(Received May 11, 2009; revised manuscript received September 4, 2009)

Transcription

1 Commun. Theor. Phys. (Beijing, China 53 (2010 pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 4, April 15, 2010 Darboux Transformation and Soliton Solutions for a Variable-Coefficient Modified Kortweg-de Vries Model from Fluid Mechanics, Ocean Dynamics, and Plasma Mechanics GAI Xiao-Ling (, 1 GAO Yi-Tian (Ô Í, 1,2, MENG De-Xin (, 1 WANG Lei (, 1 SUN Zhi-Yuan (êã, 1 LÜ Xing (ù,3 FENG Qian (ú¼, 1 WANG Ming-Zhen (, 1 YU Xin (Ù, 1 and ZHU Shun-Hui (ý ï 1 1 Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing , China 2 State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing , China 3 School of Science, Beijing University of Posts and Telecommunications, P.O. Box 122, Beijing , China (Received May 11, 2009; revised manuscript received September 4, 2009 Abstract This paper is to investigate a variable-coefficient modified Kortweg-de Vries (vc-mkdv model, which describes some situations from fluid mechanics, ocean dynamics, and plasma mechanics. By the Ablowitz Kaup Newell Segur procedure and symbolic computation, the Lax pair of the vc-mkdv model is derived. Then, based on the aforementioned Lax pair, the Darboux transformation is constructed and a new one-soliton-like solution is obtained as well. Features of the one-soliton-like solution are analyzed and graphically discussed to illustrate the influence of the variable coefficients in the solitonlike propagation. PACS numbers: Yv, Fg, Jr, Ik, Wz Key words: variable-coefficient modified Kortweg-de Vries model, Lax pair, Darboux transformation, soliton solutions 1 Introduction It is known that the constant-coefficient modified Korteweg-de Vries (mkdv model appears in some fields of mechanics. [1 5] For example, the mkdv model has been used to describe the interfacial waves in two-layer liquid with gradually varying depth, [1] van Alfvén waves in collisionless plasma, [2] ion acoustic solitons, [3] acoustic waves in certain unharmonic lattices [4] and traffic congestion problems. [5] As a completely-integrable dynamical system, the mkdv model possesses such properties [6 8] as the inverse scattering transformation, [7] Painlevé property, [8] Miura transformation, [9] Bäcklund transformation, [10] Darboux transformation [11 12] and N- soliton solutions. [12] In recent years, it has been found that the variablecoefficient nonlinear evolution equations (NLEEs can provide us with more powerful and realistic models than their constant-coefficient counterparts in describing a variety of real phenomena. [13 22] For instance, the variablecoefficient KdV equation has been studied in describing the blood vessels, Bose Einstein condensates, rods and positons, [15] modeling the propagation of water waves in a channel with an uneven bottom, [16] and depicting the solitary wave as it enters a region with decreasing depth that forms a shelf. [18] The nonlinear Schrödinger equation with variable nonlinearity and dispersion has relevant applications in optical fibers, [19 20] plasma physics, [20] arterial mechanics, [20] Bose Einstein condensates, [21] and femtosecond pulse propagation. [22] The variable-coefficient mkdv (vc-mkdv equation [16] and its higher-dimensional generalizations [17] have been discovered to model the dustion-acoustic waves in such cosmic environments as those in the supernova shells and Saturns F-ring. In this paper, with symbolic computation, [13 15] we will consider a vc-mkdv equation, [16] which can be read as u t + g(tu xxx + f(tu 2 u x + l(tu + q(tu x = 0, (1 g(t, l(t, f(t, and q(t are all analytic functions of t. Equation (1 has attracted attention in some fields including fluid mechanics, ocean dynamics, and plasma mechanics. [16,23] The soliton solutions, bilinear Bäcklund transformation, and Lax pair of Eq. (1 have been presented in Ref. [16]. When l(t = 0, some Jacobi elliptic function solutions are obtained in Ref. [24], while univariable and bi-variable traveling wave-like solutions are 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. 2005CB321901, and by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos and , Chinese Ministry of Education Corresponding author, gaoyt@public.bta.net.cn

