Integrable properties for a variable-coefficient Boussinesq equation from weakly nonlinear dynamics with symbolic computation.doc

精品论文http:/www.paper.edu.cnIntegrable 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 Telecommunications, 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, ChinaAbstractDescribing 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 counterpart 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-B¨acklund transformation, nonlinear superposition formula, Lax pair and Dar- boux transformation. In addition, some soliton-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: yaxingzhgmail.com4精品论文1. IntroductionAs an important nonlinear model in hydrodynamics, the Boussinesq equation is widely 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 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) + 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 investigated 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 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 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 + $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 along 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 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 to investigate the following variable-coefficient Boussinesq equation with the space-dependent coefficients 24utt + uxxx + µ + 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 or modal instabilities, where , µ 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 streamwise 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 B¨acklund transformation 25, 26 and Lax pair 27, 28.As shown in Refs. 18, 19 and references therein, it is a straightforward and effective 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 certain 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-B¨acklund transformation, nonlinear superposition formula, Lax pair and Darboux transformation. In Section 4, 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 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 x + b ,(6)where B(x, t), A(t), X (x) and T (t) are all analytic functions to be determined, while a andb are both constants.By virtue of symbolic computation, Eqn. (4) with the substitution of Format (6) becomesA(t) T 0(t)2 UT T + a4 A(t) UX X X X + a2 µ A(t) + a2 A(t) B + a2 A(t) f (x) UX XX+a2 A(t)2 U UX X + a2 A(t)2 U 2+ a A(t) g(x) + a A(t) f 0(x) + 2 a A(t) Bx UX+ 2 A0(t) T 0(t) + A(t) T 00(t) UT + A(t) g0(x) + A00(t) + A(t) Bxx U + B g0(x)x+Btt + g(x) Bx + f 0(x) Bx + B2 + µ Bxx + B Bxx + f (x) Bxx + Bxxxx = 0 ,(7)where the prime notes differentiation with respect to independent variable, which is Eqn. (5),provided thata4 + T 0(t)2 = 0 ,(8)(a2 µ + a2 B + a2 f (x) + T 0(x)2 = 0 ,(9)a2 A(t) + T 0(t)2 = 0 ,(10)a g(x) + a f 0(x) + 2 a Bx = 0 ,(11)2 A0(t) T 0(t) + A(t) T 00(t) = 0 ,(12)A(t) g0(x) + A00(t) + A(t) Bxx = 0 ,(13)xBtt + g(x) Bx + f 0(x) Bx + B2 + µ Bxx + f (x) Bxx+B g0(x) + Bxx + Bxxxx = 0 ,(14)from which we can getA(t) = A ,(15)T (t) = c t + ,(16)1B(x, t) = 2 where c = 0, A and are three constants.Zf (x) +g(x) dx,(17)Subsequently, with the help of symbolic computation, substituting Eqn. (17) into Eqn. (9)yields the constraints on the variable coefficients f (x) and g(x), as follows,Zf (x) =g(x) dx 2 c2 a2 2 µ ,(18)f 0(x)2 + µ f 00(x) + f (x) f 00(x) + f (4) (x) = 0 .(19)Integrating Eqn. (19) with respect to x gives rise tof (x)2 f 00(x) + µ f (x) +2= 0 ,(20)which is an elliptic equation with the following exact analytic solutions 29a1 a2 secha 2 h a1 i2 (x x0)2 h a1 i,(a1 > 0, a2 f (x) < 0)af (x) =12 a1csch2 h2a1(x x0)i,(a1 > 0, a2 f (x) > 0)(21)where a1 = µa2 sec3 and a2 = 1 .2 (x x0),(a1 < 0)Therefore, the transformation from Eqn. (4) to Eqn. (5) turns out to beu = B(x) + A U X (x), T (t) ,(22)精品论文5withhttp:/www.paper.edu.cnX (x) = a x + b ,(23)T (t) = c t + ,(24)1B(x) = 2 2Zf (x) +g(x) dx,(25)where a2 = µ , A = µ and c2 = µ , while b and are both arbitrary constants, and 2 f (x) and g(x) obey Constraints (18) and (19).It is worth noting that Eqn. (5) is solvable by the inverse scattering method 7 and possesses many remarkable properties such as the B¨acklund transformation 25, 26, Painlev´e property 26, Lax representation 27, 28, an infinite number of conservation laws 30 and Darboux transformation 27, 31. Accordingly, when