Research Article Traveling Wave Solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony Equation

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1 Abstract and Applied Analsis, Article ID , 9 pages Research Article Traveling Wave Solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahon Equation Zhengong Ouang Department of Mathematics, Foshan Universit, Foshan, Guangdong , China Correspondence should be addressed to Zhengong Ouang; zouang math@163.com Received 26 March 2014; Accepted 17 April 2014; Published 18 June 2014 Academic Editor: Junling Ma Copright 2014 Zhengong Ouang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. We use bifurcation method of dnamical sstems to stud exact traveling wave solutions of a nonlinear evolution equation. We obtain exact explicit expressions of bell-shaped solitar wave solutions involving more free parameters, and some existing results are corrected and improved. Also, we get some new exact periodic wave solutions in parameter forms of the Jacobian elliptic function. Further, we find that the bell-shaped waves are limits of the periodic waves in some sense. The results impl that we can deduce bell-shaped waves from periodic waves for some nonlinear evolution equations. 1. Introduction The Benjamin-Bona-Mahon (BBM) equation [1] u t +u x +uu x u xxt =0 (1) wasproposedasthemodelforpropagationoflongwaves where nonlinear dispersion is incorporated. The Kadomtsev- Petviashvili (KP) equation [2] (u t +auu x +u xxx ) x +u =0 (2) was given as the generalization of the KdV equation. In addition, both BBM and KdV equations can be used to describe long wavelength in liquids, fluids, and so forth. Combining the two equations, the Kadomtsev-Petviashvili- Benjamin-Bona-Mahon (KP-BBM) equation (u t +u x a(u 2 ) x bu xxt ) x +ku =0 (3) was presented in [3] for further stud. Some methods are developed and applied to find exact solutions of nonlinear evolution equations because exact solutions pla an important role in the comprehension of nonlinear phenomena. For instance, extended tanh method, extended mapping method with smbol computation, and bifurcation method of dnamical sstems are emploed to stud (3)[4 6], and some solitar wave solutions and triangle periodic wave solutions were obtained. However, there is no method which can be applied to all nonlinear evolution equations. The research on the solutions of the KP-BBM equation now appears insufficient. Further studies are necessar for the traveling wave solutions of the KP-BBM equation. The purpose of this paper is to appl the bifurcation method [7 10]of dnamical sstems to continue to seek traveling waves of (3). Firstl, we obtain bell-shaped solitar wave solutions involving more free parameters, and some results in [6] are corrected and improved. Then, we get some new periodic wave solutions in parameter forms of Jacobian elliptic function, and numerical simulation verifies the validit of these periodic solutions. The periodic wave solutions obtained in this paper are different from those in [5]. Furthermore, we find an interesting