2 674 GAI Xiao-Ling, GAO Yi-Tian, MENG De-Xin, WANG Lei, SUN Zhi-Yuan, LÜ Xing, FENG Qian, et al. Vol. 53 given in Ref. [25]. Furthermore, when l(t = 0, Eq. (1 can be used to investigate the dynamics hidden in the plasma sheath transition layer and inner sheath layer. [16] However, to our knowledge, there exist certain problems about the Lax pair of Eq. (1 presented in Ref. [16] and its Darboux transformation has not been constructed, either. Darboux transformation works as an algorithmic procedure to construct some explicit solutions of partial differential equations, [11 12,26] obtained successively by an algebraic algorithm. [27] In the present paper, the Lax pair of Eq. (1 will be constructed via the Ablowitz Kaup Newell Segur (AKNS procedure and symbolic computation, and by virtue of the Lax pair we will obtain its Darboux transformation and a new one-soliton-like solution. Finally, features of the one-soliton-like solution will be analyzed and graphically discussed to illustrate the influence of the variable coefficients in the solitonlike propagation. The structure of this paper will be organized as follows. In Sec. 2, a Lax pair of Eq. (1 will be obtained through the AKNS procedure and symbolic computation. In Sec. 3, the Darboux translation of Eq. (1 is constructed based on the Lax pair. In Sec. 4, a new one-soliton-like solution of Eq. (1 will be derived and the influence of variable coefficients will be presented. Finally, the conclusions will be given in Sec Lax Pair It has been shown that when f(t, g(t, l(t satisfy the relation f(t = 6 2 g(t exp 2 l(tdt, (2 Equation (1 can pass the Painlevé test, [28] C 0 is an arbitrary nonzero constant. Noting that there exist certain problems about the Lax pair of Eq. (1 presented in Ref. [16], we proceed to give certain modification following the AKNS procedure. The solution is Φ x = UΦ, λ e l(tdt u(x, t U = 1 2 e l(tdt, (3 u(x, t λ ( A B Φ t = V Φ, V =, (4 C A ( a1x b 1x λ c 1x d 1x ( a0x b 0x + c 0x d 0x + Φ = ( φ1 φ 2 A = 4g(tλ 3, (5 { q(t + 2 C 2 0 [ exp 2 ] } l(tdt u 2 g(t λ,(6 B = exp{ 2 l(tdt 2 exp 2 l(tdt g(tu 3 } + 4λ 2 g(tu + q(tu + 2λg(tu x + g(tu xx, (7 C = exp[ l(tdt] { 2 [ 2 2 exp 2 ] l(tdt g(tu 3 + 4λ 2 g(tu + q(tu 2λg(tu x + g(tu xx }, (8 λ is a spectral parameter independent of x and t. The compatibility condition Φ xt = Φ tx yields a zero curvature equation U t V x + [U, V ] = 0, (9 which leads to Eq. (1 by direct computation. 3 Darboux Transformation Now we introduce a gauge transformation of the spectral problems (3 and (4 as below T is defined by Φ = TΦ, (10 T x + TU = ŪT, (11 T t + TV = V T. (12 The Lax pair (3 and (4 can be transformed into Φ x = Ū Φ, (13 Φ t = V Φ, (14 Ū, V have the same forms as U and V except replacing u with u 1. As we know, a gauge transformation of a spectral problem is called a Darboux transformation of the spectral problem if it transforms the spectral problem into another spectral problem of the same type. [29] Now we assume that ( ( a1 b 1 a0 b 0 T = λ +, (15 c 1 d 1 c 0 d 0 a 1, b 1, c 1, d 1, a 0, b 0, c 0, and d 0 are functions of x and t. Substituting Eq. (15 into Eq. (11 gives ( λ(a0 + λa 1 + u(b 0 + λb 1 δ(x, t (a 0 + λa 1 β(x, t λ(b 0 + λb 1 λ(c 0 + λc 1 + u(d 0 + λd 1 δ(x, t (c 0 + λc 1 β(x, t λ(d 0 + λd 1 = 0, (16 ( λ(a0 + λa 1 + u 1 (c 0 + λc 1 β(x, t λ(b 0 + λb 1 + u 1 (d 0 + λd 1 β(x, t u 1 (a 0 + λa 1 δ(x, t λ(c 0 + λc 1 u 1 (b 0 + λb 1 δ(x, t λ(d 0 + λd 1

