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

and Applied Analysis 3 (2) If 0 < g < g0, we get a solitary wave solution u6 ( x, y, t ) c (√ 2 α − 1 β cosh θ ( x y − ct )) α − 1 ( 1 β − cosh 2θ ( x y − ct )) α ( −1 α β − α − 1 cosh 2θ ( x y − ct )) , 2.4 and two blow-up solutions u7± ( x, y, t ) c ( α ( 2 β ) − 2 − 2 α − 1 cosh θ ( x y − ct ) ± β 3/2 coth θ/2 ( x y − ct )) 2α ( −1 α β − α − 1 cosh θ ( x y − ct )) , 2.5 where β 6 − 6α α2 and θ c √ β/α. (3) If g g0, we get three blow-up solutions as follows: u8 ( x, y, t ) −12 √ 3 ( 6 6 √ 3 ) c ( x y − ct ) − ( 3 √ 3 ) c2 ( x y − ct )2 6 ( 2 √ 3 ( x y − ct ) − c ( x y − ct )2) , u9 ( x, y, t ) 12 √ 3 ( 6 6 √ 3 ) c ( x y − ct ) ( 3 √ 3 ) c2 ( x y − ct )2 6 ( 2 √ 3 ( x y − ct ) c ( x y − ct )2) , u10 ( x, y, t ) −9c 9 √ 3c ( 3 √ 3 ) c3 ( x y − ct )2 −18 6c2 ( x y − ct )2 . 2.6 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 φ ξ , v φ ξ with ξ x y − ct into the 2 1 -dimensional BKK equations 1.1 , it follows that ⎧ ⎨ ⎩ −cφ′′ − φ′′′ 2 ( φφ′ )′ 2φ′′ 0, −cφ′ φ′′ 2 ( φφ )′ 0. 3.1 Integrating the first equation of 3.1 twice and letting integral constants be zero, we have φ 1 2 ( cφ φ′ − φ2 ) . 3.2 Integrating the second equation of 3.1 once, we have −cφ φ′ 2φφ 1 2 g, 3.3 where 1/2 g is integral constant. 4 Abstract and Applied Analysis Substituting 3.2 into 3.3 , we get 1 2 φ′′ − 1 2 c2φ 3 2 cφ2 − φ3 1 2 g. 3.4 Letting ψ φ′, we get the following planar system: dφ dξ ψ, dψ dξ 2φ3 − 3cφ2 c2φ g. 3.5 Obviously, the above system 3.5 is a Hamiltonian system with Hamiltonian function H ( φ, ψ ) 1 2 ψ2 − 1 2 φ4 cφ3 − 1 2 c2φ2 − gφ. 3.6 Now, we consider the phase portraits of system 3.5 . Set f0 ( φ ) 2φ3 − 3cφ2 c2φ, f ( φ ) 2φ3 − 3cφ2 c2φ g. 3.7 f0 φ has three fixed points φ0, φ1, φ2, and their expressions are given as follows: φ0 0, φ1 c 2 , φ2 c. 3.8 It is easy to obtain the two extreme points of f0 φ as follows: φ± 3c ± √ 3c 6 . 3.9 Let g0 ∣f0 ( φ± )∣∣ c3 6 √ 3 , 3.10 then it is easily seen that g0 is the extreme values of f0 φ . Let φi, 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 φi, 0 are λ± ± √ f ′ ( φi ) . 3.11 Abstract and Applied Analysis 5 From the qualitative theory of dynamical systems, we therefore know that, i if f ′ φi > 0, φi, 0 is a saddle point; ii if f ′ φi < 0, φi, 0 is a center point; iii if f ′ φi 0, φi, 0 is a degenerate saddle point. Therefore, we obtain the phase portraits of system 3.5 in Figure 1. Now, we will obtain the explicit expressions of solutions for the 2 1 -dimensional BKK equations 1.1 . 1 If g 0, we will consider two kinds of orbits. i First, we see that there are two heteroclinic orbits Γ1 and Γ2 connected at saddle points φ0, 0 and φ2, 0 . In φ, ψ -plane, the expressions of the heteroclinic orbits are given as ψ ±φ ( φ − c ) . 3.12and Applied Analysis 5 From the qualitative theory of dynamical systems, we therefore know that, i if f ′ φi > 0, φi, 0 is a saddle point; ii if f ′ φi < 0, φi, 0 is a center point; iii if f ′ φi 0, φi, 0 is a degenerate saddle point. Therefore, we obtain the phase portraits of system 3.5 in Figure 1. Now, we will obtain the explicit expressions of solutions for the 2 1 -dimensional BKK equations 1.1 . 1 If g 0, we will consider two kinds of orbits. i First, we see that there are two heteroclinic orbits Γ1 and Γ2 connected at saddle points φ0, 0 and φ2, 0 . In φ, ψ -plane, the expressions of the heteroclinic orbits are given as ψ ±φ ( φ − c ) . 3.12 Substituting 3.12 into dφ/dξ ψ and integrating them along the heteroclinic orbits Γ1 and Γ2, it follows that ∫φ φ∗ 1 s c − s ds ∫ ξ 0 ds, ∫φ∗ φ 1 s s − c ds ∫0 ξ ds, 3.13 where φ∗ ∈ 0, c is constant and


