Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation with Symbolic Computation

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1 Commun. Theor. Phys. (Beijing, China) 49 (2008) pp c Chinese Physical Society Vol. 49, No. 4, April 15, 2008 Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation Symbolic Computation ZHANG Ya-Xing, 1 ZHANG Hai-Qiang, 1 LI Juan, 1 XU Tao, 1 ZHANG Chun-Yi, 2,3 and TIAN Bo 1,4,5, 1 School of Science, P.O. Box 122, Beijing University of Posts and Telecommunications, Beijing , China 2 Meteorology Center of Air Force Command Post, Changchun , China 3 Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing , China 4 State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing , China 5 Key Laboratory of Optical Communication and Lightwave Technologies, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing , China (Received April 13, 2007) Abstract In this paper, we put our focus on a variable-coefficient fifth-order Korteweg-de Vries (fkdv) equation, which possesses a great number of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived, such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out. PACS numbers: Nm, Wz, Yv Key words: variable-coefficient fifth-order Korteweg-de Vries equation, Lax pair, Darboux transformation, solitonic solutions, symbolic computation 1 Introduction The nonlinear evolution equations (NLEEs), particularly those that are integrable, are of great interest both from mathematical and physical points of view. [1 3] During the past decades, many scientific workers have devoted themselves to nonlinear problems and proposed various effective methods to seek the exact analytic solutions of NLEEs, such as the Darboux transformation, [4] Bäcklund transformation, [5] and generalized hyperbolicfunction method. [3,6,7] Nowadays, a wide class of NLEEs have been derived to describe a variety of nonlinear wave phenomena in different physical contexts, including nonlinear optics, [8] hydrodynamics, [9] condensed matter, [10] plasma physics, [11] quantum field theory, [12] and Bose Einstein condensates. [13] The Korteweg-de Vries (KdV) equation is one of the most important nonlinear models and has a wide range of applications in many fields of physical and engineering sciences. [13,14] As an important generalization of the KdV equation, the fifth-order KdV (fkdv) equation has the following general form [15] u t + αu 2 u x + βu x u xx + γuu xxx + u xxxxx = 0, (1) where α, β, and γ are all arbitrary nonzero and real parameters. The characteristics of the fkdv equation could be drastically changed by the parameters α, β, and γ. [7] For different choices of the parameters in Eq. (1), we list four kinds of famous nonlinear models below: (i) The Lax equation [16] u t + 30u 2 u x + 20u x u xx + 10uu xxx + u xxxxx = 0, (2) β = 2γ, α = 3 10 γ2. (ii) The Sawada Kotera (SK) equation [17] u t + 5u 2 u x + 5u x u xx + 5uu xxx + u xxxxx = 0, (3) β = γ, α = 1 5 γ2. (iii) The Kaup Kupershmidt (KK) equation [18] u t + 20u 2 u x + 25u x u xx + 10uu xxx + u xxxxx = 0, (4) β = 5 2 γ, α = 1 5 γ2. The project supported by the Key Project of the Chinese Ministry of Education under Grant No , the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No , Chinese Ministry of Education, the National Natural Science Foundation of China under Grant Nos and , the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE , Beijing University of Aeronautics and Astronautics, and by the National Basic Research Program of China (973 Program) under Grant No. 2005CB Corresponding author, gaoyt@public.bta.net.cn

