N -soliton-typed solution in terms of Wronskian.doc

精品论文大全http:/www.paper.edu.cnN -soliton-typed solution in terms of Wronskiandeterminant for a forced variable-coefficient Korteweg-deVries equationZhen-Zhi Yao1, Chun-Yi Zhang2, 3, Hong-Wu Zhu1, Xiang-Hua Meng1, Tao Xu1 and Bo Tian11. School of Science, P. O. Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China 2. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China3. Meteorology Center of Air Force Command Post, Changchun 130051, ChinaAbstractIn this paper, we first derive the bilinear form and auto-B¨acklund transformation for a variable-coefficient Korteweg-de Vries-typed (vcKdV) equation with external-force term. Then, we obtain the N -soliton-typed solution in terms of Wronskian form, which is proved to be indeed an exact solution of this equation through Wronskian technique. In addition, we show that the (N 1)- and N -soliton-typed solutions satisfy the auto- Backlund transformation by using the same method.Keywords: Variable-coefficient Korteweg-de Vries equation; Auto-B¨acklund trans- formation; N -soliton-typed solution; Wronskian determinant.Mailing address for ZZY: yzz20051418yahoo.com.cn9精品论文大全1. IntroductionWith the consideration of inhomogeneities of media and nonuniformities of boundaries, the variable-coefficient Korteweg-de Vries (vcKdV)-typed equations have attracted much attention in vessel dynamics, arterial mechanics, Bose-Einstein condensates, nonlinear exci- tations of Bose gases, etc 1, 2. Higher dimensional extensions of them are seen in space plasmas, ocean dynamics and other fields 1, 3, 4. These equations, although often hard to be investigated, are able to describe various real situations more powerfully than their constant-coefficient counterparts. In this paper, we will investigate a forced vcKdV equa- tion 5, as follows:ut + f (t) u ux + g(t) uxxx + h(t) = 0 , (1)where the wave amplitude u(x, t) is a function of the scaled “space” x and “time” t and the real functions f (t) = 0, g(t) = 0 and h(t) represent the coefficients of the nonlinear, dispersive and external-force terms, respectively. If h(t) = 0, Eq. (1) reduces to the generalized vcKdV equation investigated in 6ut + f (t) u ux + g(t) uxxx = 0 , (2)which can be used to describe the ion-acoustic waves in plasma physics and such coastal waters as those observed in the Adriatic Sea, eastern Mediterranean, north west shelf of Australia and Baltic Sea 7, 8. Corresponding to that h(t) = 0, f (t) = 6 and g(t) = 1, Eq. (1) becomes the standard KdV equation 9ut + 6 u ux + uxxx = 0 . (3) Generally speaking, the N -soliton solution for an integrable nonlinear evolution equationcould be expressed in different forms. In the inverse scatting scheme, the N -soliton solution may be written in terms of an N ×N determinant 10. On the other hand, using Hirotas direct method, we can express the N -soliton solution as an N th-order polynomial in Nexponentials 11. The two methods have been proved to be very difficult for the verification of these solutions by direct substitution into the evolution