# Periodic Wave Solutions and Their Limits for the Generalized KP-BBM Equation

Authors: Ming Song; Zhengrong Liu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP-BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow-up wave solutions, and solitary wave solutions.


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## Body

## 1. Introduction

The Benjamin-Bona-Mahony (BBM) equation [1],
(1.1)ut+ux+uux-uxxt=0,
has been proposed as a model for propagation of long waves where nonlinear dispersion is incorporated.The Kadomtsov-Petviashvili (KP) equation [2] is given by
(1.2)ut+a(ux+uxxx)x+uyy=0,
which is a weekly two-dimensional generalization of the KdV equation in the sense that it accounts for slowly varying transverse perturbations of unidirectional KdV solitons moving along the x-direction.Wazwaz [3] presented the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation
(1.3)(ut+ux+a(u2)x-buxxt)x+ruyy=0,
and the generalized KP-BBM equation
(1.4)(ut+ux+a(u3)x-buxxt)x+ruyy=0.Wazwaz [3, 4] obtained some solitons solution and periodic solutions of (1.3) by using the sine-cosine method and the extended tanh method. Abdou [5] used the extended mapping method with symbolic computation to obtain some periodic solutions of (1.3), solitary wave solution, and triangular wave solution. Song et al. [6] obtained exact solitary wave solutions of (1.3) by using bifurcation method of dynamical systems.The aim of this paper is to study the traveling wave solutions and their phase portraits for (1.4) by using the bifurcation method and qualitative theory of dynamical systems [6–15]. Through some special phase orbits, we obtain a number of smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded solutions, blow-up wave solutions, and solitary wave solutions.The remainder of this paper is organized as follows. In Section2, by using the bifurcation theory of planar dynamical systems, two phase portraits for the corresponding traveling wave system of (1.4) are given under different parameter conditions. In Section 3, we present our main results and their proofs. A short conclusion will be given in Section 4.


## 2. Phase Portraits

To derive our results, we give some preliminaries in this section. For given positive constant wave speedc, substituting u=φ(ξ) with ξ=x+y-ct into the generalized KP-BBM equation (1.4), it follows that
(2.1)(-cφ′+φ′-a(φ3)′+bcφ′′′)′+rφ′′=0.Integrating (2.1) twice and letting the first integral constant be zero, we have
(2.2)(r-c+1)φ-aφ3+bcφ′′=g1,
where g1 is the second constant of integration.Lettingϕ=φ′, we get the following planar system:
(2.3)dφdξ=ϕ,dϕdξ=αφ3-βφ+g,
where α=a/bc, β=(r+1-c)/bc and g=g1/bc.Obviously, the system (2.3) is a Hamiltonian system with Hamiltonian function
(2.4)H(φ,ϕ)=ϕ2-α2φ4+βφ2-2gφ=h,
where h is Hamiltonian.Now we consider the phase portraits of system (2.3). Set
(2.5)f0(φ)=αφ3-βφ,f(φ)=αφ3-βφ+g.Obviously,f0(φ) has three zero points, φ-, φ0 and φ+, which are given as follows:
(2.6)φ-=-βα,φ0=0,φ+=βα.It is easy to obtain two extreme points off0(φ) as follows:
(2.7)φ-*=-β3α,φ+*=β3α.Letting(2.8)g0=|f0(φ-*)|=|f0(φ+*)|=2|β|3β3α,
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 (2.3), then the characteristic values of the linearized system of system (2.3) at the singular points (φi,0) are
(2.9)λ±=±f′(φi).From the qualitative theory of dynamical systems, we therefore know that(i)
iff′(φi)>0, (φi,0) is a saddle point.(ii)
iff′(φi)<0, (φi,0) is a center point.(iii)
iff′(φi)=0, (φi,0) is a degenerate saddle point.Therefore, we obtain the phase portraits of system (2.3) in Figures 1 and 2.Figure 1
The phase portraits of system (2.3) when α>0, β>0.Figure 2
The phase portraits of system (2.3) when α<0, β<0.


## 3. Main Results and Their Proofs

In this section, we state our main results.Proposition 3.1.
For given positive constantsc and g0, (1.4) has the following periodic wave solutions when α>0 and β>0.
(1) Ifg=0, we get eight periodic blow-up wave solutions
(3.1)u1±(x,y,t)=±2βαcscβξ,(3.2)u2±(x,y,t)=±2βαsecβξ,(3.3)u3±(x,y,t)=±φ6sn(φ6α/2ξ,φ5/φ6),(3.4)u4±(x,y,t)=±φ62-φ52(sn(φ6α/2ξ,φ5/φ6))21-(sn(φ6α/2ξ,φ5/φ6))2,
two periodic wave solutions
(3.5)u5±(x,y,t)=±φ5sn(φ6α2ξ,φ5φ6),
two kink wave solutions
(3.6)u6±(x,y,t)=±βαtanhβ2ξ,
and two unbounded wave solutions
(3.7)u7±(x,y,t)=±βαcoth⁡β2ξ.