relationship between the bell-shaped waves and periodic waves; that is, the bellshaped waves are limits of the periodic waves in forms of Jacobian elliptic function as modulus approaches 1. This paper is organized as follows. First, we draw the bifurcation phase portraits of planar sstem according to the KP-BBM equation in Section 2. Second, bell-shaped solitar wave solutions to the equation under consideration are presented in Section 3. Third, periodic solitar waves are

2 2 Abstract and Applied Analsis givenintheformsofjacobianellipticfunctionandnumerical simulation is done. Finall, the relationship between the bell-shaped solitar waves and periodic waves is proved in Section Bifurcation Phase Portraits of Sstem (6) Suppose (3) possesses traveling wave solutions in the form u(x, t) = (ξ), ξ=x+r ct,wherec is the wave speed and r is a real constant. Substituting u(x, t) = (ξ), ξ=x+r ct into (3) admits to the following ODE: (1+kr 2 c) a( 2 ) + (4) =0, (4) where the derivative is for variable ξ. Integrating(4) twice with respect to ξ andlettingthefirstintegralconstanttake value zero, it follows that (1 + kr 2 c) a 2 + =g, (5) where g is the second integral constant. Equation (5) is equivalent to the following two-dimensional sstem: d dξ =, d dξ = a 2 (1+kr 2 c)+g. (6) It is obvious that sstem (6)hasthefirstintegral H (, ) = a kr2 c 2 g 2 =h, (7) where h istheconstantofintegration. Define Δ = (1+kr 2 c) 2 4ag.WhenΔ>0,therearetwo equilibrium points ( 1,0)and ( 2,0)of (6)on-axis, where 1 = ((1 + kr 2 c) Δ)/, 2 = ((1 + kr 2 c)+ Δ)/. The Hamiltonian H of ( 1,0)and ( 2,0)is denoted b h 1 = H( 1,0)and h 2 =H( 2,0). In the case of 1+kr 2 c < 0and 1+kr 2 c > 0,the bifurcation phase portraits of sstem (6) seefigures1and2 in [6], respectivel, in which there are some homoclinic and periodic orbits of sstem (6). For our purpose, we redraw the homolinic and periodic orbits in this paper(see Figures 1 3). 3. Exact Explicit Expressions of Solitar Wave Solutions In this section, we discuss bell-shaped wave solutions under g=0and g =0,respectivel The Case g=0. Sstem (6)canberewrittenas d dξ =, d dξ = a 2 (1+kr 2 c). (8) The first integral of (8)is H (, ) = a kr2 c 2 =h. (9) 2 When (1+kr 2 c)/ < 0, there are two homoclinic orbits Γ 1 and Γ 2 (see Figures 1(a) and 1(b)). In - plane, Γ 1 and Γ 2 can be described b 2 = kr2 c 2, (0, ) or (,0), where = 3(1 + kr 2 c)/.thatis, (10) =± kr2 c 2. (11) Substituting (11) intod/dξ = and integrating along homoclinc orbits Γ 1 and Γ 2,respectivel,weget ds (/3) s 3 ((1+kr 2 c) /) s 2 =± ξ. (12) Completing the above integration, it follows that u 1 (x,,t) = 3(1+kr 2 c) a[1+cosh ( ((1 + kr 2 c) /) (x + r ct))]. (13) Remark 1. u 1 (x,,t)is a bright soliton solution when /a < 0 andadarksolitonsolutionwhen/a > 0. Ifthereal number r in (13)takesvalue1,thensolution(13)is thesameto solution (1.2) in [6]. Solution (1.1) in [6]isnotarealsolution of the KP-BBM equation; it is obvious that solution (1.1) tends to infinite as ξ 0, and it does not satisf the KP-BBM equation (3). When (1+kr 2 +c)/ > 0, there are two homoclinic