@yahoo.com.cn

2 832 LI Hong-Zhe TIAN Bo LI Li-Li and ZHANG Hai-Qiang Vol. 53 those solutions will be seen in Sec. 5. Finally Sec. 6 will be our conclusions. 2 Painlevé Analysis The Painlevé analysis with the Weiss Tabor Carnevale (WTC) algorithm is one of the powerful methods for identifying the complete integrability properties of the NPDEs. [15] It is known that the Painlevé integrability is a necessary condition for an NPDE to be Lax and/or IST integrable. [16] Once the given equation passes the Painlevé property we can look for the Lax pair and Darboux transformation [17] for establishing the complete integrability properties of the system. According to the WTC procedure the solutions of system (2) can be expanded in terms of the Laurent series u(x t) = ϕ p (x t) u j (x t)ϕ j (x t) (6) v(x t) = ϕ q (x t) j=0 v j (x t)ϕ j (x t) (7) j=0 with u 0 (x t) v 0 (x t) 0. Here p and q are both negative integers u j (x t) and v j (x t) are the set of expansion coefficients which are analytic in the neighborhood of the noncharacteristic singular manifold ϕ(x t) = x+ φ(t) = 0. Looking at the leading-order behaviour we introduce into system (2) then obtain and the compatibility conditions u u 0 ϕ p v v 0 ϕ q (8) p = 1 q = 2 (9) 2γ 1 (t)u 0 β 1 (t)u 2 0 2α 1 (t)v 0 = 0 (10) [2α 2 (t) + β 2 (t)]u 0 v 0 + 6γ 2 (t)v 0 + 6p(t)u 0 = 0. (11) Substituting the full Laurent series and keeping the leading-order terms alone we find that system (2) can pass the Painlevé test when α 1 (t) = mβ 1 (t) α 2 (t) = β 1 (t) = β 2 (t) γ 1 (t) = γ 2 (t) = nβ 1 (t) p(t) = (1 16k2 n 2 )β 1 (t) 16k 2 (12) m with m n and k being arbitrary constants. Thus the resonances are found to be j = (13) As usual the resonance at j = 1 corresponds to the arbitrariness of the singularity manifold ϕ(x t). Upon substituting the full Laurent series into system (2) and collecting the coefficients of different powers of ϕ(x t) we find that system (2) admits sufficient number of arbitrary functions for the above conditions. Having identified the integrability nature of system (2) our next aim is to construct the Lax pair and Darboux transformation under Constraints (cons). 3 Lax Pair of System (2) In this section we will manage to derive the Lax pair associated with system (2). Consider the linear eigenvalue problem Ψ x = UΨ Ψ t = V Ψ Ψ = (Ψ 1 Ψ 2 ) T (14) with the Lax operator U given in the form ( ku + λ (k 4k 2 n)u x 4k 2 ) mv U = (15) 1 ku λ where λ is the constant spectral parameter m and k are arbitrary constants. At the same time we assume that the matrix V can be written as 2 V = V (j) λ j. (16) j=0 According to the compatibility condition for Eqs. (14) i.e. the zero-curvature equation U t V x + [U V ] = 0 the matrix V can be obtained ( as ) A B V = (17) C A where A = β 1(t)( 2λ 2 + 2k 2 u 2 + ku x ) 4k B = 1 4 β 1(t)[u xx 4kmv x 4knu xx + 8kmv(ku λ) + 2u x (4kn 1)(ku λ)] C = (λ ku)β 1(t). 