(2) If0<g<g0, we get four periodic blow-up wave solutions
(3.8)u8±(x,y,t)=φ7*-2γ1δ1±η1cos⁡αγ1/2ξ,(3.9)u9(x,y,t)=φ10(φ11-φ13)+φ11(φ13-φ10)(sn(ω1ξ,k1))2φ11-φ13+(φ13-φ10)(sn(ω1ξ,k1))2,u10(x,y,t)=φ13(φ12-φ10)-φ12(φ13-φ10)(sn(ω1ξ,k1))2φ12-φ10-(φ13-φ10)(sn(ω1ξ,k1))2,
where
(3.10)φ7*=12(-φ7+4βα-3φ72),γ1=-4β+3φ7(αφ7+4αβ-3α2φ72)α,δ1=-2φ7+24αβ-3α2φ72α,η1=4φ7(αφ7+4αβ-3α2φ72)α,ω1=α(φ13-φ11)(φ12-φ10)22,k1=(φ12-φ11)(φ13-φ10)(φ13-φ11)(φ12-φ10),
a periodic wave solution
(3.11)u11(x,y,t)=φ11(φ10-φ12)+φ10(φ12-φ11)(sn(ω1ξ,k1))2φ10-φ12+(φ12-φ11)(sn(ω1ξ,k1))2,
a blow-up wave solution
(3.12)u12(x,y,t)=φ7+2(β-3αφ72)2αφ7-2α(β-αφ72)cosh⁡3αφ72-βξ,
a solitary wave solution
(3.13)u13(x,y,t)=φ7+2(β-3αφ72)2αφ7+2α(β-αφ72)cosh⁡3αφ72-βξ,
and an unbounded wave solution
(3.14)u14(x,y,t)=φ7-2(β-3αφ72)csch3αφ72-βξ-2αβ+6α2φ72+2αφ7tanh(3αφ72-βξ/2).
(3) Ifg=g0, we get three blow-up wave solutions
(3.15)u15(x,y,t)=182-β3βξ3α(18ξ-3βξ3),u16(x,y,t)=182+β3βξ3α(-18ξ+3βξ3),u17(x,y,t)=3β(9+2βξ2)α(-9+6βξ2),
and a periodic blow-up wave solution
(3.16)u18(x,y,t)=-A1φ17+B1φ18+(A1φ17+B1φ18)cn(αA1B1/2ξ,k2)-A1+B1+(A1+B1)cn(αA1B1/2ξ,k2),
where
(3.17)A1=(φ18-c1+c¯12)2-(c1-c¯1)24,B1=(φ17-c1+c¯12)2-(c1-c¯1)24,k2=(A1+B1)2-(φ18-φ17)24A1B1,c1 and c¯1 are conjugate complex numbers.Proof.
(1) Ifg=0, we will consider three kinds of orbits.
(i) From the phase portrait, we note that there are two special orbitsΓ1 and Γ2, which have the same Hamiltonian with that of the center point (0,0). In (φ,ϕ) plane, the expressions of these two orbits are given as
(3.18)ϕ=±α2φ2(φ-φ1)(φ-φ2),
where φ1=-2β/α and φ2=2β/α.
Substituting (3.18) into dφ/dξ=ϕ and integrating them along the two orbits Γ1 and Γ2, it follows that
(3.19)±∫φ∞‍1s(s-φ1)(s-φ2)ds=α2∫0ξ‍ds,±∫φ2φ‍1s(s-φ1)(s-φ2)ds=α2∫0ξ‍ds.
From (3.19) and noting that u=φ(ξ) and ξ=x+y-ct, we get four periodic blow-up solutions u1±(x,y,t) and u2±(x,y,t) as (3.1) and (3.2).
(ii) From the phase portrait, we note that there are three special orbitsΓ3, Γ4, and Γ5 passing the points (φ3,0),  (φ4,0),  (φ5,0), and (φ6,0). In (φ,ϕ) plane, the expressions of the orbit are given as
(3.20)ϕ=±α2(φ-φ3)(φ-φ4)(φ-φ5)(φ-φ6),
where φ3=-φ6,φ4=-2β/α-φ62,φ5=2β/α-φ62  and  β/α<φ6<2β/α.
Substituting (3.20) into dφ/dξ=ϕ and integrating them along Γ3, Γ4, and Γ5, we have
(3.21)±∫φ∞‍1(s-φ3)(s-φ4)(s-φ5)(s-φ6)ds=α2∫0ξ‍ds,±∫φ6φ‍1(s-φ3)(s-φ4)(s-φ5)(s-φ6)ds=α2∫0ξ‍ds,±∫0φ‍1(s-φ3)(s-φ4)(s-φ5)(s-φ6)ds=α2∫0ξ‍ds.
From (3.21) and noting that u=φ(ξ) and ξ=x+y-ct, we get four periodic blow-up wave solutions u3±(x,y,t), u4±(x,y,t) as (3.3), (3.4) and two periodic solutions u5±(x,y,t) as (3.5).
(iii) From the phase portrait, we see that there are two heteroclinic orbitsΓ6 and Γ7 connected at saddle points (φ-,0) and (φ+,0). In (φ,ϕ) plane, the expressions of the heteroclinic orbits are given as
(3.22)ϕ=±α2(φ-φ-)2(φ-φ+)2.
Substituting (3.22) into dφ/dξ=ϕ and integrating them along the heteroclinic orbits Γ6 and Γ7, it follows that
(3.23)±∫0φ‍1(s-φ-)(φ+-s)ds=α2∫0ξ‍ds,±∫φ∞‍1(s-φ-)(φ+-s)ds=α2∫0ξ‍ds.