Introduction
Consider the 2 1 -dimensional Broer-Kaup-Kupershmidt BKK equations 1-10 u ty − u xxy 2 uu x y 2v xx 0, v t v xx 2 uv x 0. 1.1 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 2 1 -dimensional BKK equations 1.1 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 2 1 -dimensional BKK equations have been studied by many authors.Yomba 1, 2 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 1.1 .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 doubly-like periodic solutions of 1.1 .Abdou and Soliman 4 obtained some traveling wave solutions of 1.1 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 1.1 .Zhang 6 used the Exp-function method to seek generalized exact solutions with three arbitrary functions of 1.1 .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 1.1 by using the first integral method.Davodi et al. 9 obtained some generalized solitary solutions of 1.1 .Bai and Zhao 10 used the Repeated General Algebraic Method to obtain exact solutions of 1.1 .
In this paper, we employ the bifurcation method and qualitative theory of dynamical systems 11-19 to investigate the 2 1 -dimensional BKK equations 1.1 , and we obtain some explicit expressions of solutions for 1.1 .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 1-10 .
The remainder of this paper is organized as follows.In Section 2, 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 1/2 cϕ ϕ − ϕ 2 , n 1, 2, . . ., 10, in the entire process.
Proposition 2.1.For given constants c and g 0 , which will be given later in 3.10 , the 2 1dimensional BKK equations have the following exact solutions.
(1) If g 0, we get two kink solutions: where δ and η are constants, two blow-up solutions and four periodic blow-up solutions
(3) If g g 0 , we get three blow-up solutions as follows: 2.6

The Derivations of Main Results
In this section, we will give the derivations for our main results.
For given constant wave speed c,

3.1
Integrating the first equation of 3.1 twice and letting integral constants be zero, we have Integrating the second equation of 3.1 once, we have where 1/2 g is integral constant.

3.5
Obviously, the above system 3.5 is a Hamiltonian system with Hamiltonian function Now, we consider the phase portraits of system 3.5 .Set

3.7
f 0 ϕ has three fixed points ϕ 0 , ϕ 1 , ϕ 2 , and their expressions are given as follows: It is easy to obtain the two extreme points of f 0 ϕ as follows: 3.9 Let then it is easily seen that g 0 is the extreme values of f 0 ϕ .Let ϕ 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 ϕ i , 0 are λ ± ± f ϕ i .

3.11
From the qualitative theory of dynamical systems, we therefore know that, i if f ϕ i > 0, ϕ i , 0 is a saddle point; ii if f ϕ i < 0, ϕ i , 0 is a center point; iii if f ϕ i 0, ϕ i , 0 is a degenerate saddle point.
Therefore, we obtain the phase portraits of system 3.5 in Figure 1.Now, we will obtain the explicit expressions of solutions for the 2 1 -dimensional BKK equations 1.1 .
1 If g 0, we will consider two kinds of orbits.i First, we see that there are two heteroclinic orbits Γ 1 and Γ 2 connected at saddle points ϕ 0 , 0 and ϕ 2 , 0 .In ϕ, ψ -plane, the expressions of the heteroclinic orbits are given as 3.12 Substituting 3.12 into dϕ/dξ ψ and integrating them along the heteroclinic orbits Γ 1 and Γ 2 , it follows that

3.16
Noting that u ϕ ξ and ξ x y − ct, we get two kink-shaped solutions u 1 x, y, t , u 2 x, y, t and two blow-up solutions u 3 ± x, y, t as 2.1 , and 2.2 .ii From the phase portrait, we note 6 Abstract and Applied Analysis that there are two special orbits Γ 3 and Γ 4 , which have the same Hamiltonian with that of the center point ϕ 1 , 0 .In ϕ, ψ -plane, the expressions of these two orbits are given as where c.

3.18
Substituting 3.17 into dϕ/dξ ψ and integrating them along the two orbits Γ 3 and Γ 4 , it follows that From 3.19 , we have

3.20
At the same time, we note that if u ϕ ξ is a solution of system 3.5 , then u ϕ ξ γ is also a solution of system 3.5 .Specially, when we take γ π/2, we get other two solutions

3.21
Abstract and Applied Analysis 7 Noting that u ϕ 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 2.3 .
2 If 0 < g < g 0 , we set the largest solution of f ϕ 0 as ϕ 5 c/α 1 < α < 2 , then we can get another two solutions of f v 0 as follows:

3.22
We see that there is a homoclinic orbit Γ 5 , which passes the saddle point ϕ 5 , 0 .In ϕ, ψplane, the expressions of the homoclinic orbit are given as where

3.25
From 3.25 , we have

3.26
where β 6 − 6α α 2 and θ c β /α.Noting that u ϕ ξ and ξ x y − ct, we get a solitary wave solution u 6 x, y, t and two blow-up solutions u 7± x, y, t as 2.4 and 2.5 .
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 ϕ * , 0 .In ϕ, ψ -plane, the expressions of these two orbits are given as

3.30
Noting that u ϕ ξ and ξ x y − ct, we get three blow-up solutions u 8 x, y, t , u 9 x, y, t , and u 10 x, y, t as 2.6 .Thus, we obtain the results given in Proposition 2.1.
Remark 3.1.One may find that we only consider the case when g ≥ 0 in Proposition 2.1.In fact, we may get exactly the same solutions in the opposite case.

Conclusion
In this paper, we have obtained many new solutions for the 2 1 -dimensional BKK equations 1.1 by employing the bifurcation method and qualitative theory of dynamical systems.The explicit expressions of the solutions have been given in Proposition 2.1.The method can be applied to many other nonlinear evolution equations, and we believe that many new results wait for further discovery by this method.

Figure 1 :
Figure 1:The phase portraits of system 3.5 .