2k and n is also an arbitrary constant. It is easy to verify that system (2) can be recovered from the compatibility condition of the above Lax pair. Especially when we let β 1 (t) = 1 m = 1 n = µ k = ±(1/4 µ 2 + ν) the Lax pair found above yields Eqs. (3) i.e. the famous WBK equations. Furthermore the U V given in Eqs. (15) and (17) can also yield Eqs. (4) and (5) with suitable choices of the coefficient β 1 (t) and arbitrary constants k m and n. 4 Darboux Transformation Based on the Lax pair obtained in Sec. 3 we will concentrate on constructing the DT of system (2). Actually the DT is a special gauge transformation with the following form Ψ = DΨ (18) where D is a nonsingular matrix. It is required that Ψ should also satisfy the same form of spectral problem (14) Ψ x = Ū Ψ D x + DU = ŪD (19) Ψ t = V Ψ D t + DV = V D (20) where Ū and V have the same forms as U and V by replacing u and v with ū and v. We suppose that D has the following form ( ) a0 + λa 1 b 0 D = (21) c 0 d 0 + λd 1 where a 0 a 1 b 0 c 0 d 0 and d 1 are all undetermined functions with respect to the variables x and t. We insert D U and Ū into (19) and then collect the coefficients of λ j (j = 0 1 2). The case of j = 2 is trivial. For the case j = 1 it yields

3 No. 5 Painlevé Analysis and Darboux Transformation for a Variable-Coefficient Boussinesq System in Fluid Dynamics with kua 1 kūa 1 + a 1x = 0 (22) 2b 0 + 4mva 1 k 2 4k 2 m vd 1 (4k 2 n k)(a 1 u x d 1 ū x ) = 0 (23) d 1 a 1 + 2c 0 = 0 (24) d 1x kud 1 + kūd 1 = 0. (25) For the case j = 0 it yields 4m vc 0 k 2 + 4nc 0 ū x k 2 + ua 0 k ūa 0 k c 0 ū x k + b 0 + a 0x = 0 (26) 4m vd 0 k 2 4mva 0 k 2 4na 0 u x k 2 + 4nd 0 ū x k 2 ub 0 k ūb 0 k + a 0 u x k d 0 ū x k + b 0x = 0 (27) kuc 0 a 0 + kūc 0 + d 0 + c 0x = 0 (28) d 0x 4mvc 0 k 2 4nc 0 u x k 2 ud 0 k + ūd 0 k + c 0 u x k b 0 = 0. (29) Solving (22) (25) we get d 1 = 1 a 1 (30) c 0 = a a 1 ū = kua 1 + a 1x ka 1 (32) v = 4mva2 1 k2 + 4na 2 1 u xk 2 4nū x k 2 a 2 1 u xk + ū x k + 2a 1 b 0 4k 2 m (31). (33) With the use of the relation Tr U = Tr Ū = 0 and Eq. (21) it can be yielded that det(d) is polynomial of λ. It exists λ 1 λ 2 such that det(d) = (λ λ 1 )(λ λ 2 ) and when λ = λ i det( Ψ(x t λ i )) = 0 (i = 1 2). We set the solution of Eq. (14) with λ = λ i ϕ 1 = ϕ 1 (x t λ 1 ) ϕ 2 = ϕ 2 (x t λ 1 ) φ 1 = φ 1 (x t λ 2 ) φ 2 = φ 2 (x t λ 2 ) (34) and have (a 0 + a 1 λ 1 )ϕ 1 + b 0 ϕ 2 = 0 (35) (a 2 1 1)ϕ 1 + 2(a 1 d 0 + λ 1 )ϕ 2 = 0 (36) (a 0 + a 1 λ 2 )φ 1 + b 0 φ 2 = 0 (37) (a 2 1 1)φ 1 + 2(a 1 d 0 + λ 2 )φ 2 = 0 (38) which yield a 1 = ± φ1 ϕ 2 φ 2 [ϕ 1 + 2(λ 2 λ 1 )ϕ 2 ] φ1 ϕ 2 φ 2 ϕ 1 (39) a 0 = ± (λ 1φ 2 ϕ 1 λ 2 φ 1 ϕ 2 ) φ 1 ϕ 2 φ 2 [ϕ 1 + 2(λ 2 λ 1 )ϕ 2 ] (φ 1 ϕ 2 φ 2 ϕ 1 ) 3/2 (40) b 0 = (λ 1 λ 2 )φ 1 ϕ 1 φ1 ϕ 2 φ 2 [ϕ 1 + 2(λ 2 λ 1 )ϕ 2 ] (φ 1 ϕ 2 φ 2 ϕ 1 ) 3/2 (41) λ 2 φ 2 ϕ 1 λ 1 φ 1 ϕ 2 d 0 = ± (φ1 ϕ 2 φ 2 ϕ 1 )[φ 1 ϕ 2 φ 2 ϕ 1 + 2ϕ 2 φ 2 (λ 2 λ 1 )]. (42) Substituting Eqs. (30) (33) and (39) (42) back into Eqs. (26) (39) and (20) by direct calculations we can easily prove that all the equations are satisfied automatically. To this stage we have constructed the DT of system (2) and verified the invariance of the Lax pair under the DT. Thus from Eqs. (32) and (33) the new soliton solutions of system (2) can be generated with the trivial solution. 5 New Analytic Soliton Solutions of System (2) In this section we will apply the iterative algorithm of the DT to construct the soliton solutions of system (2). Taking u = 0 and v = 0 as the seed solutions and substituting them into the Lax pair with λ = λ i (i = 1 2) respectively we get ϕ 1 = e λ1(x+c[t]) α 1 ϕ 2 = eλ1(x+c[t]) α 1 2λ 1 + e λ1( x c[t]) α 2 (43)