Constraints (18) and (19) are satisfied, Eqn. (4) could be dealt with via soliton theory based on Eqn. (5) (see Section 3).3. Integrable propertiesGenerally speaking, the variable-coefficient NLEEs are not integrable unless the coeffi- cient functions satisfy certain constraints. In view of the current importance of variable- coefficient NLEEs, many efforts have been devoted to studying their integrable properties by transforming them into the integrable constant-coefficient counterparts 32. In what fol- lows, via the obtained transformation in Section 2, we will further investigate the integrable properties of Eqn. (4) with Constraints (18) and (19).3.1. Auto-Backlund transformation for Eqn. (4)To Eqn. (4), the auto-B¨acklund transformation exists along with each family of those solutions, under the corresponding set of the aforementioned constraints on f (x) and g(x).By virtue of Transformation (22), we will construct the auto-B¨acklund transformation forEqn. (4).Without loss of generality, we shall assume that = 1, = 6 and = 1 in Eqn. (5) sincethe following equationis equivalent to Eqn. (5).UT T UX X UX X X X 3 (U 2)X X = 0 ,(26)As we know, the auto-B¨acklund transformation for Eqn. (26) has been worked out inRef. 29. Taking U = WX , Eqn. (26) turns out to beWT T WX X 6 WX WX X WX X X X = 0 ,(27)which has the auto-B¨acklund transformation as belowWT + W0, T = + 2 (W W0) + (W W0)2 + (W W0 )3+ 3 + (W W0) (WX W0, X ) + (WX X W0, X X ) ,(28)1WT W0,T = + 3 (W W0) (WX W0,X ) + (WX X + W0,X X ) ,(29)精品论文http:/www.paper.edu.cn3where W and W0 are two different solutions of Eqn. (27), while 2 = 1 , and are botharbitrary constants.Utilizing Transformation (22) with Eqns. (28) and (29), where u = wx and U = WX ,gives rise to the following auto-B¨acklund transformation for Eqn. (4),6A c 2a c (w w0)a2 c (w w0)3wt + w0,t =+ c a (w w0 ) +A +A2 c +a+ 3 c (w w0 ) A(wx w0,x ) + c (wxx w0,xx )2a2,(30)wt w0, t =c A a 3 c (w w0 )(wx w0,x) +c (2 B0 + wxx + w0,xx )a2 ,(31)where a, c, , A, and have been defined as above, while w(x, t) satisfieswtt + wxxxx + µ + f (x) wxx + wx wxx + g(x) wx = 0 .(32)3.2. Nonlinear superposition formula for Eqn. (4)Furthermore, on the ground of the obtained auto-B¨acklund transformation, we will get the nonlinear superposition formula for Eqn. (4), which can be used to construct the multi- soliton-like solutions.Via the seed solution w0(x, t) for Eqn. (32), when 1 = 1 and 2 = 2, the solu-tions w1(x, t), w2(x, t) for Eqn. (32) are respectively obtained. Consequently, the nonlinearsuperposition formula for Eqn. (32) is expressed in the formZw3 = 21B(x) dx w0 + A A 21 a w1+ a w2Z( A2 + 2 A (w2,x w1,x)a )+ (A 1 A 2 + a w2 a w1)w1 + w2 2B(x) dx,(33)where is an arbitrary constant, from which, as u = wx, the multi-soliton-like solutions forEqn. (4) can be explicitly derived.3.3. Lax PairAs we have mentioned above, under Constraints (18) and (19), Eqn. (4) has been con- verted into the constant-coefficient one by virtue of Transformation (22). Actually, when Constraints (18) and (19) hold, Eqn. (4) can also be converted into the following Boussinesq equationUT T UX X + (U 2)X +13 UX X X X = 0 ,(34)2through Transformation (22) with A = 2 µ , a2 = µ and c2 = µ , while b and are botharbitrary constants.3 3 精品论文http:/www.paper.edu.cnIt is known that the Lax pair of Eqn. (34) is9 3L¯ = V +U 3 3+,(35)24 X2 X 3withM¯ = U X 2,(36)33VX = 4 UX X 4 UT ,(37)where L¯is the operator for the spectral problem and M¯is the operator governing theassociated time evolution of the eigenfunctions 27. Under Constraints (18) and (19), we can directly obtain the Lax pair for Eqn. (4), i.e.,L = ,(38) t = M ,(39)where t = 0, meanwhile, and (x, t) denote the eigenvalue and eigenfunction respectively, while L and M are presented in the forms3 3 (u B) 13L = v 4 a x +2 a A+ x a3 x3 ,(40)c (u B)c2withM = A a2 x2 ,(41)33 avx = 4 A a (uxx B00) 4 A c ut .(42)Then it is easy to prove that L and M satisfy the Lax equationLt M L + L M = 0 ,(43) if and only if u(x, t) is a solution of Eqn. (4), which suggests that Eqn. (4) is integrable under Constraints (18) and (19).3.4. Darboux transformation for Eqn. (4)In virtue of the Lax representation for Eqn. (4), we can further derive its Darboux transformation in this subsection.Let (x, t) be a solution of the Lax pair, i.e., (38) and (39), and h(x,