3 No. 4 Darboux Transformation and Soliton Solutions for a Variable-Coefficient Modified Kortweg-de Vries Model from 675 β(x, t = exp l(tdt, δ(x, t = exp/ l(tdt 2. Comparing the coefficients of λ j (j = 2, 1, 0, for the case of j = 2, we have For the case of j = 1, we get b 1 = 0, (17 c 1 = 0. (18 a 1x = 0, (19 d 1x = 0, (20 { u 1 = exp l(tdt exp[ } l(tdt ]u 2b 0, (21 c 0 = b 0 2. (22 For the case of j = 0, we obtain a 0 = d 0, (23 d 0x = 2b exp[ l(tdt]ub 0 2, (24 [ ] 2 exp l(tdt ud 0 + 2b 0 d 0 + b 0x = 0. (25 On the other hand, there is a constant λ = λ 1 and a solution Φ = (φ 1, φ 2 T of Eq. (3 which satisfy i.e., (a 0 + a 1 λ 1 φ 1 + (b 0 + b 1 λ 1 φ 2 = 0, (26 (c 0 + c 1 λ 1 φ 1 + (d 0 + d 1 λ 1 φ 2 = 0, (27 b 0 = 2C2 0λ 1 φ 1 φ 2 φ C2 0 φ2 2 d 0 = C2 0 λ 1φ 2 2 λ 1φ 2 1 φ C2 0 φ2 2 u 1 = 4e2(x+ζλ1 l(tdt λ 1 2 e 4ζλ1 2 + e4xλ1, (28. (29 Now it is easy to prove the following proposition. Proposition If u is a given solution of Eq. (1 and Φ = (φ 1, φ 2 T is a solution of Eq. (3 with λ = λ 1 then the unknown functions a 0, b 0, c 0, d 0, a 1, b 1, c 1, d 1, and u 1 can be defined by relations (17 (23, (28 and (29. The relations (24 and (25 can be proved to be satisfied automatically. Especially the new solution u 1 can be expressed as follows { [ } u 1 = exp l(tdt exp l(tdt ]u 2b 0, (30 b 0 = 2C2 0 λ 1φ 1 φ 2 φ C2 0 φ2 2. (31 Using the results above, it is easy to prove that Eq. (1 is satisfied automatically. 4 Soliton Solutions In this secion, we start from a trivial solution u = 0 to find a new one-soliton-like solution of Eq. (1. Substituting u = 0 into Eqs. (3 and (4, we get φ 1 = exp ( 4g(tλ 3 1 q(tλ 1dt + xλ 1, (32 [ ] φ 2 = exp (4g(tλ q(tλ 1 dt xλ 1. (33 With the aid of Eq. (31, b 0 can be determined as λ b 0 = 2e2(x+ζλ1 1 2 e 4ζλ1 2, (34 + e4xλ1 ζ = [4g(tλ q(t]dt. (35 Substituting Eq. (34 into Eq. (30, we can obtain a new one-soliton-like solution of Eq. (1 = 4λ 1{cosh[2(x + ζλ 1 l(tdt] + sinh[2(x + ζλ1 l(tdt]} 2 [cosh(4ζλ 1 + sinh(4ζλ 1 ] 2 + cosh(4xλ. ( sinh(4xλ 1 The following part of this section is devoted to analyzing the influence of the variable coefficients on the propagation of the solitonlike. Firstly, aiming to discuss the influences of l(t in Eq. (1, we assume that the coefficients g(t and q(t be constants [without loss of generality, g(t = q(t = 1], only leaving l(t to be variable. Thus we have [ u 1 = 2 exp ] l(tdt sech[2(x 5t], (37 which shows that the characteristic line is 2(x 5t = 0. (38 The velocity of the solitonlike can be obtained as v = dx/dt = 5 and the amplitude of the solitonlike solution is A 1 = 2 exp l(tdt. (39 In Figs. 1 and 2, we choose some special forms of l(t to investigate the influences of l(t. Figure 1 shows that when l(t = 0, the profile and amplitude of the solitonlike keep invariable during propagation. Figure 2 depicts the decreasing amplitude of the solitonlike with time increasing when l(t = t. At the same time, Figs. 1 and 2 illustrate that the solitonlike propagate along a straight line with time increasing. From above analysis, it can be found that l(t influences the amplitude of the solitonlike while the other variable coefficients of Eq. (1 are chosen to be constants.

