Integrable properties for a variable-coefficient Boussinesq equation from weakly nonlinear dynamics with symbolic computation.doc
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1、精品论文http:/Integrable properties for a variable-coefficient Boussinesq equation from weakly nonlinear dynamics with symbolic computationYa-Xing Zhang1 , Tao Xu1, Juan Li1, Chun-Yi Zhang2, 3, Hai-Qiang Zhang1 and Bo Tian11. School of Science, P. O. Box 122, Beijing University of Posts and Telecommunic
2、ations, Beijing 100876, China2. 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, ChinaAbstractD
3、escribing the weakly nonlinear dynamics of long waves embedded in marginally stable shear flows that vary in the streamwise direction, a variable-coefficient Boussinesq equation is investigated in this paper. With symbolic computation, such a model is transformed into its constant-coefficient counte
4、rpart under certain constraints on the coefficient functions. By virtue of the obtained transformation, some integrable properties for this equation are derived, such as the auto-Backlund transformation, nonlinear superposition formula, Lax pair and Dar- boux transformation. In addition, some solito
5、n-like solutions are obtained from the integrable properties and the relevant physical applications are also pointed out.PACS numbers: 02.30.Ik; 05.45.Yv; 02.30.Jr; 52.35.Mw; 02.70.Wz Mailing address: 4精品论文1. IntroductionAs an important nonlinear model in hydrodynamics, the Boussinesq equation is wi
6、dely employed to describe some physical and applied mathematic problems including the nonlinear Kelvin-Helmholtz instability 1, 2, nonlinear lattices 3, 4, nonlinear baroclinic instability 5,6, vibrations in a nonlinear string 7 and ion sound waves in plasmas 8. In addition, such an equation is also
7、 applicable to modelling the propagation of long waves in shallow water under gravity and in one-dimensional nonlinear lattices 10. Applications of the constant- coefficient Boussinesq equation can be seen from the following examples: (1) The originalBoussinesq equation 11tt xx2xxxxxxu m u+ 3 (u2) +
8、 n u= 0 ,(1)governs the propagation of long waves in shallow water in a uniform environment, wherem and n are both constants, while x and t represent a “long” space and a “slow” time,respectively. (2) The “good” Boussinesq equation 12 is written asutt = uxx uxxxx + (u2 )xx ,x R ,(2)which has been in
9、vestigated in detail through the analytic and numerical methods 13. It has been pointed out in Refs. 13 that Eqn. (2) exhibits some interesting behavior: the small-amplitude solitons keep their shapes and velocities unchangeable in the interaction process, while the large-amplitude ones change into
10、the so-called antisolitons after collision.In reality, the media may be inhomogeneous and the boundaries may be nonuniform, e.g., in plasmas 14, 15, superconductors 16, optical-fiber communications 17, 18, blood vessels and Bose-Einstein condensates 19. Therefore, the variable-coefficient nonlinear
11、evo- lution equations (NLEEs) are considered to be more realistic than their constant-coefficient counterparts in describing a large variety of real situations. For instance, in terrain-followingcoordinates, the second order Boussinesq equation,%$ 2M () 2 M 0() 2f M () ftt + 3 M () f tt M ()f +ft +
12、$tM 2()ft f = 0 ,(3)is a useful model in the presence of rapidly varying topographies 20, where the prime means differentiation with respect to , $ is the nonlinearity parameter, % is the dispersion parameter and M is the variable free surface coefficient. This model governs the velocitypotential al
13、ong the bottom topography and when a topography is absent, it reduces to the standard Boussinesq equation.On the other hand, the coefficient functions related to x and/or t bring about a greatamount of complicated calculations which are unmanageable manually, although the variable- coefficient NLEEs
14、 are of current importance in many fields of physics and engineering sci- ences 21, 22. Fortunately, however, symbolic computation 14, 15, 23, as a new branch of artificial intelligence, makes it exercisable to deal with those nonlinear systems with variable coefficients.In this paper, we would like
15、 to investigate the following variable-coefficient Boussinesq equation with the space-dependent coefficients 24utt + uxxx + + f (x) ux + u ux + g(x) ux = 0 ,(4)which can describe the long waves embedded in marginally stable shear flows varying in the streamwise direction and can govern new nonmodal
16、or modal instabilities, where , and are constants defined in Ref. 5; f (x) and g(x), related to the streamwise varying background flow, are both the analytic functions; f (x) uxx and g(x) ux contribute to theemergence of new (linear) modal or nonmodal instabilities. When the background flow is strea
17、mwise uniform, i.e., f (x) = 0 and g(x) = 0, Eqn. (4) reduces to the constant-coefficient Boussinesq equation, which possesses a lot of good properties such as the Backlund transformation 25, 26 and Lax pair 27, 28.As shown in Refs. 18, 19 and references therein, it is a straightforward and effectiv
18、e method for investigating the variable-coefficient NLEEs to transform them into some known equations. Thus, we organize our paper as follows. In Section 2, with the aid of symbolic computation, the transformation from Eqn. (4) to a constant-coefficient Boussinesq equation is constructed under certa
19、in constraints. In Section 3, taking advantage of the obtained transformation, a series of integrable properties are figured out when the relevant constraints are satisfied, such as the auto-Backlund transformation, nonlinear superposition formula, Lax pair and Darboux transformation. In Section 4,
20、several soliton-like solutions, differing from the existing ones, are presented through the above properties and possible applications are also discussed. Section 5 is the conclusions for this paper.2. Transformation from Eqn. (4) to a constant-coefficient Boussinesq equationIn this section, we will
21、 determine the conditions for Eqn. (4) to be transformable into the following constant-coefficient Boussinesq equationU U(U T TX X X X22)X X UX X = 0 .(5)Similar to Refs. 18, 19, the transformation from Eqn. (4) to Eqn. (5) is assumed to be of the formu = B(x, t) + A(t) U X (x), T (t) with X (x) = a
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