4 834 LI Hong-Zhe TIAN Bo LI Li-Li and ZHANG Hai-Qiang Vol. 53 eλ2 (x+c[t]) α3 + eλ2 ( x c[t]) α4 (44) 2λ2 where R λ β1 (t)dt c[t] = (45) 2k and α1 α2 α3 α4 are four arbitrary integration constants. By suitably choosing the values of αj (j = ) λ1 and λ2 and according to expressions (32) and (33) the explicit representation of the soliton solutions of system (2) can be obtained. In what follows some figures will be plotted to demonstrate the abundant structures of the soliton solutions. As displayed in Fig. 1 the typical kink-shape solitary waves can be obtained through expressions (32) and (33) while Figs. 2 5 describe some other types of the kink-shape-like solitary waves with the same parameters in Fig. 1 except that β1 (t) is chosen as some different functions of t respectively. Moreover by selecting the appropriate parameters expressions (32) and (33) can possess several different types of soliton-like structures examples of which can be seen in Fig. 6. φ1 = eλ2 (xc[t]) α3 φ2 = Fig. 1 Evolution of the kink-shape solitary waves via expressions (32) and (33) with β1 (t) = 1 λ1 = 1.3 λ2 = 1.2 α1 = 1 α2 = 1 α3 = 0 α4 = 1 k = 1 m = 1 and n = 1. Fig. 2 Evolution of the parabolic solitary waves via expressions (32) and (33) with β1 (t) = t λ1 = 1.3 λ2 = 1.2 α1 = 1 α2 = 1 α3 = 0 α4 = 1 k = 1 m = 1 and n = 1. Fig. 3 Evolution of the solitary waves via expressions (32) and (33) with β1 (t) = t2 λ1 = 1.3 λ2 = 1.2 α1 = 1 α2 = 1 α3 = 0 α4 = 1 k = 1 m = 1 and n = 1.

5 No. 5 Painleve Analysis and Darboux Transformation for a Variable-Coefficient Boussinesq System in Fluid Dynamics with... Fig. 4 Evolution of the solitary waves via expressions (32) and (33) with β1 (t) = t 3 λ1 = 1.3 λ2 = 1.2 α1 = 1 α2 = 1 α3 = 0 α4 = 1 k = 1 m = 1 and n = 1. Fig. 5 Evolution of the preiodic type solitary waves via expressions (32) and (33) with β1 (t) = sin t λ1 = 1.3 λ2 = 1.2 α1 = 1 α2 = 1 α3 = 0 α4 = 1 k = 1 m = 1 and n = 1. Fig. 6 Evolution of some solitary waves via expression (32) with λ1 = 0.2 λ2 = 1.3 k = 1 α1 = 1 α2 = 1 α3 = 1 and α4 = 1; the choice of β1 (t) of the five photographs are respectively β1 (t) = 1 in (a) β1 (t) = t in (b) β1 (t) = t2 in (c) β1 (t) = t 3 in (d) and β1 (t) = sin t in (e). 835

6 836 LI Hong-Zhe TIAN Bo LI Li-Li and ZHANG Hai-Qiang Vol. 53 To be well contrasted to Figs. 1 5 the value of β 1 (t) in each plot in Fig. 6 is also set to be the five types of function adopted in Figs. 1 6 correspondingly. For the difference of the other parameters structures demonstrated in Fig. 6 can be viewed as the bell-shape-like solitary waves. In general from all the figures given above we can say when β 1 (t) is a linear function of t expression (31) yields the parabolic-type solitary waves and when β 1 (t) is a function of sint it yields the periodic-type solitary waves. 