equation. A more convenient representation of the N -soliton solution is in terms of the Wronskian determinant of N exponential functions, which has particularly nice properties when differentiated 12, 13,14. The Wronskian technique was firstly introduced for the Korteweg-de Vries (KdV) and modified KdV equations in 12 and then extended to investigate solutions for some other soliton equations such as the Kadomtsev-Petviashvili (KP) equation, Boussinesq equation, Nonlinear Schr¨odinger (NLS) equation and Davey-Stewartson (DS) equation 13. We notice the fact that the differentiation of the Wronskian determinant results in the sum of a number of determinants depending not on the size of the determinant. Thus, inserting the N -soliton (or N -soliton-typed) solutions into those soliton equations, we can verify them easily.With symbolic computation 1, 3, 4, 15, the present paper proceeds as follows: in Section2, we will educe the bilinear form and auto-B¨acklund transformation for Eq. (1). Then, in Section 3, by means of the auto-B¨acklund transformation, we present the Wronskian form of the N -soliton-typed solution and verify it. Meanwhile, we will prove that the (N 1)- and N -soliton-typed solutions satisfy the auto-B¨acklund transformation. Section 4 will be our conclusions.2. Bilinear form and auto-B¨acklund transformation of Eq. (1)By introducing w(x, t) defined by u = wx, Eq. (1) may be integrated to give1 2wt + 2 f (t) wx + g(t) wxxx + x h(t) = (t) , (4)where (t) is an integration function of t. Under the dependent variable transformationw = x B(t) +12 g(t)f (t)ln F (x, t) , (5) xEq. (1) becomesB2(t) f (t)2+ 12 g(t) f 0 (t) Fxf 2(t) F+ 12 g0(t) Fxf (t) F12 g(t) Ft Fxf (t) F 212 B(t) g(t) F 212 g(t) Fxt12 B(t) g(t) Fxx36 g(t)2 F 2+ x xxF 2+48 g2(t) Fx Fxxxf (t) F 12 g2(t) FxxxxF +f (t) F 2f (t) F 2 + x h(t) x B0(t) = (t) . (6)f (t) F2If B(t) = R h(t) dt and (t) = B2 (t) f (t) , Eq. (6) reduces to the following bilinear formZxDx Dt + g(t) D4 + f (t)xh(t) dt D2 (F · F ) = 0 , (7)throughwith the constraintZu(x, t) = h(t) dt +12 g(t)f (t) 2 x2 ln F (x, t) , (8)g(t) = f (t) , (9)where = 0 is an arbitrary constant. Here and in the following analysis, DxDt , D2 and D4xxare all the bilinear operators, which are defined in Refs. 16Dmn m na(x, t) b(x0 , t0 ). (10)x Dt a · b x x0t t0x0 =x, t0 =tIn order to derive the Wronskian solution, we first give the auto-B¨acklund transformation in bilinear form for Eq. (1) as follows:ZxDt + g(t) D3 + f (t)h(t) dt Dx 3 g(t) Dx(F · F 0) = 0 , (11)xD2 + (F · F 0) = 0 , (12)where is an arbitrary parameter, while F (x, t) and F 0(x, t) are two different solutions forEq. (7).3. N -soliton-typed solution in terms of Wronskian determinantIf we take F 0 = 1, corresponding to the vacuum solution of Eq. (1), we obtain a pair of equations with respect to F (x, t) from Eqs. (11) and (12)ZFt + g(t) Fxxx + f (t)h(t) dt Fx 3 g(t) Fx = 0 , (13)Fxx + F = 0 , (14)from which we obtainZF = e + e with = k x 4 k3g(t) dt kZ hZf (t)h(t) dt i dt + , (15)where = k2. Substituting Eq. (15) into Transformation (8) we