From (3.23) and noting that u=φ(ξ) and ξ=x+y-ct, we get two kink wave solutions u6±(x,y,t) as (3.6) and two unbounded solutions u7±(x,y,t) as (3.7).
(2) If0<g<g0, we set the largest solution of f(φ)=0 be φ7(β/3α<φ7<β/α), then we can get another two solutions of f(φ)=0 as follows:
(3.24)φ7*=12(-φ7+4βα-3φ72),φ7⋆=12(-φ7-4βα-3φ72).
(i) From the phase portrait, we see that there are two special orbitsΓ8 and Γ9, which have the same Hamiltonian with that of the center point (φ7*,0). In (φ,ϕ)-plane, the expressions of the orbits are given as
(3.25)ϕ=±α2(φ-φ7*)2(φ-φ8)(φ-φ9),
where
(3.26)φ8=12(φ7-4βα-3φ72-2φ72+φ74βα-3φ72),φ9=12(φ7-4βα-3φ72+2φ72+φ74βα-3φ72).
Substituting (3.25) into dφ/dξ=ϕ and integrating them along the orbits, it follows that
(3.27)±∫φ8φ‍1(s-φ7*)2(s-φ8)(s-φ9)ds=α2∫0ξ‍ds,±∫φ9φ‍1(s-φ7*)2(s-φ8)(s-φ9)ds=α2∫0ξ‍ds.
From (3.27) and noting that u=φ(ξ) and ξ=x+y-ct, we get two periodic blow-up wave solutions u8±(x,y,t) as (3.8).
(ii) From the phase portrait, we note that there are three special orbitsΓ10, Γ11, and Γ12 passing the points (φ10,0),  (φ11,0), (φ12,0), and (φ13,0). In (φ,ϕ) plane, the expressions of the orbit are given as
(3.28)ϕ=±α2(φ-φ10)(φ-φ11)(φ-φ12)(φ-φ13),
where φ8<φ10<φ14<φ7⋆<φ15<φ11<φ12<φ7<φ13<φ9.
Substituting (3.28) into dφ/dξ=ϕ and integrating them along Γ10, Γ11, and Γ12, we have
(3.29)∫φφ10‍1(φ13-s)(φ12-s)(φ11-s)(φ10-s)ds=α2∫0ξ‍ds,∫φ13φ‍1(s-φ13)(s-φ12)(s-φ11)(s-φ10)ds=α2∫0ξ‍ds,∫φ11φ‍1(φ13-s)(φ12-s)(s-φ11)(s-φ10)ds=α2∫0ξ‍ds.
From (3.29) and noting that u=φ(ξ) and ξ=x+y-ct, we get two periodic blow-up wave solutions u9(x,y,t), u10(x,y,t) as (3.9) and a periodic wave solution u11(x,y,t) as (3.11).
(iii) From the phase portrait, we see that there are a spacial orbitΓ13, which passes the point (φ14,0), and a homoclinic orbit Γ14 passing the saddle point (φ7,0). In (φ,ϕ) plane, the expressions of the orbits are given as
(3.30)ϕ=±α2(φ-φ7)2(φ-φ14)(φ-φ15),
where
(3.31)φ14=-αφ7-2αβ-2α2φ72α,φ15=-αφ7+2αβ-2α2φ72α.
Substituting (3.30) into dφ/dξ=ϕ and integrating them along the orbits, it follows that
(3.32)±∫φ14φ‍1(s-φ7)2(s-φ14)(s-φ15)ds=α2∫0ξ‍ds,±∫φ15φ‍1(s-φ7)2(s-φ14)(s-φ15)ds=α2∫0ξ‍ds,±∫φ∞‍1(s-φ7)2(s-φ15)(s-φ15)ds=α2∫0ξ‍ds.
From (3.32) and noting that u=φ(ξ) and ξ=x+y-ct, we get a blow-up solution u12(x,y,t) as (3.12), a solitary wave solution u13(x,y,t) as (3.13), and an unbounded wave solution u14(x,y,t) as (3.14).
(3) Ifg=g0, we will consider two kinds of orbits.
(i) From the phase portrait, we see that there are two orbitsΓ15 and Γ16, which have the same Hamiltonian with the degenerate saddle point (φ+*,0). In (φ,ϕ) plane, the expressions of these two orbits are given as
(3.33)ϕ=±α2(φ-φ+*)3(φ-φ16),
where
(3.34)φ16=-3βα.
Substituting (3.33) into dφ/dξ=ϕ and integrating them along these two orbits Γ15 and Γ16, it follows that
(3.35)±∫φ+∞‍1(s-φ+*)3(s-φ16)ds=α2∫0ξ‍ds,±∫φ16φ‍1(s-φ+*)3(s-φ16)ds=α2∫0ξ‍ds.
From (3.35) and noting that u=φ(ξ) and ξ=x+y-ct, we get three blow-up solutions  u15(x,y,t), u16(x,y,t), and u17(x,y,t) as (3.15).