orbits Γ 3 and Γ 4 (see Figures 2(a) and 2(b)). Similarl solitar wave solutions according to Γ 3 and Γ 4 are obtained as follows: u 2 (x,,t) =((1+kr 2 c) [ 2 + cosh ( 1+kr2 c (x + r ct))] ) [ ] (a[ 1+cosh ( 1+kr2 c (x+r ct))]). [ ] (14) Remark 2. If the real number r in (14) takesvalue1,then solution (14)is the same to solution(1.4)in[6]. Solution (1.5) in [6]isnotarealsolutionoftheKP-BBMequation;itiseas to verif that it does not satisf the KP-BBM equation (3). 1

3 Abstract and Applied Analsis 3 Γ 1 Γ 2 Γ 7 Γ 8 r 1 0 r 2 r 3 r 1 r 2 0 r 3 (a) /a < 0 (b) /a > 0 r 1 0 r 2 r 3 r 1 r 2 0 r 3 (c) (d) Figure1:Thebifurcationphaseportraitsofsstem(6)withg=0and (1 + kr 2 +c)/< The Case g =0. There are two homoclinic orbits Γ 5 and Γ 6 when g =0 (see Figures 3(a) and 3(b)). Γ 5 and Γ 6 can be described b 2 = kr2 c 2 + 2g +2h. (15) When /a < 0, the corresponding homoclinic orbit Γ 5 has a double zero point 1 and a zero point 3 on -axis (see Figure 3(a)), so (15)canberewrittenas that is, 2 = 3 ( 1) 2 ( 3 ) ; (16) =± 3 ( 1) 2 ( 3 ). (17) Substituting (17)into d/dξ =and integrating along homoclinic orbits Γ 5,weget 3 ds = ξ (/3) (s 1 ) 2, (18) (s 3 ) where 1 =(1+kr 2 c Δ)/ and 3 =(1+kr 2 c+2 Δ)/ if a>0and < 0; then, completing (18)wegetthefollowing solution: u 3 (x,,t) =((1+kr 2 c Δ) cosh Δ (x+r ct)+1

4 4 Abstract and Applied Analsis Γ 3 Γ 4 r 1 r 2 0 r 3 r 1 0 r 2 r 3 Γ 9 Γ 10 (a) /a < 0 (b) /a > 0 r 1 r 2 0 r 3 r 1 0 r 2 r 3 (c) (d) Figure 2: The bifurcation phase portraits of sstem (6)withg=0and (1 + kr 2 +c)/>0. +kr 2 c+5 Δ) ((cosh 1 Δ (x+r ct)+1)). (19) In (18), 1 =(1+kr 2 c+ Δ)/ and 3 =(1+kr 2 c 2 Δ)/ if a<0and > 0;then,completing(18)wegetthe following solution: u 4 (x,, t) =((1+kr 2 c+ Δ) cosh Δ (x+r ct)+1+kr2 c 5 Δ) ((cosh 1 Δ (x+r ct)+1)). (20) Remark 3. If the real number r in (19) and(20) takesvalue 1, then solutions (19)and(20) are the same to solutions (1.6) and (1.8) in [6]. Solutions (1.7) and (1.9) in [6] arenotreal solutions of the KP-BBM equation, and it is eas to verif that the do not satisf the KP-BBM equation (3).

5 Abstract and Applied Analsis 5 Γ 5 Γ 6 Γ 11 Γ 12 c 1 1 c 2 2 c c 1 1 c 2 2 c 3 (a) /a < 0 (b) /a > 0 c 1 1 c 2 2 c 3 c 1 1 c 2 2 c 3 (c) (d) Figure 3: The bifurcation phase portraits of sstem (6) withg =0. When /a > 0, the corresponding homoclinic orbit Γ 6 has a double zero point 2 and a zero point 4 on -axis (see Figure 3(b)), so (15)canberewrittenas 2 = 3 ( 2) 2 ( 4 ); (21) that is, =± 3 ( 2) 2 ( 4 ). (22) Substituting (22) intod/dξ = and integrating along homoclinic orbits Γ 6,weget 4 ds = ξ (/3) (s 2 ) 2, (23) (s 4 ) where 2 =(1+kr 2 c Δ)/ and 4 =(1+kr 2 c+2 Δ)/ if a<0and < 0;then,completing(23)wegetthesolution u 3 (x,,t).in(23), 2 = (1 + kr 2 c+ Δ)/ and 4 = (1 + kr 2 c 2 Δ)/ if a>0and > 0;then,completing (23)wegetthesolutionu 4 (x,,t). 4. Periodic Wave Solutions So as to explain our work convenientl, in this section the Jacobian elliptic function sn(l, m) with modulus m will be expressed b snl. We discuss the periodic wave solutions under conditions g=0and g =0,respectivel The Case g=0. When (1 + kr 2 + c)/ < 0, sstem(6) has periodic orbits Γ 7 and Γ 8 (see Figures 1(a) and 1(b)). Their