derive the single-soliton- typed solution of Eq. (1).Next, we suppose that the N -soliton-typed solution in the sense of 11 has the form of the following Wronskian determinant(1)(2)(N 1) 1 1 1 · · ·1 2(1)(2)(N 1) F (N ) = W (1, 2, · · ·, N ) = 2 2 · · ·2 , (16) · · ·· · ·· · ·· · ·· · · N(1)(2)(N 1) iwith (i) = j (i = 1, 2, · · ·, N 1) , N N· · ·N= e j + (1)j+1 e j ,jxiZjZ hZjj = kj x 4 k3g(t) dt kjf (t)h(t) dtidt + j ,where kj and j (j = 1, 2, · · ·, N ) are all arbitrary real parameters.It is obvious that the entries j (j = 1, 2, · · ·, N ) satisfy the following conditionsZj j, xx = k2 j, j, t= 4 g(t) j, xxx f (t)h(t) dt j, x. (17)If we introduce the notation (N k) = W (1, 2, · · ·, N k ), Fobtained as(N )and its derivatives can bex= (N 2, N ) ,F (N ) = (N 1) ,(18)F (N )(19)F (N )(20)F (N )(21)xx= (N 3, N 1, N ) + (N 2, N + 1) ,xxx = (N 4, N 2, N 1, N ) + 2 (N 3, N 1, N + 1) + (N 2, N + 2) ,F (N ) xxxx = (N 5, N 3, N 2, N 1, N ) + 3 (N 4, N 2, N 1, N + 1)+ 2 (N 3, N, N + 1) + 3 (N 3, N 1, N + 2) + (N 2, N + 3) , (22)F (N )t = 4 g(t)h(N 4, N 2, N 1, N ) (N 3, N 1, N + 1) + (N 2, N + 2)iZF (N ) f (t)hh(t) dt (N 2, N ) , (23)ixt = 4 g(t)(N 5, N 3, N 2, N 1, N ) (N 3, N, N + 1) + (N 2, N + 3)Z f (t)h(t) dth(N 3, N 1, N ) + (N 2, N + 1)i. (24)Substituting the above derivatives of F (N ) into Eq. (7), we havexZxDx Dt + g(t) D 4 + f (t)h(t) dt D 2 (F (N ) · F (N ) )= 2 F (N ) F (N ) 2 F (N ) F (N ) + 2 g(t) F (N ) F (N ) 4 F (N ) F (N ) + 3 F (N ) F (N ) xtxtZxxxxxxxxxxxx+ 2 f (t)h(t) dt F (N ) F (N ) F (N ) F (N )xxxx= 6 g(t) N 30N 2N 1N N + 1 = 0 . (25)0N 3N 2N 1N N + 1 Thus, we have verified that F (N ) is an exact solution of Eq. (7).We have already presented that the N -soliton-typed solution of Eq. (7) may be expressed in terms of Wronskian determinant. In the following analysis, we will use the ideas as above to verify that the auto-B¨acklund transformation between the (N 1)- and N -soliton-typed solutions is satisfied indeed.Similarly, we choose the (N 1)-soliton-typed solution expressed in terms of F (N 1)(1)(2)(N 2) 1 1 1 · · ·1 0 (1)(2)(N 2) 2 2 2 · · ·2 0 F (N 1) = W (1, 2 , · · ·, 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, ) , (26)where = (0, · · ·, 0, 1)T . Thus, the derivatives of F (N 1) can be written as:F (N 1) F (N 1) F (N 1) x= (N 3, N 1, ) , (27) xx= (N 4, N 2, N 1, ) + (N 3, N, ) , (28) xxx= (N 5, N 3, N 2, N 1, ) + 2 (N 4, N 2, N, )+ (N 3, N + 1, ) , (29)F (N 1)t = 4 g(t)h(N 5, N 3, N 2, N 1, ) (N 4, N 2, N, )Z+ (N 3, N + 1, )i f (t)h(t) dt (N 3, N 1, ) . (30)Substituting all the derivatives of F (N ) and F (N 1) into the auto-B¨acklund transforma- tion, i.e., the set of Eqs. (11) and (12), we obtainZxDt + g(t) D3 + f (t)h(t) dt Dx 3 g(t) Dx(F (N ) · F (N 1) )= F (N 1) F (N )(N 1)t FtF (N )+ g(t)F (N 1) F (N )(N 1)(N )(N 1) N(N 1)(N )Z+ f (t)xxx 3 FxFxx + 3 Fxx Fx Fxxx Fh(t) dt F (N 1) F (N ) F (N 1) F (N ) 3 g(t) F (N 1) F (N ) F (N 1) N x x x x F= 3 g(t) N 30N 2N 1N + 1 0N 3N 2N 1N + 1 N 4N 200N 3N 1N 3 g(t) = 0 , (31)00N 4N 2N 3N 1N xD2 + (F N · F (N 1) )= F (N 1) F N(N 1) N(N 1) N(N 