Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 984791, 7 pages http://dx.doi.org/10.1155/2014/984791
Research Article New Exact Explicit Nonlinear Wave Solutions for the Broer-Kaup Equation Zhenshu Wen School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China Correspondence should be addressed to Zhenshu Wen;
[email protected] Received 27 November 2013; Accepted 10 January 2014; Published 20 February 2014 Academic Editor: Yongkun Li Copyright Β© 2014 Zhenshu Wen. 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 study the nonlinear wave solutions for the Broer-Kaup equation. Many exact explicit expressions of the nonlinear wave solutions for the equation are obtained by exploiting the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which are new. Some previous results are extended.
1. Introduction
2. Bifurcation of Phase Portraits
In 1975, Broer [1] obtained a dispersive equation as follows:
In this section, we give the process of obtaining the bifurcation of phase portraits corresponding to (1). For given constant π, substituting π’(π₯, π‘) = π(π), π(π₯, π‘) = π(π) with π = π₯ β ππ‘ into (1), it follows
1 π’π‘ + ππ₯ + (π’2 )π₯ = 0, 2
(1)
ππ‘ + (π’π + π’ + π’π₯π₯ )π₯ = 0, which describes the evolution of horizontal velocity component π’(π₯, π‘) of water waves of height π(π₯, π‘) propagating in both directions in an infinite narrow channel of finite constant depth. Equation (1) plays an important role in nonlinear physics and gains considerate attention [2β4]. The traveling wave solutions for (1) have been studied by many works, such as [5β10]. In this paper, we employ the bifurcation method and qualitative theory of dynamical systems [11β21] to investigate the nonlinear wave solutions for (1), and we obtain many exact explicit expressions of nonlinear wave solutions for (1). These nonlinear wave solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which, to our knowledge, are newly obtained. The remainder of this paper is organized as follows. In Section 2, we show the bifurcation of phase portraits corresponding to (1). We state our main results and the theoretical derivation for the main results in Section 3. A short conclusion will be given in Section 4.
σΈ 1 βππσΈ + πσΈ + (π2 ) = 0, 2 σΈ
σΈ σΈ σΈ
(2)
βππ + (ππ + π + π ) = 0. Integrating (2) once leads to 1 βππ + π + π2 = π, 2
(3)
βππ + ππ + π + πσΈ σΈ = πΊ, where both π and πΊ are integral constants, respectively. From the first equation of system (3), we obtain 1 π = π + ππ β π2 . 2
(4)
Substituting (4) into the second equation of system (3) leads to 1 3π πσΈ σΈ = π3 β π2 + (π2 β π β 1) π + ππ + πΊ. 2 2
(5)
2
Journal of Applied Mathematics y
y
y
π
π
G < c β G0
π
y π7
Ξ3
y
Ξβ5
π6 π
π1
π3 π5
Ξ+4
βπ0
c β G0 < G < c
G = c β G0
Ξβ4
y + Ξ1
Ξβ2
π0
π
G=c y
Ξ+5
π8
Ξβ1
Ξ+2
π
π
π4 π2
c < G < G0 + c
G = G0 + c
G > G0 + c
Figure 1: The phase portraits of system (7).
By setting π = π + π, (5) becomes 1 1 πσΈ σΈ = π3 β (π2 + 2π + 2) π + πΊ β π. 2 2
Let 3
(6)
dπ = π¦, dπ (7)
1 1 1 π» (π, π¦) = π¦2 β π4 + (π2 + 2π + 2) π2 β (πΊ β π) π. 2 8 4 (8) Now, we study the bifurcation of phase portraits of system (7). Set
1 1 π (π) = π3 β (π2 + 2π + 2) π + πΊ β π. 2 2
(9)
Obviously, π0 (π) has three zero points, which can be expressed as π = 0, Β±π0 ,
(10)
where π0 = βπ2 + 2π + 2, when π2 + 2π + 2 > 0. Additionally, it is easy to obtain the two extreme points of π(π) as follows: 1 πΒ±β = Β±β (π2 + 2π + 2). 3
π Β± = Β±βπσΈ (ππ ).
