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1、精品论文Wronskian Form of N -Solitonic Solution fora Variable-Coefficient Korteweg-de VriesEquation with NonuniformitiesKe-Jie Cai a, Cheng Zhang a, Huan Zhang a, Xiang-Hua Meng a, Xing Lu a, Tao Geng a, Wen-Jun Liu a and Bo Tiana, b, c,a School of Science, P. O. Box 122, Beijing University of Posts and
2、Telecommunications, Beijing 100876, Chinab State Key Laboratory of Software Development Environment, BeijingUniversity of Aeronautics and Astronautics, Beijing 100083, Chinac Key Laboratory of Optical Communication and Lightwave Technologies, Ministry of Education, Beijing University of Posts and Te
3、lecommunications, Beijing 100876, ChinaAbstractBy the symbolic computation and Hirota method, the bilinear form and an auto- Backlund transformation for a variable-coefficient Korteweg-de Vries equation with nonuniformities are given. Then, the N -solitonic solution in terms of Wronskian form is obt
4、ained and verified. In addition, we show that the (N 1)- and N -solitonic solu- tions satisfy the auto-Backlund transformation through the Wronskian technique.Keywords: Variable-coefficient KdV equation; Bilinear auto-Backlund transforma- tion; N -solitonic solution; Wronskian determinantCorrespondi
5、ng author, with e-mail address as gaoyt8精品论文1. IntroductionRecently, in line with the development of symbolic computation 14 , there has been much interest in the study of the variable-coefficient Korteweg-de Vries (vcKdV) equations in different forms 58. A number of physical problems have been foun
6、d to be modelled by vcKdV equations such as the propagation of nonlinear waves in a fluid-filled tube 9, Bose- Einstein condensates in the weakly-interacting atomic gases 10, 11, evolution of internal gravity waves in lakes of changing cross-section 12, cylindrical dust-acoustic and dust-ion- acoust
7、ic waves in an un-magnetized dusty plasma 13, 14. These equations, though often hard to be investigated, are able to describe various real models more powerfully than their constant-coefficient counterparts. Higher-dimensional cases can be found, e.g., in Refs. 1, 2.In this paper, we consider a vcKd
8、V equation with nonuniformities in the form 7:ut + g(t) (6 uux + uxxx) + 6 f (t) g(t) u = x ( f 0(t) + 12 g(t) f 2 (t) ) , (1)where the wave amplitude u(x, t) is a function of the scaled “ space ” x and “ time ” t, andf (t), g(t) are both functions of the variable t only. Eq. (1) contains many speci
9、al forms: (a) For g(t) = 1, Eq. (1) reduces to the generalized KdV (GKdV) equation 15ut + 6 uux + uxxx + 6 f u = x (f 0 + 12 f 2) . (2)1(b) For g(t) = 1 and f (t) =12 t, Eq. (1) becomes the cylindrical KdV equation 15, 161ut + 6 uux + uxxx + 2t u = 0 . (3)(c) For f (t) = 0, Eq. (1) turns to the vari
10、able-coefficient KdV equationut + g(t) (6 uux + uxxx) = 0 . (4) Specially, for g(t) = 1, Eq. (4) reduces to the well known KdV equation.Eq. (1) is a model describing the propagation of weakly nonlinear, weakly dispersive waves in the inhomogeneous media. Ref. 7 has discussed Eq. (1) and obtained one
11、- and two-solitonic solutions by homogeneous balance principle.In this paper, with symbolic computation, we will investigate Eq. (1) using the Hirota method and Wronskian technique.Up to now, there exist many powerful methods to find exact solutions for nonlinear evo- lution equations, one of which
12、is the Hirota method 17. It can transform certain nonlinear evolution equations into the bilinear forms, which are relatively easy to be solved. Addition- ally, the Backlund transformation (BT) is another effectively method to deal with soliton problems. Generally, with the BT, we can derive a new s
13、olution from the known one and iteratively obtain the N -soliton solution in this manner. However, the verification of the N -soliton solution is very difficult and tedious, if not impossible. With the developmentof the Wronskian technique 18, 19, the difficulty can be solved to a great extent. The
14、Wronskian technique provides an alternative formulation of the N -soliton solution, in terms of Wronskian determinant, which makes the verification of the solution more easily by direct substitution because differentiation of a Wronskian determinant can be easily handled and its derivatives take sim
15、ilar compact forms.This paper is organized as follows: In Section 2, we present the bilinear form and an auto- BT for Eq. (1). By using the auto-BT, one-solitonic solution is obtained and corresponding solutions of Eqs. (2)(4) are given respectively. In Section 3, we construct the N -solitonic solut
16、ion in the Wronskian form and prove its validity via the Wronskian technique. Mean-while, we prove that the (N 1)- and N -solitonic solutions satisfy the auto-BT. Conclusionsare given in Section 4.2. Bilinear form and an auto-BT of Eq. (1)Through the dependent variable transformationu = xf (t) + 2ln
17、 F (x, t)xx , (5)Eq. (1) is rewritten in the bilinear formDxDt + g(t) D4 + 6f (t) g(t) x D2+ 6f (t) g(t) (F F ) = 0 , (6)xx xwhere the operators DxDt , D4 and D2 2022 are defined byDmnx xm na(x, t) b(x0 , t0 ). (7)x Dt a b x x0t t0x0 =x, t0 =tBased on Eq. (6), we derive an auto-BT of Eq. (1) as foll
18、ows:x Dt + g(t) D3+ 6 f (t) g(t) x Dx + 3 (t) g(t) Dx + (t) F 0 F = 0 , (8a)x D2 (t) F 0 F = 0 , (8b)where F (x, t) and F 0(x, t) are two different solutions of Eq. (6), and (t) and (t) are both arbitrary functions of t.Without loss of generality, the function (t) in Eq. (8a) can be assumed to zero.