2 834 ZHANG Ya-Xing, ZHANG Hai-Qiang, LI Juan, XU Tao, ZHANG Chun-Yi, and TIAN Bo Vol. 49 (iv) The Ito equation [19] u t + 2u 2 u x + 6u x u xx + 3uu xxx + u xxxxx = 0, (5) β = 2γ, α = 2 9 γ2. The Lax, SK, and KK equations all possess an infinite number of conservation laws, [20] while the Ito equation only has a limited number of conservation laws [7] because it is not completely integrable. The fkdv equation, arising from quantum mechanics and nonlinear optics, [7] can be used to describe the motion of long waves in shallow water under gravity and in a onedimensional nonlinear lattice. [15 20] By using the Hirota method [21] and inverse scattering method, [22] the analytical and numerical solutions of Eq. (1) have been figured out relevant physical significance discussed. When the inhomogeneities of media and nonuniformities of boundaries are taken into account in various real physical situations, the variable-coefficient NLEEs are considered to be more realistic than their constantcoefficient counterparts in describing a large variety of real phenomena, for example, in optical-fiber communications, [23,24] Bose Einstein condensates, [25] blood vessels and dynamics in the circulatory system [26] and arterial mechanics. [27] On the other hand, the coefficient functions related to x and/or t bring about a great amount of complicated calculations which are unmanageable manually, however, symbolic computation as a new branch of artificial intelligence, makes it exercisable to deal those nonlinear systems variable coefficients (see Refs. [1] [3] and references therein). In this paper, we mainly investigate the variablecoefficient SK equation, [28] u t + f(t)u 2 u x + g(t)u x u xx + h(t)uu xxx + k(t)u xxxxx = 0, (6) where the coefficients f(t), g(t), h(t), and k(t) are all analytic functions, which has recently brought about a certain interest of researchers. With homogeneous balance method, some exact solutions of Eq. (6) have been presented. [28] This paper is organized as follows. In Sec. 2, we will present a transformation and relevant constraints, under which equation (6) will be transformed into Eq. (1). In what follows, the Lax pair and Darboux transformation for Eq. (6) will be proposed under the constraints. In Sec. 3, the aid of the obtained Darboux transformation, we will get the one- and two-solitonic solutions and discuss their applications. Section 4 will be our conclusions for this paper. 2 Integrable Study for Eq. (6) Symbolic Computation As we mentioned above, equation (6) has been proposed in Ref. [28] and some valuable outcomes have been figured out, including the auto-bäcklund transformation and soliton solutions. In view of the current importance of variable-coefficient NLEEs, many efforts have been devoted to studying their integrable properties. In this section, we would like to investigate Eq. (6) from the integrable point of view. 2.1 Transformation from Eq. (6) to Eq. (1) Once we transform Eq. (6) into Eq. (1) under some constraints, then it illustrates that equation (6) is integrable if equation (1) is completely integrable. Subsequently, to construct this transformation, we suppose the following transformation u = B(x, t) + A(x, t)u[x(x, t), T(t)], (7) where A(x, t), B(x, t), X(x, t), and T(t) are all analytic functions and u satisfies Eq. (6). Through transformation (7), equation (6) could be converted into Eq. (1), provided that 5kA x X 4 x + 10kX 3 xax xx = 0, (8) ABhX 3 x + 10kX 3 xa xx + 30kA x X 2 xx xx + 15AkX x X 2 xx + 10AkX 2 xx