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-Backlund 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-Backlund trans- formation; N -soliton-typed solution; Wronskian determinant.Mailing address for ZZY: yzz20051418yahoo.com.cn1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 Schrodinger (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-Backlund transformation for Eq. (1). Then, in Section 3, by means of the auto-Backlund 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-Backlund transformation. Section 4 will be our conclusions.2. Bilinear form and auto-Backlund transformation of Eq. (1)By introducing w(x, t) defined by u = wx, Eq. (1) may be integrated to give1 2wt + 2 f (t