(13)
From the qualitative theory of dynamical systems, we therefore know that
with first integral
1 1 π0 (π) = π3 β (π2 + 2π + 2) π, 2 2
(12)
which is the absolute value of extreme values of π0 (π). Let (ππ , 0) be one of the singular points of system (7). Then the characteristic values of the linearized system of system (7) at the singular point (ππ , 0) are
Letting π¦ = πσΈ , we obtain a planar system
dπ¦ 1 3 1 2 = π β (π + 2π + 2) π + πΊ β π, dπ 2 2
2 σ΅¨ σ΅¨ β (π + 2π + 2) , π0 = σ΅¨σ΅¨σ΅¨π0 (πΒ±β )σ΅¨σ΅¨σ΅¨ = 27
(11)
(i) if πσΈ (ππ ) > 0, then (ππ , 0) is a saddle point; (ii) if πσΈ (ππ ) < 0, then (ππ , 0) is a center point; (iii) if πσΈ (ππ ) = 0, then (ππ , 0) is a degenerate saddle point. Therefore, based on the above analysis, we obtain the bifurcation of phase portraits of system (7) in Figure 1.
3. Main Results and the Theoretic Derivations of the Main Results In this section, we state our results about solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions for the first component of system (7). To relate conveniently, we omit π = π + π and the expression of the second component of system (7), that is, π = π + ππ β (1/2)π2 , in the following theorem. Theorem 1. For given constants π and πΌ (1 < πΌ < 3), which will be given later, the Broer-Kaup equation (1) has the following exact explicit nonlinear wave solutions. (1) When πΊ = π, one obtains two kink-shaped solutions π1Β± (π₯, π‘) = Β±βπ2 + 2π + 2 tanh (
βπ2 + 2π + 2 2
(π₯ β ππ‘)) , (14)
Journal of Applied Mathematics
3
two singular solutions
+ β6 β 2πΌ sinh (β
π2Β± (π₯, π‘) = Β±βπ2 + 2π + 2 coth (
βπ2 + 2π + 2 2
π2 + 2π + 2 (3 β πΌ) πΌ 2
(π₯ β ππ‘) ) , Γ |π₯ β ππ‘| ) + 4 β 2πΌ)
(15) and four periodic singular solutions
π2 + 2π + 2 (3 β πΌ) Γ (2 cosh (β (π₯ β ππ‘)) πΌ 2
π3Β± (π₯, π‘) = Β±β2 (π2 + 2π + 2) Γ sec (β
π2 + 2π + 2 (π₯ β ππ‘)) , 2
+ β6 β 2πΌ sinh (β (16)
π4Β± (π₯, π‘) =
Β±β2 (π2
π2 + 2π + 2 (3 β πΌ) πΌ 2
+ 2π + 2)
β1
Γ |π₯ β ππ‘| ) β 2) ,
π2 + 2π + 2 Γ csc (β (π₯ β ππ‘)) . 2
(18) and two periodic singular solutions
(2) When π < πΊ < πΊ0 + π, one gets two solitary wave solutions π5Β± (π₯, π‘) = Β±β
π7Β± (π₯, π‘) = Β± (π2 (π6 β π7 ) sin (arcsin (
π2 + 2π + 2 πΌ
β
π2 + 2π + 2 (3 β πΌ) Γ (β2πΌ β 2 cosh (β (π₯ β ππ‘)) πΌ 2
2π2 β π6 β π7 ) π6 β π7
ββ (π2 β π6 ) (π2 β π7 ) 2
Γ |π₯ β ππ‘| ) (19) + 2π6 π7 β π2 π6 β π2 π7 )
+2πΌ β 4)
Γ ((π6 β π7 ) sin (arcsin (
π2 + 2π + 2 (3 β πΌ) Γ (β2πΌ β 2 cosh (β πΌ 2
β
β1
Γ (π₯ β ππ‘) ) + 2) ,
β1
where π2 , π6 , and π7 will be given in the proof of the theorem.