19、 TakingF 0 = 1, we obtain a pair of equations with respect to F (x, t) from Eqs. (8a) and (8b)Ft + g(t) Fxxx + 6 f (t) g(t) x Fx + 3 (t) g(t) Fx = 0 . (9a)Fxx (t) F = 0 . (9b) Then, the solution of Eq. (9a) and (9b) can be expressed in the following form:F = e + e (10)Z1with = k x e6R f (t) g(t) dt
20、4 k3e18R f (t) g(t)dt g(t)dt + c , (11)1where k is an arbitrary constant. Then the exact one-solitonic solution of Eq. (1) can beobtainedu(x, t) = xf (t) + 2k2e12R f (t) g(t)dt kxSech2 e 6R f (t) g(t)dtZ 4 k3e18R f (t) g(t)dt g(t)dt + c . (12)Hereby, we can also derive the corresponding exact soluti
21、ons of Eq. (2)-(4) respectively: (a) When g(t) = 1, we can obtain the one-solitonic solution of GKdV Eq. (2):u(x, t) = xf (t) + 2k2 e12R f (t)dt Sech2 kxe 6R f (t)dt Z14 k3 e18R f (t)dt dt + c. (13)(b) When g(t) = 1 and f (t) = KdV Eq. (3):112 t, we can obtain the one-solitonic solution of cylindric
22、alu(x, t) =x12 t2k2+tSech2 kx +t8k3 + c1t. (14)(c) When f (t) = 0, we can obtain the one-soliton solution of vcKdV Eq. (4):u(x, t) = 2k2 Sech2 Zkx 4k3g(t)dt + c1. (15)(d) When f (t) = 0 and g(t) = 1, we can obtain the one-soliton solution of the well knownKdV equation:u(x, t) = 2k2 Sech2 kx 4k3t + c
23、1 . (16)3. N -solitonic solution in terms of Wronskian determinantIn the present section, we show that the Eq. (6) has the N -solitonic solution in the following Wronskian form(1)(2)(N 1) 1 1 1 1 (1)(2)(N 1) 2 2 2 2 F (N ) = W (1, 2, , N ) = , (17)(1)(2)(N 1) N 1 N 1 N 1 N 1 N(1)(2)(N 1)N N N jwith
24、(j) = i(j = 1, 2, , N 1) , = e i + (1)i+1 e i ,ixjiRZRi = ki x e6f (t) g(t) dt 34 kie18f (t) g(t)dt g(t)dt + ci ,精品论文and i (i = 1, 2, , N ) enjoy the following conditionsi xx = i (t) i , (18a)i t = 4 g(t) i xxx 6f (t)g(t) x i x , (18b)iRwhere i (t) = k2 e12f (t) g(t)dt (i = 1, 2, , N ). Introducing
25、the notation W (1, 2 , , N ) =(0, 1, 2, , N 1) = (N 1), we observe thatF (N ) F (N ) F (N ) F (N ) F (N ) = (N 1) , (19) x= (N 2, N ) , (20) xx= (N 3, N 1, N ) + (N 2, N + 1) , (21) xxx = (N 4, N 2, N 1, N ) + 2 (N 3, N 1, N + 1) + (N 2, N + 2) , (22) xxxx = (N 5, N 3, N 2, N 1, N ) + 3 (N 4, N 2, N
26、 1, N + 1)+ 2 (N 3, N, N + 1) + 3 (N 3, N 1, N + 2) + (N 2, N + 3) . (23)From Eq. (18b), we haveF (N )t = 4 g(t)h(N 4, N 2, N 1, N ) (N 3, N 1, N + 1) + (N 2, N + 2)i 6f (t)g(t) x (N 2, N ) 3N (N + 1) f (t)g(t) (N 1) , (24)F (N )xt = 4 g(t)h(N 5, N 3, N 2, N 1, N ) (N 3, N, N + 1) + (N 2, N + 3)i 6f
27、 (t)g(t) (N 2, N ) 6f (t)g(t) x (N 3, N 1, N ) 6f (t)g(t) x (N 2, N + 1) 3N (N + 1) f (t)g(t) (N 2, N ) . (25)Substituting Eqs. (19)(25) into Eq. (6), we can work out thatxhDx Dt + g(t) D4x(N )+ 6f (t)g(t) x D2+ 6f (t)g(t) i (F (N ) F (N ) ) x= 2 F (N ) F (N ) 2 F F (N )+ g(t) h2 F (N ) F (N ) 8 F (