4 676 GAI Xiao-Ling, GAO Yi-Tian, MENG De-Xin, WANG Lei, SUN Zhi-Yuan, LÜ Xing, FENG Qian, et al. Vol. 53 Fig. 1 (a The propagation of the solitonlike with the parameters: λ 1 = 1, g(t = 1, q(t = 1, and l(t = 0; (b The wave profile with selected time as t = 0, t = 1, and t = 1.5. Fig. 2 (a The propagation of the solitonlike with the parameters: λ 1 = 1, g(t = 1, q(t = 1, and l(t = t; (b The wave profile with selected time as t = 0, t = 1, and t = 1.5. Secondly, the influence of the coefficient q(t will be taken into consideration. Following the analysis of l(t, the coefficient g(t is assumed as constant, i.e., g(t = 1 without loss of generality. Besides, in order to avoid the influences brought by the term l(t, we can set l(t = 0. Based on Eq. (36, we have { } u 1 = 2sech 2x 2[q(t + 4]dt, (40 which shows that the characteristic line is x [q(t + 4]dt = 0, (41 The amplitude of the solitonlike is A = 2 and the velocity of the solitonlike can be obtained as v = dx/dt = q(t + 4. (42 When q(t = 0, the velocity of the solitonlike can be obtained as v = dx/dt = 4. From Fig. 3, it is obvious to find that the velocity and amplitude of the solitonlike keep unchangeable during propagation. If q(t is a function of t, the velocity of the solitonlike will change with time increasing. For example, when q(t = t, the results are shown in Fig. 4. Comparing Fig. 3 with Fig. 4, it can be found that the main difference between the two conditions is focused on the propagation trace of the solitonlike. The propagation trace of the solitonlike takes the form of straight line and parabola curve, corresponding to q(t = 0 and q(t = t respectively. Finally, our attention will be paid to the coefficient of the high order derivative term u xxx in Eq. (1. Based on the analysis of the terms l(t and q(t, we turn to consider g(t, the variable coefficient of u xxx, with assumption of q(t = l(t = 0. From Eq. (36, we derive u 1 = 2sech 2x 8 g(tdt, (43 which indicates that the characteristic line should be x 4 g(tdt = 0. (44 The amplitude of the solitonlike is A = 2 and the velocity of the solitonlike is v = 4g(t. (45 When q(t = 1, the discussion reveals the same results as those shown in Fig. 3. From Eq. (45, it can be found that the choice of function g(t has ability to change the propagation velocity of the solitonlike. At the same time, the propagation trace of the solitonlike will be influenced by the term g(t. For example, Fig. 5 illustrates the parabolatype and cosine-type propagation trace of the solitonlike corresponding to g(t = t and g(t = sin(t.

5 No. 4 Darboux Transformation and Soliton Solutions for a Variable-Coefficient Modified Kortweg-de Vries Model from 677 Fig. 3 The propagation of the solitonlike with the parameters: λ 1 = 1, g(t = 1, q(t = 0, and l(t = 0; (b The wave profile with selected time as t = 0, t = 1, and t = 1.5. Fig. 4 (a The propagation of the solitonlike with the parameters: λ 1 = 1, g(t = 1, q(t = t, and l(t = 0; (b The wave profile with selected time as t = 0, t = 1, and t = 1.5. Fig. 5 (a The propagation of the solitonlike with the parameters: λ 1 = 1, g(t = t, q(t = 0, and l(t = 0; (b The propagation of the solitonlike with the parameters: λ 1 = 1, g(t = sin(t, q(t = 0, and l(t = 0. 5 Conclusions In this paper, we have studied a vc-mkdv equation which is a model in fluid mechanics, ocean dynamics, and plasma mechanics. Based on the AKNS procedure and symbolic computation, the Lax pair of Eq. (1 has been constructed. Furthermore, the Darboux transformation and a one-soliton-like solution of Eq. (1 have been derived. By analyzing the influence of the variable coefficients on the motion of the solitonlike for Eq. (1, we have revealed that l(t, q(t, and g(t have close connections with the amplitude, velocity, and propagation trace of the solitonlike. Acknowledgments The authors are very grateful to Prof. B. Tian, Mr. H.Q. Zhang, Ms. Y. Liu, Mr. R. Guo and other members of our discussion group for their valuable comments.

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