xxx = 0, AgA x X 2 x + 3AhA x X 2 x + 3A 2 hx x X xx = 0, (9) 3BhA x X 2 x + AgB x X 2 x + 3ABhX x X xx + 30kX x A xx X xx + 15kA x X 2 xx + 10kX 2 xa xxx + 20kA x X x X xxx + 10AkX xx X xxx + 5AkX x X xxxx = 0, (10) 2AgA x X 2 x + A 2 gx x X xx = 0, (11) 2A 2 BfX x + 2gA 2 x X x + AgX x A xx + 3AhX x A xx + AgA x X xx + 3AhA x X xx + A 2 hx xxx = 0, (12) AX x + AB 2 fx x + 2gA x B x X x +3BhX x A xx +AgX x B xx + 3BhA x X xx + AgB x X xx +10kX xx A xxx +ABhX xxx + 10kA xx X xxx + 5kX x A xxxx + 5kA x X xxxx + AkX xxxxx = 0, (13) A 2 fa x = 0, (14) 2ABfA x + A 2 fb x + ga x A xx + AhA xxx = 0, (15) A t + B 2 fa x + 2ABfB x + gb x A xx + ga x B xx + BhA xxx + AhB xxx + ka xxxxx = 0, (16) B t + B 2 fb x + gb x B xx + BhB xxx + kb xxxxx = 0. (17) Thus, we derive that A(x, t) = ε, (18) B(x, t) = 0, (19) X(x, t) = ax + b, (20) where ε, a, and b are all arbitrary constants. Now, substituting transformation (7) into Eq. (6) yields εt U t + aε 3 fu 2 U x + a 3 ε 2 gu x U xx + a 3 ε 2 huu xxx

3 No. 4 Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation a 5 εku xxxxx = 0, (21) where the prime notes differentiation respect to t, which becomes Eq. (1), when the following conditions are satisfied, i.e., aε 2 f(t) α = a3 εg(t) β = a3 εh(t) γ = a 5 k(t). (22) Therefore, in what follows, we will put our focus on u t + f(t)u 2 u x + g(t)u x u xx + h(t)uu xxx + k(t)u xxxxx = 0 (23) aε 2 f(t) α = a3 εg(t) β = a3 εh(t) γ = a 5 k(t), (24) where a, ε, α, β, and γ are all arbitrary constants. Some integrable properties for Eq. (23) will be derived in the following section. 2.2 Lax Pair for Eq. (6) The Lax pair guarantees the complete integrability of a nonlinear evolution equation by using the Ablowitz Kaup Newell Segur procedure, [30] so we suppose the following linear eigenvalue problem Ψ x = ŪΨ, Ψ t = V Ψ, (25) ( ) ( ) a λ au/ε Ā B Ū =, V =, (26) a a λ C Ā where λ is the isospectral parameter, which is independent of x and t, and u = u(x, t) satisfies Eq. (23). For simplicity, we set α = 30, β = 20, and γ = 10. Then, according to the integrability condition Ūt V x + Ū V V Ū = 0, it can be determined that ( Ā = a 5 k(t) 16 λ λ 3 u + 4 λ 2 u x ε aε + 2 λu xx a 2 ε + 6uu x aε 2 + u ) xxx a 3, ε ( B = a 5 k(t) 16 λ4 u + 8 λ 3 u x + 8 λ 2 u 2 ε aε ε λu 2 ε λ 2 u xx a 2 ε + 12 λuu x aε λu xxx a 3 + 6u3 ε ε 3 + 6u x a 2 ε 2 + 8uu xx a 2 ε 2 + u ) xxxx a 4, ε ( C = a 5 k(t) 16 λ λ 2 u + 6u2 ε ε 2 + 2u ) xx a 2, ε where ε is an arbitrary constant. As equation (23) is a generalization of Eq. (2), the Lax pair for Eq. (2) can be derived directly the aid of the above outcome, i.e., ( ) λ p Φ x = UΦ, U =, (27) q λ ( ) A B Φ t = V Φ, V = (28) C A 2 p = u, (29) q = 1, (30) A = 16λ 5 8λ 3 u 4λ 2 u x 2λ(3u 2 + u xx ) 6uu x u xxx, (31) B = 16λ 4 u 8λ 3 u x 4λ 2 (2u 2 +u xx ) 2λ(u xxx +6uu x ) 6u 3 6u x 2 8uu xx u xxxx, (32) C = 16λ 4 + 8λ 2 u + 6u 2 + 2u xx, (33) where λ is the isospectral parameter and u = u(x, t) satisfies Eq. (2). Through calculation Mathematica and Eqs. (29) (33), we can prove that equation (2) is recovered by the compatibility condition U t V x +UV V U = 0. To our knowledge, this Lax pair is obviously different from the one obtained in Ref. [31]. 