The Exact Solitary Wave Solutions for a Family of BBM Equation

Transcription

1 ISSN (print), (online) International Journal of Nonlinear Science Vol. (2006) No., pp The Exact Solitary Wave Solutions f a Family of BBM Equation Lixia Wang, Jiangbo Zhou, Lihong Ren Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University Zhenjiang, Jiangsu, 2203,P.R.China (Received 2 September 2005, accepted 30 November 2005) Abstract: In this paper, the family of BBM equation with strong nonlinear dispersive B(m,n) equation is introduced. By using a simple and effective algebraic method,some abundant exact solutions f this class of BBM equations are gotten and call its solutions with solitary patterns, solitary wave solution with singular point, smooth solitary wave solution, kink solution, antikink solution, floating solitary wave solution, peakon solution, compacton solution. Keywds: BBM equation; solitary wave; Peakon ; Compacton; strong dispersion Introduction To study the role of nonlinear dispersive in the fmation of patterns in liquid drops, Rosenau and Hyman (see]) studied the genuinely nonlinear dispersive equations K(m,n) given by u t + (u m ) x + (u n ) xxx = 0, m > 0, < n 3, and introduced a class of solitary wave solutions with compact suppt that are solutions of a two parameter family of fully nonlinear dispersive partial differential equations such as K(2,2) equation and u t + (u 2 ) x + (u 2 ) xxx = 0, u t + (u 3 ) x + (u 3 ) xxx = 0. Recently, Wazwaz (see2])gave exact special solutions with solitary patterns f the nonlinear dispersive K(m,n) equations: u t (u m ) x + (u n ) xxx = 0, m, n >. In 3], Wazwaz studied the genuinely nonlinear dispersive K(m,n) equations u t + (u m ) x + (u n ) xxx = 0, m, n >, and obtained the new solitary wave special solutions with compact suppt of the nonlinear dispersive K(m,n) equation. Yan and Bluman (see4]), Yan (see5]) and Zhu (see6]) investigated a family of Boussinesq-like equations with fully nonlinear dispersion B(m,n) equations: u tt = (u m ) xx + (u n ) xxx and u tt (u m ) xx + (u n ) xxx = 0. New families of solitons with compact suppt and solitary patterns solutions f Boussinesqlike B(m,n) equations with fully nonlinear dispersion are developed by the Adomian decomposition method, respectively. Cresponding auth. address: wanglixia@63.com Tel: Copyright c Wld Academic Press, Wld Academic Union IJNS /09

2 L.Wang.J.Zhou,L.Ren:The Exact Solitary Wave Solutions f a Family of BBM Equation 59 Tian, Gui and Liu (see7]) solved the well-posedness problem, the isospectral problem and the scattering problem f the Dullin-Gottwald-Holm (DGH) equation. Meanwhile, the scattering data of the scattering problem f the equation was explicitly expressed. Tian and Yin (see8]) introduced a fifth-der K(m,n) equation with nonlinear dispersion to obtain multicompaction solutions by the Adomian decomposition method. And in 9], Fan and Tian proved that solitary wave of mkdv-ks equation persisted when the perturbation parameter was suitably small. Tian, Yin (see0]]) considered the nonlinear generalized Camassa-Holm equation and derived some new compacton and floating compacton solutions by using four direct ansatzs. Tian and Song (see2]) considered generalized Camassa-Holm equations and generalized weakly dissipative Cammassa-Holm equations, and derived some new exact peaked solitary wave solutions. Ding and Tian (see3]) studied the existence of global solution and got the existence of the global attract on dissipative Cammassa-Holm equation. BBM equation the regularized long-wave equation: u t + uu x u xxt = 0, was put fward by Peregine (see4]5]) and Benjamin et al.(see6]) as an alternative model to the Kteweg-de VriesKdVequation f small-amplitude, long wavelength surface water waves. Shang (see7]) studied the BBM-like equations with fully nonlinear dispersion, B(m,n) equations: u t + (u m ) x (u n ) xxt = 0, m, n >. The exact solitary-wave solutions with compact suppt and exact special solutions with solitary patterns of the equations are derived. In this paper, we introduce and study a family of BBM equation with strong nonlinear dispersive B(m,n) equation : u t + u x + a(x m ) x + (u n ) xxt = 0. (.) Some abundant exact solutions f the equations are gotten. The rest of this paper is ganized as follows: In section 2, we give many types of solitary pattern solutions of B(m,n) equations. In section 3 and section 4, we derive peakon solution and compacton solution f the family of equations. 2 Solitary pattern solutions of B(m,n) equations First of all, we make the traveling wave transfmation as follows: u(x, t) = u(ξ), ξ = λ x + λ 2 t, (2.2) where λ i (i =, 2, 3) are parameters to be determined later. Integrating () and setting integration constants to zero, we have λ 2 u + λ u + aλ u m + λ 2 λ 2 (u n ) ξξ = 0. (2.3) To seek solitary pattern solutions we assume that3has the solution in the fm where A,B are parameters to be determined later. 2. The first type of solitary pattern solution Substituting (2.3)into (2.2), we get the following polynomial equations u(x, t) = u(ξ) = A sin h B (ξ). (2.4) u(x, t) = u(ξ) = A cos h B (ξ). (2.5) u(x, t) = u(ξ) = A tan h B (ξ). (2.6) λ 2 A sin h B (ξ) + λ sin h B (ξ) + aλ A m sin h mb (ξ) + λ 2 λ 2 A n nb(nb ) sin h nb 2 (ξ) +λ 2 λ 2 A n n 2 B 2 sin h nb (ξ) = 0. (2.7) IJNS homepage:

3 60 International Journal of Nonlinear Science,Vol,(2006),No.,pp Comparing the coefficients of similar terms we have n =, B = 2 m, Am = 2λ 2 ( + m) a + ( m)2 ], λ 2 = λ ( m) 2 + ( m)2, (2.8) m = n, B = 2 n, An = 4λ2 n2 a(n ) 2 a(n )2 2aλ 2 n(n + ), λ 2 = 4λ n 2. (2.9) By combing (2.3),(2.7) and (2.8), we obtain u (x, t) = 2λ 2 ( + m) a + ( m)2 ] 4λ 2 u 2 (x, t) = n 2 a(n ) 2 2aλ 2 n(n + ) m 2 sin h m λ x λ ( m) 2 ] 4λ 2 t + ( m)2 n 2 sin h n λ x ] a(n )2 4λ n 2 t, (2.0). (2.) (2.9),(2.0) describe solitary wave solution with singular point and solitary pattern solutions f the equation ()respectively. (see Fig. and Fig.2 ) Figure : Graphics of solution (0) Figure 2: Graphics of solution () 2.2 The second type of solitary pattern solution Substituting (2.4)into (2.2) λ 2 A cos h B (ξ) + λ A cos h B (ξ) + aλ A m cos h mb (ξ) +λ 2 λ 2 A n n 2 B 2 cos h nb (ξ) λ 2 λ 2 A n nb(nb ) cos h nb 2 (ξ) = 0. (2.2) m = n, B = 2 n, An = a(n )2 n2 a(n )2 2aλ 2 (n + ), λ 2 = 4λ n 2, (2.3) n =, B = 2 m, Am = 2 a λ 2 ( + m). (2.4) + ( m)2 IJNS eamil f contributiom: edit@nonlinearscience.g.uk

4 L.Wang.J.Zhou,L.Ren:The Exact Solitary Wave Solutions f a Family of BBM Equation 6 By combing (2.4),(2.2) and (2.3), we obtain a(n ) 2 u 3 (x, t) = n2 2aλ 2 n(n + ) u 4 (x, t) = 2λ2 ( + m) a + ( m)2 ] n 2 cos h n λ x ] a(n )2 4λ n 2 t m 2 cos h m λ x λ ( m) 2 ] 4λ 2 t + ( m)2, (2.5). (2.6) (2.4)(2.5) describe solitary pattern solutions and smooth solitary wave solution f the equation ()respectively. (see Fig.3) 2.3 The third type of solitary pattern solution Substituting (2.5)into (2.2): Figure 3: Graphics of solution (6) (λ 2 + λ )A tan h B (ξ) + aλ A m tan h mb (ξ) + λ 2 λ 2 A n nb(nb ) tan h nb 2 (ξ) 2λ 2 λ 2 A n n 2 B 2 tan h nb (ξ) + λ 2 λ 2 A n (nb + )nb tan h nb+2 (ξ) = 0. (2.7) B =, n =, m = 3, A 2 = 2λ 2 a(2λ 2 ), λ 2 = λ 2λ 2, (2.8) m = n, nb =, B = 3, A n = a + 2λ2 2aλ 2, λ 2 = a, (2.9) 2λ n =, B = 2, m = 2, A = a( ), λ 2 = λ, (2.20) B = 4, m = 2, m = n, An = 4λ2 + a 4aλ 2, λ 2 = a. (2.2) 4λ By combing (2.5),(2.7),(2.8),(2.9)and (2.9), we obtain u 5 (x, t) = u 6 (x, t) = 2λ 2 a(2λ 2 ) 2 tan h ( λ x + λ 2λ 2 t a + ( 2λ2 n tan h 3 2aλ 2 λ x + u 7 (x, t) = a(4λ 2 tan h2 ) u 8 (x, t) = ) a ) t 2λ ) ( λ x + λ t 4λ2 + a ( n tan h 4 4aλ 2 λ x + a ) t 4λ, (2.22), (2.23), (2.24). (2.25) (2.2),(2.22),(2.23)and (2.24) describe kink solution, anti-kink solution, floating solitary wave solution f the equation ()respectively.(see Fig.4,Fig.5,Fig.6,Fig.7) IJNS homepage:

5 62 International Journal of Nonlinear Science,Vol,(2006),No.,pp Figure 4: Graphics of solution (22) Figure 5: Graphics of solution (23) Figure 6: Graphics of solution (24) 3 Peakon solution of B(m,n) equations First of all we make the traveling wave transfmation as follows: Integrating Eq.(.) and setting integration constants to zero, we have Let u(ξ) = λe b ξ, (3.2) reduces to be u(x, t) = u(ξ), ξ = x Dt. (3.26) Du + u + au m D(u n ) ξξ = 0. (3.27) e bξ λ De bξ λ + (e bξ ) m aλ m + b 2 De 2bnξ nλ n b 2 De bnξ nλ n b 2 De 2bnξ n 2 λ n = 0. (3.28) a m = n, D =, b = ± n, then u 9 (x, t) = λe ± a n x t. (3.4)describes peakon solution f the equation (.). (see Fig.8) IJNS eamil f contributiom: edit@nonlinearscience.g.uk

6 L.Wang.J.Zhou,L.Ren:The Exact Solitary Wave Solutions f a Family of BBM Equation 63 Figure 7: Graphics of solution (25) Figure 8: Graphics of solution (29) 4 Compacton solution of B(m,n) equations Let u(ξ) = { A cos q (Bξ), Bξ π 2, 0, Bξ π 2. (4.29) Substituting (4.)into (3.2): ( D)A cos q (Bξ)+aA m cos mq (Bξ)+DA n B 2 n 2 q 2 cos nq (Bξ) DA n B 2 nq(nq ) cos nq 2 (Bξ) = 0. where B is another arbitrary constant. m = n, q = 2 ( n, A = a D ) n, ( ) u 0 (ξ) = a n D cos 2 n, Bξ π 2, (4.30) 0, Bξ π 2. (4.2) describes compacton solution f the equation (.). (see Fig.0) Figure 9: Graphics of solution (3) IJNS homepage:

7 64 International Journal of Nonlinear Science,Vol,(2006),No.,pp Acknowledgements The auth thanks Profess Lixin Tian f his valuable discussion and suggestions. This research was suppted by the National Nature Science Foundation of China (No:007033) and the Nature Science Foundation of Jiangsu Province (No: BK ) References ] P Rosenau,J M Hyman : Compactons: solitons with finite wavelengths. Phys Rev Lett.70(5), (993) 2] A M Wazwaz :Exact special solutions with solitary patterns f the nonlinear dispersive K(m,n) equations.chaos Solitons and Fractals. 3,6-70(2002) 3] A M Wazwaz :New solitary wave special solutions with compact suppt the nonlinear dispersive K(m,n)equations. Chaos Solitons and Fractals.3,32-330(2002) 4] Z Y Yan,G Bluman : New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations.comput Phys Commun.49,-8(2002) 5] Z Y Yan:New families of solitons with compact suppt f Boussinesq-like B(m,n) equations with fully nonlinear dispersion. Chaos,Solitons and Fractals.4,5-58(2002) 6] Y G Zhu :Exact special solutions with solitary patterns f Boussinesq-like B(m,n) equations with fully nonlinear dispersion. Chaos, Solitons and Fractals.22,23-220(2004) 7] Lixin Tian,Guilong Gui,Yue Liu:On the well-posedness problem and the scattering problem f DGH equation.comm.math.phys.257(3),667-70(2005) 8] Lixin Tian,Jiuli Yin:Stability of multi-compacton and Backlund transfmation in K(m,n,l).Chaos Solitons and Fractals. 23(),59-69( ] Xinghua Fan,Lixin Tian:The existence of solitary waves of singularly perturbed mkdv-ks equation.chaos Solitons and Fractals. 26,-8(2005) 0] Lixin Tian,Jiuli Yin:New compacton solutions and solitary wave solutions of fully nonlinear generalized Camassa-Holm equations.chaos,solitons and Fractal.20(3), (2004) ] Jiuli Yin,Lixin Tian:Compacton solutions and floating compacton solutions of one type of nonlinear equations.acta Physica Sinica.53(9),827(2004) 2] Lixin Tian,Xiuying Song.New peaked solitary wave solutions of the generalized Camassa-holm equation.chaos,solitons and Fractals. 9(3),62-639(2004) 3] Danping Ding,Linxin Tian:The attract in dissipative Camassa-Holm equation.acta Physica Sinica.27(3), (2004) 4] D H Peregine :Calculations of the development of an undular be. J Fluid Mech.25,32-330(964) 5] D H Peregine :Long waves on a beach.j Fluid Mech. 27,85-827(967) 6] T B Benjamin,JL Bona, J Mahony :Model equations f long wave in nonlinear dispersive systems.philos Trans R Soc London A.272,47-78(972) 7] Shang Yadong:Explicit and exact special solutions f BBM-like B(m,n equations with fully nonlinear dispersion.chaos Solitons and Fractals.25,083-09(2005) IJNS eamil f contributiom: edit@nonlinearscience.g.uk