Research Article New Exact Solutions for the 2 1 -Dimensional Broer-Kaup-Kupershmidt Equations

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1 Hindawi Publishing Corporation Abstract and Applied Analysis Volume 00, Article ID 549, 9 pages doi:0.55/00/549 Research Article New Exact Solutions for the -Dimensional Broer-Kaup-Kupershmidt Equations Ming Song, Shaoyong Li, and Jun Cao Department of Mathematics, Faculty of Sciences, Yuxi Normal University, Yuxi 5300, China College of Mathematics and Information Sciences, Shaoguan University, Shaoguan 5005, China Correspondence should be addressed to Ming Song, songming 5@3.com Received August 00; Accepted November 00 Academic Editor: Narcisa C. Apreutesei Copyright q 00 Ming Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the -dimensional Broer-Kaup-Kupershmidt equations. Some explicit expressions of solutions for the equations are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain kink-shaped solutions, blowup solutions, periodic blow-up solutions, and solitary wave solutions. Some previous results are extended.. Introduction Consider the -dimensional Broer-Kaup-Kupershmidt BKK equations 0 u ty u xxy uu x y v xx 0, v t v xx uv x 0.. These equations have been widely applied in many branches of physics like plasma physics, fluid dynamics, nonlinear optics, and so forth. So a good understanding of more solutions of the -dimensional BKK equations. might be very helpful, especially for coastal and civil engineers to apply the non-linear water models in a harbor and coastal design. Recently, the -dimensional BKK equations have been studied by many authors. Yomba, used the modified extended Fan subequation method to obtain soliton-like solutions, triangular-like solutions, and single and combined nondegenerate Jacobi elliptic wave function-like solutions of.. Zhang and Xia 3 used the further improved extended Fan sub-equation method to obtain soliton-like solutions, triangular-like solutions, single and combined non-degenerate Jacobi elliptic wave function-like solutions, and Weierstrass elliptic

2 Abstract and Applied Analysis doubly-like periodic solutions of.. Abdou and Soliman 4 obtained some traveling wave solutions of. by using the modified extended tanh-function method. Song et al. 5 used the new extended Riccati equation rational expansion method to study multiple exact solutions of.. Zhang used the Exp-function method to seek generalized exact solutions with three arbitrary functions of.. El-Wakil and Abdou 7 obtained exact travelling wave solutions by using improved tanh-function method. Lu et al. 8 obtained some exact traveling wave solutions of. by using the first integral method. Davodi et al. 9 obtained some generalized solitary solutions of.. Bai and Zhao 0 used the Repeated General Algebraic Method to obtain exact solutions of.. In this paper, we employ the bifurcation method and qualitative theory of dynamical systems 9 to investigate the -dimensional BKK equations., and we obtain some explicit expressions of solutions for.. These solutions contain kink-shaped solutions, blow-up solutions, periodic blow-up solutions, and solitary wave solutions, most of which are new by comparing with the solutions of the references 0. The remainder of this paper is organized as follows. In Section, we present our main results. Section 3 gives the theoretical derivation for our main results. A short conclusion will be given in Section 4.. Main Results In this section, we state our main results. For ease of exposition, we have omitted the expressions of v with v n x, y, t φ n ξ /c, n,,...,0, in the entire process. Proposition.. For given constants c and g 0, which will be given later in 3.0, the dimensional BKK equations have the following exact solutions. () If g 0, we get two kink solutions: u ( x, y, t ) δc ( tanhc/ ( x y ct )) δ δ tanhc/ ( x y ct ), u ( x, y, t ) c ( tanhc/ ( x y ct )) η ( η ) tanhc/ ( x y ct ),. where δ and η are constants, two blow-up solutions ( ) c ( u 3± x, y, t ± coth c ) x y ct ( ),. and four periodic blow-up solutions ( ( ) c u 4± x, y, t ± csc c( x y ct ) ), ( ( ) c u 5± x, y, t ± sec c( x y ct ) )..3

3 Abstract and Applied Analysis 3 () If 0 <g<g 0, we get a solitary wave solution ( ( ) ) ( ) c α β cosh θ x y ct α ( β cosh θ ( x y ct )) u x, y, t α ( α β α cosh θ ( x y ct )),.4 and two blow-up solutions u 7± ( x, y, t ) c ( α ( β ) α cosh θ ( x y ct ) ± β 3/ cothθ/ ( x y ct )) α ( α β α cosh θ ( x y ct )),.5 where β α α and θ c β/α. (3) If g g 0, we get three blow-up solutions as follows: ( ) ( 3 ) 3 c ( x y ct ) ( 3 ) 3 c ( x y ct ) u 8 x, y, t ( 3 ( x y ct ) c ( x y ct ) ), ( ) ( 3 ) 3 c ( x y ct ) ( 3 ) 3 c ( x y ct ) u 9 x, y, t ( 3 ( x y ct ) c ( x y ct ) ), ( ) 9c 9 ( 3c 3 ) 3 c 3( x y ct ) u 0 x, y, t 8 c ( x y ct ).. 3. The Derivations of Main Results In this section, we will give the derivations for our main results. For given constant wave speed c, substituting u ξ, v φξ with ξ x y ct into the -dimensional BKK equations., it follows that c ( ) φ 0, cφ φ ( φ ) Integrating the first equation of 3. twice and letting integral constants be zero, we have φ ( c ). 3. Integrating the second equation of 3. once, we have cφ φ φ g, 3.3 where /g is integral constant.