(ii) From the phase portrait, we see that there are two special orbitsΓ17 and Γ18 passing the points (φ17,0) and (φ18,0). In (φ,ϕ) plane, the expressions of the orbits are given as
(3.36)ϕ=±α2(φ-φ18)(φ-φ17)(φ-c1)(φ-c¯1),
where φ17<φ16<φ+*<φ18, c1 and c¯1 are conjugate complex numbers.
Substituting (3.36) into dφ/dξ=ϕ and integrating them along Γ17 and Γ18, we have
(3.37)±∫φ18φ‍1(s-φ18)(s-φ17)(s-c1)(s-c¯1)ds=α2∫0ξ‍ds.
From (3.37) and noting that u=φ(ξ) and ξ=x+y-ct, we get a periodic blow-up wave solutions u18(x,y,t) as (3.16).
Thus, we obtain the results given in Proposition3.1.Proposition 3.2.
For given positive constantsc and g0, (1.4) has the following periodic wave solution when α<0 and β<0.
(1) Ifg=0, we get four periodic wave solutions
(3.38)u19(x,y,t)=φ~1cn(β-αφ~12ξ,-φ~1α-2β+2αφ12),(3.39)u20(x,y,t)=φ~2cn(β-αφ~22ξ,φ~2α-2β+2αφ22),(3.40)u21(x,y,t)=-κ1φ~6+λ1φ~6sn2(ω2ξ,k3)κ1+λ1sn2(ω2ξ,k3),(3.41)u22(x,y,t)=φ~62-(2φ~62-2βα)sn2(φ~6-α2ξ,k4),
where
(3.42)κ1=φ~6+2βα-φ~62,λ1=φ~6-2βα-φ~62,ω2=-αφ~6+-2β+αφ~6222,k3=λ1κ1,k4=2φ~62-2β/αφ~6,
and two solitary wave solutions
(3.43)u23±(x,y,t)=±2βαsech-βξ.
(2) If-g0<g<0, we get six periodic wave solutions
(3.44)u24(x,y,t)=A2φ~10+φ~11B2+(A2φ~10-φ~11B2)cn(-αA2B2/2ξ,k5)A2+B2+(A2-B2)cn(-αA2B2/2ξ,k5),(3.45)u25(x,y,t)=φ~9*+2θ-μ+qcos⁡(αθ/2)ξ,(3.46)u26(x,y,t)=φ~14(φ~17-φ~15)+φ~17(φ~15-φ~14)sn2(ω3ξ,k6)φ~17-φ~15+(φ~15-φ~14)sn2(ω3ξ,k6),(3.47)u27(x,y,t)=φ~17(-φ~16+φ~14)-φ~14(φ~17-φ~16)sn2(ω3ξ,k6)-φ~16+φ~14-(φ~17-φ~16)sn2(ω3ξ,k6),(3.48)u28(x,y,t)=A3φ~18+φ~19B3+(A3φ~18-φ~19B3)cn(-αA3B3/2ξ,k7)A3+B3+(A3-B3)cn(-αA3B3/2ξ,k),(3.49)u29(x,y,t)=A3φ~18+φ~19B3-(A3φ~18-φ~19B3)cn(-αA3B3/2ξ,k7)A3+B3-(A3-B3)cn(-αA3B3/2ξ,k7),
where
(3.50)A2=(φ~11-c3+c¯32)2-(c3-c¯3)24,B2=(φ~10-c3+c¯32)2-(c3-c¯3)24,A3=(φ~19-c4+c¯42)2-(c4-c¯4)24,B3=(φ~18-c4+c¯42)2-(c4-c¯4)24,k5=(φ~11-φ~10)2-(A2-B2)24A2B2,k6=(φ~17-φ~16)(φ~15-φ~14)(φ~17-φ~15)(φ~16-φ~14),ω3=-α(φ~17-φ~15)(φ~16-φ~14)22,k7=(φ~19-φ~18)2-(A3-B3)24A3B3,φ~9*=-αφ~9-4αβ-3α2φ~922α,θ=-4βα+3φ~92+3φ~94βα-3φ~92,μ=2φ~9-24βα-3φ~92,q=4φ~92+4φ~94βα-3φ~92,c3, c¯3, c4 and c¯4 are complex numbers.
And two solitary wave solutions(3.51)u30(x,y,t)=φ~9+2β-6αφ~922αφ~9+2αβ-2α2φ~92cosh⁡-β+3αφ~92ξ,(3.52)u31(x,y,t)=φ~9+2β-6αφ~922αφ~9-2αβ-2α2φ~92cosh⁡-β+3αφ~92ξ.
(3) Ifg=-g0, we get two periodic wave solutions as follows:
(3.53)u32(x,y,t)=A4φ~22+φ~23B4+(A4φ~22-φ~23B4)cn(-αA4B4/2ξ,k8)A4+B4+(A4-B4)cn(-αA4B4/2ξ,k8),(3.54)u33(x,y,t)=A5φ~24+φ~25B5+(A5φ~24-φ~25B5)cn(-αA5B5/2ξ,k9)A5+B5+(A5-B5)cn(-αA5B5/2ξ,k9),
where
(3.55)A4=(φ~23-c5+c¯52)2-(c5-c¯5)24,B4=(φ~22-c5+c¯52)2-(c5-c¯5)24,A5=(φ~25-c6+c¯62)2-(c6-c¯6)24,B6=(φ~24-c6+c¯62)2-(c6-c¯6)24,k8=(φ~23-φ~22)2-(A4-B4)24A4B4,k9=(φ~25-φ~24)2-(A5-B5)24A5B5,c5, c¯5, c6 and c¯6 are complex numbers.