1) Nxx 2 FxFx + Fxx F+ F F= N 30N 2N 1N = 0 , (32)0N 3N 2N 1N Nwhere = k2 . Therefore, we have proved that the (N 1)- and N -soliton-typed solutions satisfy the auto-B¨acklund transformation, i.e., the set of Eqs. (11) and (12).4. ConclusionsIn this paper, we have derived the N -soliton-typed solution for a forced vcKdV equation and verified it using the Wronskian technique. In addition, we have also shown that the (N 1)- and N -soliton-typed solutions satisfy the auto-B¨acklund transformation. Furthermore,via the same technique, we may obtain the N -soliton-typed solution of some other variable- coefficient equations.精品论文大全AcknowledgmentsWe express our thanks to Prof. Y. T. Gao, Ms. J. Li, Ms. L. L. Li, Ms. Y. X. Zhang, Mr. X. Lu¨, Mr. Z. C. Yang, Mr. C. Zhang and Mr. H. Q. Zhang for their valuable comments. This work has been supported by the Key Project of Chinese Ministry of Education (No.106033), by the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20060006024), Chinese Ministry of Education, and by the National Natural Science Foundation of China under Grant No. 60372095.References1 M. P. Barnett, J. F. Capitani, J. Von Zur Gathen and J. Gerhard, Int. J. Quantum Chem. 100 (2004) 80; B. Tian and Y. T. Gao, Phys. Lett. A 340 (2005) 449; Phys. Plasmas 12 (2005) 054701.2 H. Demiray, J. Comput. Appl. Math. (2006) in press; B. Tian, G. M. Wei, C. Y. Zhang, W. R. Shan and Y. T. Gao, Phys. Lett. A 356 (2006) 8; Y. T. Gao and B. Tian, Phys. Lett. A 349 (2006) 314; Q. Wang, Phys. Lett. A 358 (2006) 91; X. Y. Tang, F. Huang and S. Y. Lou, Chin. Phys. Lett. 23 (2006) 887.3 W. P. Hong, Phys. Lett. A 361 (2007) 520; Y. T. Gao and B. Tian, Phys. Plasmas 13 (2006) 112901; Phys. Plasmas (Lett.) 13 (2006) 120703; Phys. Lett. A 361 (2007) 523; Europhys. Lett. 77 (2007) 15001.4 G. Das and J. Sarma, Phys. Plasmas 6 (1999) 4394;B. Tian and Y. T.Gao, Eur. Phys. J. D 33 (2005) 59;Phys. Lett. A, published online, 2007, DOI:10.1016/j.physleta.2006.10.094.5 R. Sabry, M. A. Zahranb and E. G. Fan, Phys. Lett. A 326 (2004) 93.6 R. H. Grimshaw, Proc. R. Soc. A 386 (1979) 359; N. Nirmala, M. J. Vedan and B. V.Baby, J. Math. Phys. 27 (1986) 2640; M. L. Wang and Y. M. Wang, Phys. Lett. A 287(2001) 211; E. G. Fan, Phys. Lett. A 294 (2002) 26.7 A. Osborne, Chaos, Solitons and Fractals 5 (1995) 2623; P. Holloway, E. Pelinovsky, T.Talipova and B. Barnes, J. Phys. Oceanogr. 27 (1997) 871; T. Talipova, E. Pelinovsky and T. Kouts, Oceanology 38 (1998) 33; P. Holloway, E. Pelinovsky and T. Talipova, J. Geophys. Res. C 104 (1999) 18333.8 S. Yi, J. L. Cooney, H. S. Kim, A. Amin, Y. El-Zein and K. E. Lonngren, Phys. Plasmas3 (1996) 529; G. Das and J. Sarma, Phys. Plasmas 5 (1998) 3918; H. Hong and J. Lee, Phys. Plasmas 6 (1999) 3422.9 D. J. Korteweg and G. de Vries, Philos. Mag. 39 (1895) 422.10 G. L. Lamb, Elements of Soliton Theory (Wiley-Interscience, New York, 1980).11 R. Hirota, in: Solitons, eds. R. K. Bullough and P. J. Caudrey (Springer, Berlin, 1980). 12 J. Satsuma, J. Phys. Soc. Jpn. 46 (1979) 359.13 N. C. Freeman and J. J. Nimmo, Phys. Lett. A 95