6 Conclusions In this paper we have carried out the Painlevé test to the variable-coefficient Boussinesq system and gotten the constraints on the coefficients. Based on the constraints we have successfully derived the Lax pair associated with system (2). Furthermore utilizing the obtained Lax pair we have constructed the Darboux transformation for the model and accomplished applying it to generate some new types of soliton solutions. It is notable that starting from the analytic solutions (32) and (33) we can apply the Darboux transformation once again then some other new solutions of system (2) could be obtained. In other words with the help of the symbolic computation complex analytic solutions of system (2) can be generated by iterating the Darboux transformation. Those solutions might be of some value in fluid dynamics. Acknowledgments We express our sincere thanks to Prof. Y.T. Gao for his valuable comments. References [1] J. Boussinesq Comptes Rendus 72 (1871) 755. [2] N.J. Zabusky in Nonlinear Partial Differential Equations Academic New York (1967). [3] M. Toda Phys. Rep. 18 (1975) 1. [4] V.E. Zakharov Sov. Phys. JETP 38 (1974) 108. [5] G. Das and J. Sarma Phys. Plasmas 6 (1999) 4394; B. Tian and Y.T. Gao Phys. Plasmas (Lett.) 12 (2005) ; Eur. Phys. J. D 33 (2005) 59; Phys. Plasmas 12 (2005) ; Phys. Lett. A 340 (2005) 449; 342 (2005) 228; 359 (2006) 241; 362 (2007) 283. [6] W.P. Hong Phys. Lett. A 361 (2007) 520; Y.T. Gao and B. Tian Phys. Plasmas 13 (2006) ; Phys. Plasmas (Lett.) 13 (2006) ; Phys. Lett. A 361 (2007) 523; Europhys. Lett. 77 (2007) 15001; B. Tian W.R. Shan C.Y. Zhang G.M. Wei and Y.T. Gao Eur. Phys. J. B 47 (2005) 329. [7] M.P. Barnett J.F. Capitani J. Von Zur Gathen and J. Gerhard Int. J. Quantum Chem. 100 (2004) 80; B. Tian G.M. Wei C.Y. Zhang W.R. Shan and Y.T. Gao Phys. Lett. A 356 (2006) 8; B. Tian Y.T. Gao and H.W. Zhu Phys. Lett. A 366 (2007) 223. [8] E.G. Fan and Y.C. Hon Chaos Solitons and Fractals 15 (2003) 559; M.H.M. Moussa and R. El Shikh Phys. Lett. A 372 (2008) 1429; Z.Y. Yan and H.Q. Zhang Phys. Lett. A 252 (1999) 291; M.L. Wang Phys. Lett. A 199 (1995) 169; D.Z. Lü Chaos Solitons and Fractals 24 (2005) [9] K. Singh and R.K. Gupta Int. J. Eng. Sci. 44 (2006) [10] G.B. Whitham Proc. R. Soc. A 299 (1967) 6; L.J. Broer Appl. Sci. Res. 31 (1975) 377; D.J. Kaup Prog. Theor. Phys. 54 (1975) 369. [11] B.A. Kupershmidt Commun. Math. Phys. 99 (1985) 51. [12] X.M. Li and A.H. Chen Phys. Lett. A 342 (2005) 413. [13] T. Xia Appl. Math. E-Notes 3 (2003) 171. [14] Z. Fu and S. Liu Phys. Lett. A 299 (2002) 507; J.F. Zhang Appl. Math. Mech. 21 (2000) 171; G. Xu Z. Li and Y. Liu Chin. J. Phys. 41 (2003) 232. [15] J. Weiss M.T. Tabor and G. Carnevale J. Math. Phys. 24 (1983) 552; R. Conte (Ed.) The Painlevé Property One Century Later Springer New York (1999). [16] S.L. Zhang B. Wu and S.Y. Lou Phys. Lett. A 300 (2002) 40; A. Bekir Chaos Solitons and Fractals 32 (2007) 449. [17] V.B. Matveev and M.A. Salle Darboux Transformations and Solitons Springer Berlin (1991); P.G. Estévez Inverse Problems 17 (2001) 1043; H.Z. Li B. Tian L.L. Li H.Q. Zhang and T. Xu Phys. Scr. 78 (2008) ; H.Q. Zhang B. Tian J. Li T. Xu and Y.X. Zhang IMA J. Appl. Math. 75 (2009) 46.