And a solitary wave solution(3.56)u34(x,y,t)=3βα9+2βξ2-9+6βξ2.Proof.
(1) Ifg=0, we set
(3.57)φ~2>2βα,βα<φ~6<2βα.
(i) From the phase portrait, we see that there are a closed orbitΓ~1 passing the points (φ~1,0) and (φ~2,0). In (φ,ϕ) plane, the expressions of the closed orbits are given as
(3.58)ϕ=±-α2(φ~2-φ)(φ-φ~1)(φ-c2)(φ-c¯2),
where φ~1=-φ~2, c2=iφ~22-2β/α and c¯2=-iφ~22-2β/α.
Substituting (3.58) into dφ/dξ=ϕ and integrating them along the orbit Γ~1, we have
(3.59)±∫φ~1φ‍1(φ~2-s)(s-φ~1)(s-c2)(s-c¯2)ds=-α2∫0ξ‍ds,±∫φ~2φ‍1(φ~2-s)(s-φ~1)(s-c2)(s-c¯2)ds=-α2∫0ξ‍ds.
From (3.59) and noting that u=φ(ξ) and ξ=x+y-ct, we obtain the periodic wave solutions u19(x,y,t) as (3.38) and u20(x,y,t) as (3.39).
(ii) From the phase portrait, we see that there are two closed orbitsΓ~2 and Γ~3 passing the points (φ~3,0), (φ~4,0), (φ~5,0), and (φ~6,0). In (φ,ϕ) plane, the expressions of the closed orbits are given as
(3.60)ϕ=±-α2(φ-φ~3)(φ-φ~4)(φ-φ~5)(φ~6-φ),
where φ~3=-φ~6, φ~4=-2β/α-φ~62 and φ~5=2β/α-φ~62.
Substituting (3.60) into dφ/dξ=ϕ and integrating them along Γ~2 and Γ~3, we have
(3.61)±∫φ~3φ‍1(φ~6-s)(φ~5-s)(φ~4-s)(s-φ~3)ds=-α2∫0ξ‍ds,±∫φ~6φ‍1(φ~6-s)(s-φ~5)(s-φ~4)(s-φ~3)ds=-α2∫0ξ‍ds.
From (3.61) and noting that u=φ(ξ) and ξ=x+y-ct, we obtain the periodic wave solutions u21(x,y,t) as (3.40) and u22(x,y,t) as (3.41).
(iii) From the phase portrait, we see that there are two symmetric homoclinic orbitsΓ~4 and Γ~5 connected at the saddle point (0,0). In (φ,ϕ) plane, the expressions of the homoclinic orbits are given as
(3.62)ϕ=±φ-α2(φ-φ~7)(φ~8-φ),
where φ~7=-2β/α and φ~8=2β/α.
Substituting (3.62) into dφ/dξ=ϕ and integrating them along the orbits Γ~4 and Γ~5, we have
(3.63)±∫φ~7φ‍1s(s-φ~7)(φ~8-s)ds=-α2∫0ξ‍ds,±∫φ~8φ‍1s(s-φ~7)(φ~8-s)ds=-α2∫0ξ‍ds.
From (3.63) and noting that u=φ(ξ) and ξ=x+y-ct, we obtain the solitary wave solutions u23±(x,y,t) as (3.43).
(2) If-g0<g<0, we set the middle solution of f(φ)=0 be φ~9(0<φ~9<β/3α), then we can get another two solutions of f(φ)=0 as follows:
(3.64)φ~9*=-αφ~9-4αβ-3α2φ~922α,φ~9⋆=-αφ~9+4αβ-3α2φ~922α.
(i) From the phase portrait, we see that there are a closed orbitΓ~6 passing the points (φ~10,0) and (φ~11,0). In (φ,ϕ) plane, the expressions of the closed orbits are given as
(3.65)ϕ=±-α2(φ~11-φ)(φ-φ~10)(φ-c3)(φ-c¯3),
where φ~12<φ~10<φ~9⋆<φ~11<φ~13, c3 and c¯3 are conjugate complex numbers.
Substituting (3.37) into dφ/dξ=ϕ and integrating them along Γ~6, we have
(3.66)±∫φ~10φ‍1(φ~11-s)(s-φ~10)(s-c3)(s-c¯3)ds=-α2∫0ξ‍ds.
From (3.66) and noting that u=φ(ξ) and ξ=x+y-ct, we get a periodic wave solution u24(x,y,t) as (3.44).
(ii) From the phase portrait, we note that there is a special orbitΓ~7, which has the same Hamiltonian with that of (φ9*,0). In (φ,ϕ) plane, the expressions of the orbits are given as
(3.67)ϕ=±-α2(φ-φ~9*)2(φ-φ~12)(φ~13-φ),
where
(3.68)φ~12=αφ~9+4αβ-3α2φ~92+2αφ~9(αφ~9-4αβ-3α2φ~92)2α,φ~13=αφ~9+4αβ-3α2φ~92-2αφ~9(αφ~9-4αβ-3α2φ~92)2α.