6 6 Abstract and Applied Analsis expressions are (9)on- plane, where h 1 <h<h 2 (or h 2 < h<h 1 ). Let f 1 () = kr2 c 2 +2h; (24) then, we have the following results. Claim 1. In the case of g=0, (1 + kr 2 + c)/ < 0 and h 1 < h<h 2 (or h 2 <h<h 1 ); then, the function f 1 () must have three different real zero points. Proof. Since f 1 () is a cubic polnomial about,wecanuse the Shengjin Theorem [11] to distinguish its solutions. We onl discuss the case /a < 0, andthecase/a > 0 is the same. Under the above conditions, h 1 =H( 1,0)=H(0, 0) =0, h 2 =H( 2,0)=H( 1+kr2 c,0)= (1 + kr2 c) 3 a 6a 2 <0. (25) So (1 + kr 2 c)/(6a 2 ) < h < 0. Inf 1 () the coefficients /3 < 0, (1 + kr 2 c)/ < 0, B Shengjing Theorem [11], it follows that the function f 1 () has three different real zero points. Let A = ((1 + kr 2 c)/) 2, B = 9(/3) 2h,and C = 3((1+kr 2 c)/) 2h;then,B 2 4AC = 144(a/) 2 h[h ((1 + kr 2 c) 3 /6a 2 )] < 0. Let r 1 < r 2 < r 3 be three different real zero points of f 1 (). ThenClaim 1 means that (9) hasthreeintersection points (r 1,0), (r 2,0),and(r 3,0)on -axis. Therefore, (9)can be rewritten as 2 = 3 ( r 1)( r 2 )( r 3 ), (26) where r 1 <0<r 2 <<r 3 when /a < 0 and r 1 <<r 2 < 0<r 3 when /a > 0. When /a < 0, theorbitγ 7 is according to a periodic solution of (6) and its expression is given b =± 3 ( r 1)( r 2 )( r 3 ), (r 1 <r 2 r 3 ). (27) Substituting (27)intod/dξ =and integrating along orbit Γ 7,weget r 3 ds (r 3 s)(s r 1 )(s r 2 ) = 3 ξ, (r 1 <r 2 <r 3 ). B formula (236) in [12], we have (28) g 1 sn 1 (sin ψ 1,m 5 )= 3 ξ, (29) where g 1 = 2/ r 3 r 1,sinψ 1 = (r 3 )/(r 3 r 2 ),and m 5 = (r 3 r 2 )/(r 3 r 1 ).Solving(29), we get That is, =r 3 (r 3 r 2 ) sn 2 a(r 3 r 1 ) ξ. (30) 6 u 5 (x,,t)=r 3 (r 3 r 2 ) sn 2 a(r 3 r 1 ) 6 where the modulus of sn is m 5 = (r 3 r 2 )/(r 3 r 1 ). Similarl, when /a > 0, the expression Γ 8 is (x+r ct), (31) =± 3 ( r 1)( r 2 )( r 3 ), (r 1 r 2 <r 3 ). (32) Substituting (32)intod/dξ =and integrating along orbit Γ 8,wehave r 1 ds (r 3 s)(r 2 s)(s r 1 ) = 3 ξ, (r 1 < r 2 <r 3 ). The according periodic solution of Γ 8 can be obtained as u 6 (x,,t)=r 1 +(r 2 r 1 ) sn 2 a(r 3 r 1 ) 6 (33) (x+r ct), (34) where the modulus of sn is m 6 = (r 2 r 1 )/(r 3 r 1 ). When (1 + kr 2 + c)/ > 0,sstem(6) has periodic orbits Γ 9 and Γ 10 (see Figure 2). Their expressions are (9), where h 1 < h<h 2 (or h 2 <h<h 1 ). Similarl, we can get the according periodic solutions of Γ 9 and Γ 10 as u 5 and u 6. To verif validit of the periodic wave solutions, we take a = b = k = r = 1, c = 1,andh = 9/4 to make the conditions in Claim 1 satisfied. B simple calculation, we get that r 1 = , r 2 = 1.5, andr 3 = The specific periodic wave solution is u (x,, t) = sn where the modulus of sn is 2/2. (x + + t), (35) 4.2. The Case g =0. Sstem (6) has periodic orbits Γ 11 and Γ 12 (see Figure 3). Their expressions are (7)onthe-plane, where h 1 <h<h 2 (or h 1 <h<h 2 ). Let f 2 () = kr2 c 2 + 2g +2h; (36) then, we have the following results about f 2 ().