2.3 Darboux Transformation for Eq. (23) The Darboux transformation is a special gauge transformation, which is useful and powerful for constructing the soliton solutions for integrable nonlinear evolution equations from a trivial seed solution ([32] and references therein). In this subsection, based on the Lax pair obtained in subsection 2.2, we will construct the corresponding Darboux transformation for Eq. (23) in the following form Ψ = DΨ, (34) where D is a nonsingular matrix and Ψ should satisfy the linear eigenvalue problem Ψ x = Ũ Ψ, Ψt = Ṽ Ψ, (35) where Ũ and Ṽ have the same forms as Ū and V by replacing u ũ, and satisfy Ũ = (D t + DŪ)D 1, Ṽ = (D x + D V )D 1. (36) In other words, the linear eigenvalue problem (25) could be invariant under transformation (34), provided that equations (36) are satisfied. Therefore, the Darboux transformation establishes the relationship between the initial potential function u in Ū and V and the new potential function ũ in Ũ and Ṽ. Transformation (34) maps the Lax pair (25) into another Lax pair of the same form (35) and both of the Lax pairs lead to Eq. (23) conditions (24). Analogous to Ref. [31], we specially choose the Darboux transformation as the following form D = Ĩ D = Ĩ( λi 2 2 S) (37) Ĩ = diag (1, 1), (38) S = HΛH 1, (39)

4 836 ZHANG Ya-Xing, ZHANG Hai-Qiang, LI Juan, XU Tao, ZHANG Chun-Yi, and TIAN Bo Vol. 49 where Λ = diag (λ 1, λ 2 ) and H = (Ψ 1, Ψ 2 ) is a nonsingular matrix while Ψ i (i = 1, 2) are two different solutions for Eqs. (25) the eigenvalues λ 1 and λ 2, respectively. We suppose ( ) S11 S 12 S =. (40) S 21 S 22 If Ψ 1 = (ϕ 1, ) T is a solution of Eq. (25) when λ = λ 0, then Ψ 2 = (ϕ 1 + 2λ 0, ) T is another solution for Eq. (25) λ = λ 0. Thus, we take ( ) ( ) ϕ1 ϕ 1 + 2λ 0 λ0 0 H =, Λ =, (41) 0 λ 0 and employ symbolic computation to work out the matrix S as follows: S 11 = ϕ 1 + λ 0, (42) S 12 = ϕ 1(ϕ 1 + 2λ 0 ), ϕ 2 2 (43) S 21 = 1, (44) S 22 = ϕ 1 + λ 0. (45) To this point, we can explicitly give the relationship between the initial potential function u and the new potential function ũ, i.e., ũ = u ε 2ϕ 1(ϕ 1 + 2λ 0 ) 2, (46) where the new potential ũ also satisfies Eq. (23) conditions (24). 3 Solitonic Structures and Possible Applications In this section, we will construct the exact analytic one- and two-solitonic solutions for Eq. (23) by virtue of the Darboux transformation. Taking a trivial seed solution u = 0 and solving the Lax pair (25) λ = λ 0, we can obtain two basic solutions ( ) ϕ1 Ψ = (47) ϕ 1 = e axλ 0 16a 5 λ 5 0 k(t)dt+η 1, (48) = e axλ 0+16a 5 λ 5 0 eaxλ 0 16a 5 λ 5 0 k(t)dt+η 2 k(t)dt+η 1 2λ 0, (49) where η 1 and η 2 are both arbitrary constants. Via transformation (46), the one-solitonic solution for Eq. (23) can be generated as ũ 1 = 2λ 2 0 sech 2[ 2axλ a 5 λ 5 0 k(t)dt η1 + η 2 + K ], (50) 2 where K = ln( 2λ 0 ) λ 0 < 0. (51) With suitable values for parameters in solution (50), we can graphically analyze the solitonic structures and discuss the relevant physical applications. By fixing k(t) as a constant, figure 1 shows that the solitary wave propagates stably out change of shape, velocity or amplitude. When k(t) constant in solution (50), it can be seen that the function k(t) leads to the propagation of solitary waves variable velocities. Figure 2 displays that the solitonic wave propagates a uniform acceleration, while figure 3 illustrates a solitonic structure periodic oscillations, which imply that the acceleration is nonuniform. For the arbitrariness of k(t), as a matter of fact, more abundant solitonic structures can be obtained from solution (50). Fig. 1 Solution surface u(x, t) via solution (50) a = 0.8, λ 0 = 1, η 1 = 1, η 2 = 1 and k(t) = Fig. 2 Solution surface u(x, t) via solution (50) the same parameters as in Fig. 1 except that k(t) = 0.09t Fig. 3 Solution surface u(x, t) via solution (50) the same parameters as in Fig. 1 except that k(t) = sin(0.65t ).