Substituting (3.67) into dφ/dξ=ϕ and integrating them along Γ~7, it follows that
(3.69)±∫φ~12φ‍1(φ~13-s)(s-φ~9*)2(s-φ~12)ds=-α2∫0ξ‍ds.
From (3.69) and noting that u=φ(ξ) and ξ=x+y-ct, we get a periodic wave solution u25(x,y,t) as (3.45).
(iii) From the phase portrait, we note that there are two closed orbitsΓ~8 and Γ~9 passing the points (φ~14,0), (φ~15,0), (φ~16,0), and (φ~17,0). In (φ,ϕ) plane, the expressions of the orbits are given as
(3.70)ϕ=±-α2(φ-φ~14)(φ-φ~15)(φ-φ~16)(φ~17-φ),
where φ~20<φ~14<φ~12<φ~10<φ~9⋆<φ~11<φ~13<φ~15<φ~9<φ~16<φ~9*<φ~17<φ~21.
Substituting (3.70) into dφ/dξ=ϕ and integrating them along Γ~8 and Γ~9, we have
(3.71)±∫φ~14φ‍1(φ~17-s)(φ~16-s)(φ~15-s)(s-φ~14)ds=-α2∫0ξ‍ds,±∫φ~17φ‍1(φ~17-s)(s-φ~16)(s-φ~15)(s-φ~14)ds=-α2∫0ξ‍ds.
From (3.71) and noting that u=φ(ξ) and ξ=x+y-ct, we get two periodic wave solutions u26(x,y,t) as (3.46) and u27(x,y,t) as (3.47).
(iv) From the phase portrait, we note that there is a special orbitΓ~10 passing the points (φ~18,0) and (φ~19,0). In (φ,ϕ) plane, the expressions of the orbit are given as
(3.72)ϕ=±-α2(φ~19-φ)(φ-φ~18)(φ-c4)(φ-c¯4),
where φ~18<φ~20<φ~21<φ~19, c4 and c¯4 are conjugate complex numbers.
Substituting (3.72) into dφ/dξ=ϕ and integrating it along Γ~10, we have
(3.73)±∫φ~18φ‍1(φ~19-s)(s-φ~18)(s-c4)(s-c¯4)ds=-α2∫0ξ‍ds.
From (3.73) and noting that u=φ(ξ) and ξ=x+y-ct, we get a periodic wave solution u28(x,y,t) as (3.48).
Ifφ(ξ) is a traveling wave solution, then φ(ξ+q) is a traveling wave solution too. Taking q=2K and noting that cn(u+2K)=-cnu, we get a periodic wave solution u29(x,y,t) as (3.49).
(v) From the phase portrait, we note that there are two homoclinic orbitsΓ~11 and Γ~12 connected at the saddle point (φ~9,0). In (φ,ϕ) plane, the expressions of the orbits are given as
(3.74)ϕ=±-α2(φ-φ~9)2(φ-φ~20)(φ~21-φ),
where
(3.75)φ~20=-αφ~9+2αβ-2α2φ~92α,φ~21=-αφ~9-2αβ-2α2φ~92α.
Substituting (3.74) into dφ/dξ=ϕ and integrating them along Γ~11 and Γ~12, it follows that
(3.76)±∫φ~20φ‍1(s-φ~9)2(s-φ~20)(φ~21-s)ds=-α2∫0ξ‍ds,±∫φ~21φ‍1(s-φ~9)2(s-φ~20)(φ~21-s)ds=-α2∫0ξ‍ds.
From (3.76) and noting that u=φ(ξ) and ξ=x+y-ct, we get two solitary wave solutions u30(x,y,t) as (3.51) and u31(x,y,t) as (3.52).
(3) Ifg=-g0, we will consider two kinds of orbits.
(i) From the phase portrait, we note that there is a closed orbitΓ~13 passing the points (φ~22,0) and (φ23,0). In (φ,ϕ) plane, the expressions of the orbit are given as
(3.77)ϕ=±-α2(φ~23-φ)(φ-φ~22)(φ-c5)(φ-c¯5),
where -3β/α<φ~22<-2β/3α<φ~23<β/3α, c5 and c¯5 are conjugate complex numbers.
Substituting (3.77) into dφ/dξ=ϕ and integrating it along Γ~13, we have
(3.78)±∫φ~22φ‍1(φ~23-s)(s-φ~22)(s-c5)(s-c¯5)ds=-α2∫0ξ‍ds.
From (3.78) and noting that u=φ(ξ) and ξ=x+y-ct, we get a periodic wave solutions u32(x,y,t) as (3.53).
(ii) From the phase portrait, we note that there is a closed orbitΓ~14 passing the points (φ~24,0) and (φ~25,0). In (φ,ϕ) plane, the expressions of the orbit are given as
(3.79)ϕ=±-α2(φ~25-φ)(φ-φ~24)(φ-c6)(φ-c¯6),
where φ~24<-3β/α<φ~22<-2β/3α<φ~23<β/3α<φ~25, c6 and c¯6 are conjugate complex numbers.
Substituting (3.79) into dφ/dξ=ϕ and integrating them along Γ14, we have
(3.80)±∫φ~24φ‍1(φ~25-s)(s-φ~24)(s-c6)(s-c¯6)ds=-α2∫0ξ‍ds.