π8Β± (π₯, π‘) = Β±β
2
π + 2π + 2 πΌ
Γ (2 cosh (β
2
(3) When πΊ = πΊ0 + π, one obtains four singular solutions as follows:
π6Β± (π₯, π‘) = Β±β
ββ (π2 β π6 ) (π2 β π7 )
Γ |π₯ β ππ‘|) β 4π2 ) , (17)
two singular solutions
2π2 β π6 β π7 ) π6 β π7
π2 + 2π + 2 3 2
π2 + 2π + 2 (3 β πΌ) (π₯ β ππ‘)) πΌ 2
Γ
(1 + β(π2 + 2π + 2) /3|π₯ β ππ‘|) + 3 (1 +
β(π2
2
+ 2π + 2) /3|π₯ β ππ‘|) β 1
,
4
Journal of Applied Mathematics
Γ
At the same time, we note that if π = π(π) is a solution of system, then π = π(π+πΎ) is also a solution of system. Specially, we take πΎ = π/2; we obtain another two solutions
π2 + 2π + 2 3
π9Β± (π₯, π‘) = Β±β
(π2 + 2π + 2) (π₯ β ππ‘)2 + 9 (π2 + 2π + 2) (π₯ β ππ‘)2 β 3
.
π = Β±β2 (π2 + 2π + 2)csc (β (20)
Proof. (1) When πΊ = π, we consider the following two kinds of orbits. (i) First, we see that there are two heteroclinic orbits Ξ1Β± connected as two saddle points (π0 , 0) and (βπ0 , 0) from Figure 1. In (π, π¦)-plane, from (8), the expressions of the heteroclinic orbits are given as 1 π¦ = Β± (π02 β π2 ) . 2
π
π(π) = 0 to be π1 = β(π2 + 2π + 2)/πΌ (1 < πΌ < 3), and then we can obtain another two solutions of π(π) = 0 as follows: 1 π2 + 2π + 2 β π2 = β ( 4πΌ β 3 β 1) , 2 πΌ
(21)
0
β«
dπ 1σ΅¨ σ΅¨ = σ΅¨σ΅¨πσ΅¨σ΅¨ , (π02 β π 2 ) 2 σ΅¨ σ΅¨
+β
π
(22)
dπ 1σ΅¨ σ΅¨ = σ΅¨σ΅¨σ΅¨πσ΅¨σ΅¨σ΅¨ . 2 2 (π β π0 ) 2
(27)
Noting that π = π(π) and π = π₯ β ππ‘, we get four periodic singular solutions π3Β± (π₯, π‘) and π4Β± (π₯, π‘) as (16). (2) When π < πΊ < πΊ0 + π, we set the largest solution of
(28)
2
1 π + 2π + 2 β π3 = β β ( 4πΌ β 3 + 1) . 2 πΌ
Substituting (21) into the first equation of system (7) and integrating along the heterclinic orbits, it follows that β«
π2 + 2π + 2 π) . 2
(i) First, we see that there is a homoclinic orbit Ξ3 , which passes the saddle point (π1 , 0). In (π, π¦)-plane, from (8), the expressions of the homoclinic orbit are given as 1 2 π¦ = Β± β(π1 β π) (π β π4 ) (π β π5 ), 2
From (22), we have
(29)
where π = Β±βπ2 + 2π + 2 tanh (
βπ2 + 2π + 2 2
π) ,
π4 = β (23)
π = Β±βπ2 + 2π + 2 coth (
βπ2 + 2π + 2 2
π5 = ββ
π) .