5 No. 4 Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation 837 According to the iterative algorithm of the Darboux transformation, the multi-solitonic solutions for Eq. (23) can be obtained in a recursive manner. For instance, the aid of symbolic computation, we iterate the Darboux transformation once again based on solution (50). Now we choose two basic solutions for the Lax pair (25) λ = λ 1 λ 0 ( ) ϕ3 Φ = (52) ϕ 4 ϕ 3 = e axλ 1 16a 5 λ 5 1 k(t)dt+η 3, (53) ϕ 4 = e axλ 1+16a 5 λ 5 1 eaxλ 1 16a 5 λ 5 1 k(t)dt+η 4 k(t)dt+η 3, (54) 2λ 1 where η 3 and η 4 are both arbitrary constants. We suppose ( ) α = D 1 Φ (55) β D 1 = Ĩ D 1 = Ĩ(λ 1I 2 2 S), (56) where Ĩ and S have been defined in subsection 2.3, then we get the two-solitonic solution as ũ 2 = ũ1 ε 2α(α + 2λ 1β) β 2. (57) Similarly, iterating the Darboux transformation n times analogous to the above procedure, we can further obtain the nth-iterated potential function for Eq. (23) in the explicit form (including the n-solitonic solution). Fig. 4 Solution surface u(x, t) via solution (57) ε = 2, a = 0.9, λ 0 = 1, λ 1 = 1.24, η 1 = 1.5, η 2 = 1, η 3 = 1, η 4 = 1.2 and k(t) = With the constant value for k(t), it can be seen in Fig. 4 that the two solitonic waves propagate stably out change of shape, velocity and amplitude. Meanwhile, figure 4 illustrates the two-solitonic elastic collision, where the two separate solitonic waves retain their original characteristics after collision out perturbation only a phase shift at the moment of collision. Figure 5 implies the significant effects of k(t), as a nonconstant on the interaction of solitonic waves. In Fig. 5, the two solitonic waves progress variable velocity respectively, while neither of their accelerations is uniform and the interaction between these two-solitary waves is definitely weak. Analogy to these, more abundant solitonic structures for describing the realistically physical phenomena will be obtained different choices of k(t) from solution (57). Fig. 5 Solution surface u(x, t) via solution (57) ε = 0.6, a = 0.8, λ 0 = 1.1, λ 1 = 1.3, η 1 = 0, η 2 = 0, η 3 = 0, η 4 = 0 and k(t) = 0.22sin3t. 4 Conclusions Compared constant-coefficient NLEEs, the variable-coefficient ones can be used to describe some real physical situations powerfully. [23 27] With the development of computing technology, the investigations on the variable-coefficient cases have becoming more feasible and exercisable. [1 3] Considering the practical importance of variable coefficients (see Ref. [13] and references therein), we in this paper have put our focus on a variablecoefficient fkdv equation. With the aid of symbolic computation, a transformation from Eq. (6) to Eq. (1) has been proposed the relevant constraints on their coefficient functions presented. Based on the constant-coefficient equation, this transformation provides the feasibility to investigate the variablecoefficient fkdv equation and a rich class of exact analytical solutions for Eq. (6) could be constructed. Under those constraints, we have researched the integrable properties for Eq. (6) including the Lax pair and Darboux transformation. Furthermore, the explicit one- and two-solitonic solutions of Eq. (6) have been figured out. Through the graphical analysis of these solitonic solutions, we have demonstrated some observable inhomogeneous effects related to variable coefficients. Acknowledgments We would like to thank Prof. Y.T. Gao, Ms. X.H. Meng, Mr. W. Hu and other members of our discussion group for their valuable comments.

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