From (3.80) and noting that u=φ(ξ) and ξ=x+y-ct, we get a periodic wave solutions u33(x,y,t) as (3.54).
(iii) From the phase portrait, we see that there is a homoclinic orbitΓ~15, which passes the degenerate saddle point (φ+*,0). In (φ,ϕ) plane, the expressions of the homoclinic orbit are given as
(3.81)ϕ=±-α2(φ+*-φ)3(φ-φ~26),
where
(3.82)φ~26=-3βα.
Substituting (3.81) into dφ/dξ=ϕ and integrating them along Γ15, it follows that
(3.83)±∫φ~26φ‍1(s-φ+*)(s-φ+*)(φ~26-s)ds=-α2∫0ξ‍ds.
From (3.83) and noting that u=φ(ξ) and ξ=x+y-ct, we get a solitary wave solution u34(x,y,t) as (3.56).
Thus, the derivation of Proposition3.2 has been finished.Proposition 3.3.
For these solutions, the following are their relations.
(1) Whenφ6 tends to 2β/α, the periodic blow-up wave solutions u3± and u4± tend to periodic blow-up wave solutions u1± and u2±, that is,
(3.84)lim⁡φ6→2β/αu3±(x,y,t)=u1±(x,y,t),lim⁡φ6→2β/αu4±(x,y,t)=u2±(x,y,t).
(2) Whenφ6 tends to β/α, the periodic wave solutions u5± tend to kink wave solutions u6±, that is,
(3.85)lim⁡φ6→β/αu5±(x,y,t)=u6±(x,y,t).
(3) Whenφ6 tends to β/α, the periodic blow-up wave solutions u3± tend to unbounded wave solutions u7±, that is,
(3.86)lim⁡φ6→β/αu3±(x,y,t)=u7±(x,y,t).
(4) Whenφ13 tends to φ9, the periodic blow-up wave solution u9 tends to periodic blow-up wave solution u8-, that is,
(3.87)lim⁡φ13→φ9u9(x,y,t)=u8-(x,y,t).
(5) Whenφ13 tends to φ9, the periodic blow-up wave solution u10 tends to periodic blow-up wave solution u8+, that is,
(3.88)lim⁡φ13→φ9u10(x,y,t)=u8+(x,y,t).
(6) Whenφ13 tends to φ7, the periodic wave solution u11 tends to solitary wave solution u13, that is,
(3.89)lim⁡φ13→φ7u11(x,y,t)=u13(x,y,t).
(7) Whenφ13 tends to φ7, the periodic blow-up wave solution u10 tends to unbounded wave solution u14, that is,
(3.90)lim⁡φ13→φ7u10(x,y,t)=u14(x,y,t).
(8) Whenφ18 tends to β/3α, the periodic blow-up wave solution u17 tends to blow-up wave solution u16, that is,
(3.91)lim⁡φ18→β/3αu17(x,y,t)=u16(x,y,t).
(9) Whenφ~2 tends to 2β/α, the periodic wave solutions u19 and u20 tend to solitary wave solutions u23±, that is,
(3.92)lim⁡φ~2→2β/αu19(x,y,t)=u23-(x,y,t),lim⁡φ~2→2β/αu20(x,y,t)=u23+(x,y,t).
(10) Whenφ~6 tend to 2β/α, the periodic wave solutions u21 and u22 tends to solitary wave solutions u23±, that is,
(3.93)lim⁡φ~6→2β/αu21(x,y,t)=u23-(x,y,t),lim⁡φ~6→2β/αu22(x,y,t)=u23+(x,y,t).
(11) Whenφ~11 tends to φ~13, the periodic wave solution u24 tends to periodic wave solution u25, that is,
(3.94)lim⁡φ~11→φ~13u24(x,y,t)=u25(x,y,t).
(12) Whenφ~17 tends to φ~9*, the periodic wave solution u26 tends to periodic wave solution u25, that is,
(3.95)lim⁡φ~17→φ~9*u26(x,y,t)=u25(x,y,t).
(13) Whenφ~17 tends to φ~21, the periodic wave solutions u26 and u27 tend to solitary wave solutions u30 and u31, that is,
(3.96)lim⁡φ~17→φ~21u26(x,y,t)=u30(x,y,t),lim⁡φ~17→φ~21u27(x,y,t)=u31(x,y,t).
(14) Whenφ~19 tends to φ~21, the periodic wave solutions u28 and u29 tend to solitary wave solutions u30 and u31, that is,
(3.97)lim⁡φ~19→φ~21u28(x,y,t)=u30(x,y,t),lim⁡φ~19→φ~21u29(x,y,t)=u31(x,y,t).
(15) Whenφ~22 and φ~24 tend to 3β/α, the periodic wave solutions u32 and u33 tend to solitary wave solution u34, that is,
(3.98)lim⁡φ~22→3β/αu32(x,y,t)=lim⁡φ~24→3β/αu33(x,y,t)=u34(x,y,t).Proof.
