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