4 4 Abstract and Applied Analysis Substituting 3. into 3.3, weget c 3 c 3 g. 3.4 Letting, we get the following planar system: d, dξ d dξ 3 3c c g. 3.5 Obviously, the above system 3.5 is a Hamiltonian system with Hamiltonian function H (, ) 4 c 3 c g. 3. Now, we consider the phase portraits of system 3.5.Set f 0 ( ) 3 3c c, f ( ) 3 3c c g. 3.7 f 0 has three fixed points 0,,, and their expressions are given as follows: 0 0, c, c. 3.8 It is easy to obtain the two extreme points of f 0 as follows: ± 3c ± 3c. 3.9 Let g 0 f0 ( ± ) c 3 3, 3.0 then it is easily seen that g 0 is the extreme values of f 0. Let i, 0 be one of the singular points of system 3.5. Then the characteristic values of the linearized system of system 3.5 at the singular points i, 0 are λ ± ± f ( ) i. 3.

5 Abstract and Applied Analysis 5 From the qualitative theory of dynamical systems, we therefore know that, i if f i > 0, i, 0 is a saddle point; ii if f i < 0, i, 0 is a center point; iii if f i 0, i, 0 is a degenerate saddle point. Therefore, we obtain the phase portraits of system 3.5 in Figure. Now, we will obtain the explicit expressions of solutions for the -dimensional BKK equations.. If g 0, we will consider two kinds of orbits. i First, we see that there are two heteroclinic orbits Γ and Γ connected at saddle points 0, 0 and, 0. In, -plane, the expressions of the heteroclinic orbits are given as ± ( c ). 3. Substituting 3. into d/dξ and integrating them along the heteroclinic orbits Γ and Γ, it follows that sc s ds ξ 0 ds, ss c ds 0 ξ ds, 3.3 where 0,c is constant and ± ds ss c ξ 0 ds. 3.4 From 3.3, we have ξ ξ δc tanhc/ξ δ δ tanhc/ξ, c tanhc/ξ η ( η ) tanhc/ξ, 3.5 where δ / c,η c/. From 3.4, we have ξ c ( ± coth c ξ ). 3. Noting that u ξ and ξ x y ct, we get two kink-shaped solutions u x, y, t, u x, y, t and two blow-up solutions u 3 ±x, y, t as.,and.. ii From the phase portrait, we note

6 Abstract and Applied Analysis Γ 3 Γ Γ Γ g< g 0 g g 0 g 0 <g<0 g 0 Γ Γ5 Γ 7 Γ <g<g 0 g g 0 Figure : The phase portraits of system 3.5. g>g 0 that there are two special orbits Γ 3 and Γ 4, which have the same Hamiltonian with that of the center point, 0. In, -plane, the expressions of these two orbits are given as ± ( ) ( 3 )( 4 ), 3.7 where 3 c, 4 c. 3.8 Substituting 3.7 into d/dξ and integrating them along the two orbits Γ 3 and Γ 4,it follows that ξ ± ( ) ( )( ) ds ds. 3.9 s s 3 s 4 0 From 3.9, we have ξ c ( ± ) csc cξ. 3.0 At the same time, we note that if u ξ is a solution of system 3.5, then u ξ γ is also a solution of system 3.5. Specially, when we take γ π/, we get other two solutions ξ c ( ± ) sec cξ. 3.

7 Abstract and Applied Analysis 7 Noting that u ξ and ξ x y ct, we get four periodic blow-up solutions u 4 ±x, y, t and u 5 ±x, y, t as.3. If 0 <g<g 0, we set the largest solution of f 0as 5 c/α <α<, then we can get another two solutions of fv 0 as follows: 7 c ( 3α ) α α, 4α c ( 3α ) α α. 4α 3. We see that there is a homoclinic orbit Γ 5, which passes the saddle point 5, 0. In, plane, the expressions of the homoclinic orbit are given as ± ( 5 ) ( 8 )( 9 ), 3.3 where 8 cα c α, α 9 cα c α. α 3.4 Substituting 3.3 into d/dξ and integrating them along the homoclinic orbit, it follows that ξ ± ( ) ( )( ) ds ds, 9 s 5 s 8 s 9 0 ξ ± ( ) ( )( ) ds ds. s 5 s 8 s From 3.5, we have ξ ( ) c α β cosh θξ α ( β cosh θξ ) α ( α β α cosh θξ ), ξ c( α ( β ) α cosh θξ ± β 3/ cothθ/ξ ) α ( α β α cosh θξ ), 3. where β α α and θ c β/α. Noting that u ξ and ξ x y ct, we get a solitary wave solution u x, y, t and two blow-up solutions u 7± x, y, t as.4 and.5.