(1) Lettingφ6→2β/α, it follows that φ5→0 and sn(φ6α/2ξ,φ5/φ6)→sn(βξ,0)=sinβξ, and we have
(3.99)lim⁡φ6→2β/αu3±(x,y,t)=lim⁡φ6→2β/α±φ6sn(φ6α/2ξ,φ5/φ6)=±2β/αsin⁡βξ=u1±(x,y,t),lim⁡φ6→2β/αu4±(x,y,t)=lim⁡φ6→2β/α±φ62-φ52(sn(φ6α/2ξ,φ5/φ6))21-(sn(φ6α/2ξ,φ5/φ6))2=±2β/αcos⁡βξ=u2±(x,y,t).
Therefore, the periodic blow-up solutionsu1±(x,y,t) are the limit of the elliptic function periodic blow-up solutions u3±(x,y,t) and the periodic blow-up solutions u2±(x,y,t) are the limit of the elliptic function periodic blow-up solutions u4±(x,y,t).
(2) Lettingφ6→β/α, it follows that φ5→β/α and sn(φ5α/2ξ,φ5/φ6)→sn(β/2ξ,1)=tanhβ/2ξ, and we have
(3.100)lim⁡φ6→β/αu5±(x,y,t)=lim⁡φ6→β/α±φ5sn(φ6α2ξ,φ5φ6)=±βαtanhβ2ξ=u6±(x,y,t).
Therefore, the kink wave solutionsu6±(x,y,t) are the limit of the elliptic function periodic solutions u5±(x,y,t).
(4) Lettingφ13→φ9, it follows that φ12→φ7*, φ11→φ7*, φ10→φ8 and sn(ω1ξ,k1)→sn(ω1ξ,0)=sinω1ξ, and we have
(3.101)lim⁡φ13→φ9u9(x,y,t)=lim⁡φ13→φ9φ10(φ11-φ13)+φ11(φ13-φ10)(sn(ω1ξ,k1))2φ11-φ13+(φ13-φ10)(sn(ω1ξ,k1))2=φ7*-2γ1δ1-η1cos⁡αγ1/2ξ=u8-(x,y,t).
Therefore, the trigonometric function periodic blow-up wave solutionu8-(x,y,t) is the limit of the elliptic function periodic blow-up solution u9(x,y,t).
(9) Lettingφ~2→2β/α, it follows that φ~1→-2β/α,  β-αφ~12→-β and -φ~1α/(-2β+2αφ12)→1, and we have
(3.102)lim⁡φ~2→2β/αu19(x,y,t)=lim⁡φ~2→2β/αφ~1cn(β-αφ~12ξ,-φ~1α-2β+2αφ12)=-2βαsech-βξ=u23-(x,y,t).
Therefore, the solitary wave solutionu23-(x,y,t) is the limit of the elliptic function periodic wave solution u19(x,y,t).
(15) Lettingφ~22→-3β/α, it follows that φ~23→β/3α,c5→β/3α,c¯5→β/3α,  A4→0,B4→4β/3α and cn(-αA4B4/2,k8)→cn(0,k8)=1, and we have
(3.103)lim⁡φ~22→-3β/αu32(x,y,t)=lim⁡φ22→-3β/αA4φ~22+φ~23B4+(A4φ~22-φ~23B4)cn(-αA4B4/2ξ,k8)A4+B4+(A4-B4)cn(-αA4B4/2ξ,k8)=lim⁡A4→0A4φ~22+φ~23B4+(A4φ~22-φ~23B4)cn(-αA4B4/2ξ,k8)A4+B4+(A4-B4)cn(-αA4B4/2ξ,k8)=lim⁡A4→02-2αA4B4(φ~22+φ~22χ1)+B4(A4φ~22-B4φ~23)αξχ2χ32-2αA4B4(1+χ1)+B4(A4-B4)αξχ2χ3=3βα9+2βξ2-9+6βξ2=u34(x,y,t),
where  χ1=cn(-αA4B4/2,k8),χ2=dn(-αA4B4/2,k8),χ3=sn(-αA4B4/2,k8).
Therefore, the fractional function solitary wave solutionu34(x,y,t) is the limit of the elliptic function periodic wave solution u32(x,y,t).
Similarly, we can derive the others case. This has proved Proposition3.3.Finally, we will show that the periodic wave solutionu5+(x,y,t) evolutes into the kink wave solution u6+(x,y,t) when φ6→β/α. We take some suitable choices of the parameters, such as
(3.104)a=1,b=1,r=5c=2,
as an illustrative sample and draw their plots (see Figure 3).The periodic wave solutionu5+(x,y,t) evolute into the kink wave solution u6+(x,y,t) with the conditions (3.104). (a) φ6 = 2.2; (b) φ6 = 2.01; (c) φ6 = 2.
(a)
(b)
(c)Remark 3.4.
One may find that we only consider the case wheng≥0 in Proposition 3.1 (when g≤0 in Proposition 3.2). In fact, we may get exactly the same solutions in the opposite case.


## 4. Conclusion

In this paper, we have obtained many traveling wave solutions for the generalized KP-BBM equation (1.4) by employing the bifurcation method and qualitative theory of dynamical systems. The traveling wave solutions have been given in Propositions 3.1 and 3.2. On the other hand, in Proposition 3.3, we prove that the periodic wave solutions, kink wave solutions, blow-up wave solutions, unbounded solutions, and solitary wave solutions can be obtained from the limits of the smooth periodic wave solutions or periodic blow-up solutions. 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.



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*Source: 363879-2012-07-16.xml*