7 Abstract and Applied Analsis 7 Claim 2. If g =0and h 1 <h<h 2 (or h 2 <h<h 1 ), then the function f 2 () must have three different real zero points. Proof. We onl prove the case /a < 0,andthecase/a > 0 is the same. Under the above conditions, h 1 =H( 1,0)= a kr2 c g 1 = 1 2 f 2 ( 1 )+h, h 2 =H( 2,0)= a kr2 c g 2 = 1 2 f 2 ( 2 )+h. (37) So f 2 ( 1 ) f 2 ( 2 )=4(h h 1 )(h h 2 )<0.Forf 2 (),wehave f 2 ( ) > 0, f 2 ( 1 )<0, f 2 ( 2 )>0,andf 2 (+ ) < 0. Again, f 2 () = (/)( 1)( 2 ), which is monotonous in the intervals (, 1 ), ( 1, 2 ),and( 2,+ ).Bzeropoint theorem of continuous function, there must be one real zero point of f 2 () that lies in each of the three intervals, proving the claim. Let c 1 <c 2 <c 3 be three different real zero points of f 2 (). Then Claim 2 means that (7) has three intersection points on -axis denoted b (c 1,0), (c 2,0),and(c 3,0).Then(7) canbe rewritten as 2 = 3 ( c 1)( c 2 )( c 3 ), (38) where c 1 < 1 <c 2 < 2 <c 3. When /a < 0, the expression of periodic orbit Γ 11 is =± 3 ( c 1)( c 2 )( c 3 ), (c 1 <c 2 c 3 ). (39) When /a > 0, the expression of periodic orbit Γ 12 is =± 3 ( c 1)( c 2 )( c 3 ), (c 1 c 2 <c 3 ). (40) Substituting (39)and(40)into d/dξ =and integrating along orbits Γ 11 and Γ 12,respectivel,itisthesametothe proceeding for solving u 5 and u 6 and we can get the according periodic solutions of Γ 11 and Γ 12 as follows: u 7 (x,,t)=c 3 (c 3 c 2 ) sn 2 a(c 3 c 1 ) 6 u 8 (x,,t)=c 1 +(c 2 c 1 ) sn 2 a(c 3 c 1 ) 6 (x + r ct), (x+r ct), (41) For example, we take a=b=k=r=1, c= 1, g=2, and h=3/2such thatthe conditionsin Claim2 are satisfied. B simple calculation, we get that c 1 = , c 2 = 1.5,and c 3 = The according periodic wave solution is u (x,, t) = sn where the modulus of sn is / (x++t), (42) 5. Relationship between Bell-Shaped Waves and Periodic Waves In Sections 3 and 4,thebell-shapedsolitarwaveandperiodic wave solutions are obtained. Via further stud, we find that there exists an interesting relationship between these two kinds of solutions; that is, the bell-shaped solutions are limits of the periodic waves in some sense. The results are detailed as follows. Proposition 4. Let u i (i = 1,2,...,8) be solutions of (3), let k, r, a, b, c, andg be parameters in (5), andletm i (i = 5, 6, 7, 8) be modulus of the Jacobian elliptic function sn; then, one has the following. Case 1. When g=0and (1 + kr 2 c)/ < 0, formodulus m i 1 (i = 5, 6), theperiodicwavesu 5 and u 6 degenerate bell-shaped wave u 1. Case 2. When g=0and (1 + kr 2 c)/ > 0, formodulus m i 1 (i = 5, 6), theperiodicwavesu 5 and u 6 degenerate bell-shaped wave u 2. Case 3. When g =0and /a < 0,formodulusm 7 1,the periodic wave u 7 degenerates bell-shaped wave u 3. Case 4. When g =0and /a > 0,formodulusm 8 