Noting that π = π(π) and π = π₯ β ππ‘, we get two kinkshaped solutions π1Β± (π₯, π‘) and two singular solutions π2Β± (π₯, π‘) as (14) and (15). (ii) Second, from the phase portrait in Figure 1, we note that there are two special orbits Ξ2Β± , which have the same Hamiltonian as that of the center point (0, 0). In (π, π¦)-plane, from (8), the expressions of these two orbits are given as 1 π¦ = Β± πβπ2 β 2π02 . 2
+β
π
dπ π βπ 2 β
2π02
=
1 σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨πσ΅¨ . 2σ΅¨ σ΅¨
π + 2π + 2 (β2πΌ β 2 + 1) . πΌ
π
dπ
π4
(π1 β π ) β(π β π4 ) (π β π5 )
β«
+β
β«
(24)
=
1 σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨πσ΅¨ , 2σ΅¨ σ΅¨
1σ΅¨ σ΅¨ = σ΅¨σ΅¨σ΅¨πσ΅¨σ΅¨σ΅¨ . (π β π1 ) β(π β π4 ) (π β π5 ) 2 dπ
From (31), we have π = Β±β
π2 + 2π + 2 πΌ
(25) Γ (β2πΌ β 2 cosh (β
From (25), we have π2 + 2π + 2 π = Β±β2 (π2 + 2π + 2)sec (β π) . 2
(26)
(30)
2
Substituting (29) into the first equation of system (7) and integrating along the homoclinic orbit, it follows that
π
Substituting (24) into the first equation of system (7) and integrating along the two orbits Ξ2Β± , it follows that β«
π2 + 2π + 2 (β2πΌ β 2 β 1) , πΌ
+ 2πΌ β 4)
π2 + 2π + 2 (3 β πΌ) π) πΌ 2
(31)
Journal of Applied Mathematics
5 30
β1
π2 + 2π + 2 (3 β πΌ) π) + 2) , Γ (β2πΌ β 2 cosh (β πΌ 2 π = Β±β
π2 + 2π + 2 πΌ
Γ (2 cosh (β
20 10 β15
β10
β5
5
π2 + 2π + 2 (3 β πΌ) π) πΌ 2
β30
Figure 2: The graphic of π8+ (π) when π = 2, π = 1, and πΌ = 2.
From (35), we have π = Β± (π2 (π6 β π7 ) sin (arcsin (
+ 4 β 2πΌ) π2 + 2π + 2 (3 β πΌ) Γ (2 cosh (β π) πΌ 2 + β6 β 2πΌ sinh (β
β
π2 + 2π + 2 (3 β πΌ) σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨πσ΅¨) πΌ 2 σ΅¨ σ΅¨
(32) Noting that π = π(π) and π = π₯ β ππ‘, we get two solitary solutions π5Β± (π₯, π‘) and two singular solutions π6Β± (π₯, π‘) as (17) and (18). (ii) Second, from the phase portrait in Figure 1, we note that there are another two special orbits Ξ4Β± , which have the same Hamiltonian as that of the center point (π2 , 0). In (π, π¦)plane, from (8), the expressions of these two orbits are given as (33)
where π6 = βπ2 + β2 (π2 + 2π + 2) β 2π22 ,
(34)
π7 = βπ2 β β2 (π2 + 2π + 2) β 2π22 . Substituting (33) into the first equation of system (7) and integrating along these two special orbits Ξ4Β± , it follows that
π
ββ (π2 β π6 ) (π2 β π7 )
Γ ((π6 β π7 ) sin ( arcsin (
β 2) .
1 2 π¦ = Β± β(π β π2 ) (π β π6 ) (π β π7 ), 2
dπ (π β π2 ) β(π β π6 ) (π β π7 )
=
1 σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨πσ΅¨ . 2σ΅¨ σ΅¨
2π2 β π6 β π7 ) π6 β π7
2
σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨πσ΅¨σ΅¨)
+ 2π6 π7 β π2 π6 β π2 π7 )
β1
+β
15
β20
π2 + 2π + 2 (3 β πΌ) σ΅¨σ΅¨ σ΅¨σ΅¨ + β6 β 2πΌ sinh (β σ΅¨πσ΅¨) πΌ 2 σ΅¨ σ΅¨
β«
10
β10
(35)
β
2π2 β π6 β π7 ) π6 β π7
ββ (π2 β π6 ) (π2 β π7 ) 2
σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨πσ΅¨σ΅¨)
β1
β 4π2 ) . (36) Noting that π = π(π) and π = π₯ β ππ‘, we get two periodic singular solutions π7Β± (π₯, π‘) as (19). To illustrate, we give the graphic of π8+ (π) in Figure 2 by taking π = 2, π = 1, and πΌ = 2. (3) When πΊ = πΊ0 + π, from the phase portrait in Figure 1, we note that there are two orbits Ξ5Β± , which have the same Hamiltonian as the degenerate saddle point (π8 , 0). In (π, π¦)plane, from (8), the expressions of these two orbits are given as 1 3 (37) π¦ = Β± β(π β π8 ) (π + 3π8 ), 2 where 1 (38) π8 = β (π2 + 2π + 2). 3 Substituting (37) into the first equation of system (7) and integrating along these two orbits Ξ5Β± , it follows that β«