28、N ) F (N )+ 6 (F (N )2 ixtt xxxxxxxxxxx+6f (t)g(t) x h2 F (N ) F (N ) 2 (F (N )2i + 12 f (t)g(t) F (N ) F (N )xxxx= 24 g(t) h(N 1) (N 3, N, N + 1) (N 2, N ) (N 3, N 1, N + 1)+ (N 3, N 1, N ) (N 2, N + 1) i= 12 g(t) N 30 N 2N 1N N + 1 = 0 . (26)0 N 3N 2N 1N N + 1 That is to say, F (N ) is an exact so
29、lution of Eq. (6).Next, we use the idea as above to verify that the auto-BT (8a) and (8b) between the(N 1)- and N -solitonic solutions is indeed satisfied.Similarly constructing the N -solitonic solution, we may suppose F (N ) = (N 1) and(1)(2)(N 2) 1 1 1 1 0 (1)(2)(N 2) 2 2 2 2 0 F (N 1) = W (1, 2,
30、 , N1 , ) = (1)(2)(N 2) N 1 N 1 N 1 N 1 0 N(1)(2)(N 2) N N N 1 = (N 2, ) , (27)where = (0, , 0, 1)T . Then, the derivatives of F (N 1) can be written as:F (N 1) F (N 1) F (N 1) x= (N 3, N 1, ) , (28) xx= (N 4, N 2, N 1, ) + (N 3, N, ) , (29) xxx= (N 5, N 3, N 2, N 1, ) + 2 (N 4, N 2, N, ) + (N 3, N
31、+ 1, ) , (30)F (N 1)t = 4 g(t)h(N 5, N 3, N 2, N 1, ) (N 4, N 2, N, )+ (N 3, N + 1, )i 6f (t)g(t) x (N 3, N 1, )3 (N 1)(N 2)f (t)g(t) (N 2, ) . (31)Substitution of Eqs. (27)(31) into the BT (8a) and (8b) givesx Dt + g(t) D3= F (N )+ 3 (t) g(t) Dx + 6f (t)g(t) x Dx + (t) (F (N ) F (N 1) )(N 1)tF (N 1
32、) F (N ) Ft + (t) F (N )F (N 1)xxxxx Fx+ 3 FxFxx FFxxx + g(t) F (N ) F (N 1) 3 F (N )(N 1)(N )(N 1)(N )(N 1)+ 3 (t) g(t) F (N ) F (N 1) F (N ) F (N 1) + 6f (t)g(t) x F (N ) F (N 1) F (N ) F (N 1) xxxx= 6 g(t) N 4 N 20 0N 3 N 1 N 0 0N 4 N 2 N 3 N 1 N 0 0+ 6 g(t) N 30 N 2N 1N + 1 = 0 , (32)andN 3N 2N
33、1N + 10 x D2 (t) (F (N ) F (N 1) )= F (N )(N 1)(N )(N 1)(N )(N 1)(N )(N 1)xx F 2 FxFx+ FFxx (t) F F(33)= 2 (N 2, ) (N 3, N 1, N ) 2 (N 2, N ) (N 3, N 1, )+ 2 (N 1) (N 3, N, )= N 30 N 2N 1N = 0 , (34)0N 3N 2N 1N Nprovided that (t) = k2 e12R f (t) g(t)dt and (t) = 6 (N 1) f (t)g(t). Therefore, we have
34、 proved that the (N 1)- and N -solitonic solutions satisfy the auto-BT (8a) and (8b).4. ConclusionsIn this paper, we have derived the exact N -solitonic solution and an auto-BT for a vcKdV equation with nonuniformities and proved it via the Wronskian technique. Then, we have shown that the (N 1)- an
35、d N -solitonic solutions satisfy the obtained auto-BT. Further, we may obtain N -solitonic solution of other variable-coefficient equations through the similar process.AcknowledgmentsWe express our sincere thanks to Prof. Y. T. Gao, Ms. L. L. Li, Mr. T. Xu and Mr. H. Q. Zhang for their valuable comm
36、ents. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 60772023 and 60372095, by the Key Project of Chinese Ministry of Education (No. 106033), by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE-07-
37、001, 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 (No. 20060006024), Chinese Ministry of Education.References1 W. P. Ho
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