1,the periodic wave u 8 degenerates bell-shaped wave u 4. Here, we onl prove Cases 1 and 3 for simplicit. The remaining cases are the same. In the following proofs, we use the propert of elliptic function that sn tanh when the modulus m 1[5, 13]. Proof of Case 1. When m 5 = (r 3 r 2 )/(r 3 r 1 ) 1,it means r 1 =r 2 and sn = tanh; then, we calculate r 1 =r 2 =0, r 3 = 3(1+kr2 c). (43) Substituting r i (i = 1, 2, 3) into u 5 admits to u 1 as follows: u 5 (x,,t) =r 3 (r 3 r 2 ) sn 2 a(r 3 r 1 ) 6 (x + r ct) where the moduli for sn are m 7 m 8 = (c 2 c 1 )/(c 3 c 1 ) in (41). = (c 3 c 2 )/(c 3 c 1 ) and = 3(1+kr2 c) 3(1+kr2 c)

8 8 Abstract and Applied Analsis tanh 2 (1 + kr2 c) 4 = 3(1+kr2 c) (x+r ct) [ 1 tanh 2 (1 + kr2 c) (x + r ct)] 4 [ ] = 3(1+kr2 c) 1 cosh 2 ((1+kr 2 c)/4)(x+r ct) = 3(1+kr2 c) 1 (1/2) [cosh ((1 + kr 2 c)/)(x+r ct)+1] 3(1+kr 2 c) = a[1+cosh ((1+kr 2 c)/)(x+r ct)] =u 1 (x,,t). (44) When m 6 = (r 2 r 1 )/(r 3 r 1 ) 1,itmeansr 2 =r 3 ; then, we calculate r 2 =r 3 =0and r 1 = 3(1 + kr 2 c)/,and substituting r i (i = 1, 2, 3) into u 6 we get u 6 =u 1. Proof of Case 3. When m 7 = (c 3 c 2 )/(c 3 c 1 ) 1,it means c 1 =c 2 and sn = tanh; then, we calculate c 1 =c 2 = (1 + kr 2 c Δ)/ and c 3 =(1+kr 2 c+2 Δ)/, and substituting c i (i = 1, 2, 3) into u 7 admits to u 3 as follows: u 7 (x,,t) =c 3 (c 3 c 2 ) sn 2 a(c 3 c 1 ) 6 = 1+kr2 c+2 Δ =(1+kr 2 c Δ (x+r ct) 3 Δ tanh2 Δ (x + r ct) 4 +3 Δ(1 tanh 2 Δ 4 (x+r ct))) () 1 = 1+kr2 c Δ 3 Δ + a[1+cosh Δ/ (x + r ct)] =((1+kr 2 c Δ) cosh Δ (x+r ct)+1 +kr 2 c+5 Δ) ([ cosh Δ (x+r ct)+1 ]) [ ] =u 3 (x,,t). 1 (45) The results provide a manner that we can get bell-shaped waves from periodic waves for some nonlinear development equations. Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work was supported b the National Natural Science Foundation of China (no ) and Guangdong Province (no. 2013KJCX0189 and no. Yq ). The author would like to thank the editors for their hard working and the anonmous reviewers for helpful comments and suggestions. References [1] T. B. Benjamin, J. L. Bona, and J. J. Mahon, Model equations for long waves in nonlinear dispersive sstems, Philosophical Transactions of the Roal Societ of London A. Mathematical and Phsical Sciences,vol.272,no.1220,pp.47 78,1972. [2]B.B.KadomtsevandV.I.Petviashvili, Onthestabilitof solitar waves in weakl dispersive media, Soviet Phsics Doklad, vol. 15, pp , [3] A.-M. Wazwaz, Exact solutions of compact and noncompact structures for the KP-BBM equation, Applied Mathematics and Computation,vol.169,no.1,pp ,2005. [4] A.-M. Wazwaz, The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equations, Chaos, Solitons and Fractals,vol.38,no.5,pp , 2008.

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