+β
π
dπ (π β π8 ) β(π β π8 ) (π + 3π8 )
=
1 σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨πσ΅¨ , 2σ΅¨ σ΅¨
6
Journal of Applied Mathematics β«
β3π8
dπ (π8 β π ) β(π8 β π ) (β3π8 β π )
π
=
π = π + π,
1 σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨πσ΅¨ . 2σ΅¨ σ΅¨ (39)
From (3), we have 2 β(π2 + 2π + 2) /3 σ΅¨σ΅¨σ΅¨σ΅¨πσ΅¨σ΅¨σ΅¨σ΅¨) + 3 (1 + π + 2π + 2 π = Β±β , 3 σ΅¨σ΅¨ σ΅¨σ΅¨ 2 2 β (1 + (π + 2π + 2) /3 σ΅¨σ΅¨πσ΅¨σ΅¨) β 1
π = Β±β
(41) (40)
π + 2π + 2 (π + 2π + 2) (π₯ β ππ‘) + 9 . 3 (π2 + 2π + 2) (π₯ β ππ‘)2 β 3
Noting that π = π(π) and π = π₯ β ππ‘, we get four singular solutions π8Β± (π₯, π‘) and π9Β± (π₯, π‘) as (20). Remark 2. One may find that we only consider some special orbits in Figure 1 when πΊ β₯ π. In fact, we may obtain exactly the same results when πΊ < π. Remark 3. We employ the software Mathematica to check the correctness of the above nonlinear wave solutions. To illustrate, we show the commands of verifying π8+ (π₯, π‘), 1 π2 + 2π + 2 β π2 = β ( 4πΌ β 3 β 1) , 2 πΌ
2
π7 = βπ2 β β2 (π2 + 2π + 2) β 2(π2 ) ,
[1] L. J. F. Broer, βApproximate equations for long water waves,β Applied Scientific Research, vol. 31, no. 5, pp. 377β395, 1975.
] (π₯ β ππ‘)] ]
+ 2π6 π7 β π2 π6 β π2 π7 )
β 4π2 ) ,
[2] D. J. Kaup, βA higher-order water-wave equation and the method for solving it,β Progress of Theoretical Physics, vol. 54, no. 2, pp. 396β408, 1975. [3] B. A. Kupershmidt, βMathematics of dispersive water waves,β Communications in Mathematical Physics, vol. 99, no. 1, pp. 51β 73, 1985. [4] M. Ablowitz and H. Segur, βSolitons, nonlinear evolution equations and inverse scattering,β Journal of Fluid Mechanics, vol. 244, pp. 721β725, 1992. [5] M. Wang, J. Zhang, and X. Li, βApplication of the πΊσΈ /πΊexpansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations,β Applied Mathematics and Computation, vol. 206, no. 1, pp. 321β326, 2008.
2π β π6 β π7 [ Γ ((π6 β π7 ) Sin [ArcSin [ 2 ] π6 β π7 [
β1
The author declares that there is no conflict of interests regarding the publication of this paper.
References
2π β π6 β π7 [ π = (π2 (π6 β π7 ) Sin [ArcSin [ 2 ] π6 β π7 [
2
Conflict of Interests
This research was supported by Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (no. ZQN-PY119) and the Foundation of Huaqiao University (no. 12BS223).
2
ββ (π2 β π6 ) (π2 β π7 )
In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we study the nonlinear wave solutions for the Broer-Kaup equation (1) and obtain exact explicit expressions of the various kinds of nonlinear wave solutions, which include solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions. To the best of our knowledge, most of the nonlinear wave solutions are newly obtained.
Acknowledgments
π6 = βπ2 + β2 (π2 + 2π + 2) β 2(π2 ) ,
β
4. Conclusions
2
2
β
1 Simplify [ππ‘ π + ππ₯ π + ππ₯ (π2 )] , 2 Simplify [ππ‘ π + ππ₯ (ππ + π + ππ₯,π₯ π)] .
2
2
1 π = π + ππ β π2 , 2
ββ (π2 β π6 ) (π2 β π7 ) 2
] (π₯ β ππ‘)] ]
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Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014