8 8 Abstract and Applied Analysis 3 If g g 0, from the phase portrait, we see that there are two orbits Γ 7 and Γ 8, which have the same Hamiltonian with the degenerate saddle point, 0. In, -plane, the expressions of these two orbits are given as ± ( ) ( )( ), 3.7 where ( c ) 3c. 3.8 Substituting 3.7 into d/dξ and integrating them along these two orbits Γ 7 and Γ 8, it follows that ( s ) ( s )( s ) ds ξ 0 ds, ( s ) ( s )( s ) ds ξ 0 ds 3.9 From 3.9, we have ( 3 ) ( 3 cξ 3 ) 3 c ξ ( ), 3ξ cξ ( 3 ) ( 3 cξ 3 ) 3 c ξ ( ), 3ξ cξ 9c 9 ( 3c 3 ) 3 c 3 ξ. 8 c ξ 3.30 Noting that u ξ and ξ x y ct, we get three blow-up solutions u 8 x, y, t, u 9 x, y, t, and u 0 x, y, t as.. Thus, we obtain the results given in Proposition.. Remark 3.. One may find that we only consider the case when g 0inProposition.. In fact, we may get exactly the same solutions in the opposite case. 4. Conclusion In this paper, we have obtained many new solutions for the -dimensional BKK equations. by employing the bifurcation method and qualitative theory of dynamical systems. The explicit expressions of the solutions have been given in Proposition.. The method can be

9 Abstract and Applied Analysis 9 applied to many other nonlinear evolution equations, and we believe that many new results wait for further discovery by this method. Acknowledgment Research is supported by the Natural Science Foundation of Yunnan Province no. 00ZC54. References E. Yomba, The extended Fan s sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations, Physics Letters A, vol. 33, no., pp , 005. E. Yomba, The modified extended Fan sub-equation method and its application to the dimensional Broer-Kaup-Kupershmidt equation, Chaos, Solitons and Fractals, vol. 7, no., pp. 87 9, S. Zhang and T. Xia, Further improved extended Fan sub-equation method and new exact solutions of the -dimensional Broer-Kaup-Kupershmidt equations, Applied Mathematics and Computation, vol. 8, no., pp. 5 0, M. A. Abdou and A. A. Soliman, Modified extended tanh-function method and its application on nonlinear physical equations, Physics Letters A, vol. 353, no., pp , L.-N. Song, Q. Wang, Y. Zheng, and H.-Q. Zhang, A new extended Riccati equation rational expansion method and its application, Chaos, Solitons and Fractals, vol. 3, no. 3, pp , 007. S. Zhang, Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer-Kaup-Kupershmidt equations, Physics Letters A, vol. 37, no., pp , S. A. El-Wakil and M. A. Abdou, New exact travelling wave solutions of two nonlinear physical models, Nonlinear Analysis: Theory, Methods & Applications, vol. 8, no., pp , B. Lu, H. Zhang, and F. Xie, Travelling wave solutions of nonlinear partial equations by using the first integral method, Applied Mathematics and Computation, vol., no. 4, pp , A. G. Davodi, D. D. Ganji, A. G. Davodi, and A. Asgari, Finding general and explicit solutions dimensional Broer-Kaup-Kupershmidt system nonlinear equation by exp-function method, Applied Mathematics and Computation, vol. 7, no. 4, pp , C.-L. Bai and H. Zhao, A new General Algebraic Method and its applications to the dimensional Broer-Kaup-Kupershmidt equations, Applied Mathematics and Computation, vol. 7, no. 4, pp , 00. J. Li and Z. Liu, Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Applied Mathematical Modelling, vol. 5, no., pp. 4 5, 000. Z. Liu and T. Qian, Peakons and their bifurcation in a generalized Camassa-Holm equation, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol., no. 3, pp , Z. Liu and C. Yang, The application of bifurcation method to a higher-order KdV equation, Journal of Mathematical Analysis and Applications, vol. 75, no., pp., M. Tang, R. Wang, and Z. Jing, Solitary waves and their bifurcations of KdV like equation with higher order nonlinearity, Science in China A, vol. 45, no. 0, pp. 55 7, Z. Liu and B. Guo, Periodic blow-up solutions and their limit forms for the generalized Camassa- Holm equation, Progress in Natural Science, vol. 8, no. 3, pp. 59, 008. M. Song, C. Yang, and B. Zhang, Exact solitary wave solutions of the Kadomtsov-Petviashvili- Benjamin-Bona-Mahony equation, Applied Mathematics and Computation, vol. 7, no. 4, pp , M. Song and J. Cai, Solitary wave solutions and kink wave solutions for a generalized Zakharov- Kuznetsov equation, Applied Mathematics and Computation, vol. 7, no. 4, pp , Z. Wen, Z. Liu, and M. Song, New exact solutions for the classical Drinfel d-sokolov-wilson equation, Applied Mathematics and Computation, vol. 5, no., pp , M. Song and C. Yang, Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona- Mahony equation, Applied Mathematics and Computation, vol., no., pp , 00.