Weiguo Rui. 1. Introduction

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1 Hindawi Pblishing Corporation Mathematical Problems in Engineering Volme 215 Article ID pages Research Article Frobenis Idea Together with Integral Bifrcation Method for Investigating Eact Soltions to a Water Wave Model of the Generalized mkdv Eqation Weigo Ri College of Mathematics Chongqing Normal University Chongqing China Correspondence shold be addressed to Weigo Ri; weigorhh@aliyn.com Received 19 May 214; Accepted 3 Jly 214 Academic Editor: Salvatore Alfonzetti Copyright 215 Weigo Ri. This is an open access article distribted nder the Creative Commons Attribtion License which permits nrestricted se distribtion and reprodction in any medim provided the original work is properly cited. By sing Frobenis idea together with integral bifrcation method we stdy a third order nonlinear eqation of generalization form of the modified KdV eqation which is an important water wave model. Some eact traveling wave soltions sch as smooth solitary wave soltions nonsmooth peakon soltions kink and antikink wave soltions periodic wave soltions of Jacobian elliptic fnction type and rational fnction soltion are obtained. And we show their profiles and discss their dynamic properties aim at some typical soltions. Thogh the types of these soltions obtained in this work are not new and they are familiar types they did not appear in any eisting literatres becase the eqation t + + t +β + α + (1/3) α( +2 )+3μα μα 2 ( ) + 2 μα 2 ( ) = is very comple. Particlarly compared with the cited references all reslts obtained in this paper are new. 1. Introdction It has come to light that many problems in nonlinear science associated with mechanical strctral aeronatical oceanic electricalandcontrolsystemscanbesmmarizedassolving nonlinear partial differential eqations (PDEs) which arise from important models with mathematical and physical significances. Finding their eact soltions has etensive applications in many scientific fields sch as hydrodynamics condensed matter physics solid-state physics nonlinear optics nerodynamics crystal dislocation model of meteorology water wave model of oceanography and fibre-optic commnication. The research methods for solving nonlinear PDEs deal with the inverse scattering transformation 1 2 the Darbo transformation 3 5 the Bäcklnd transformation 5 8 the bilinear method and mltilinear method 9 1 the classical and nonclassical Lie grop approaches theclarkson-krskaldirectmethod13 15 the miing eponential method 16 the geometrical method the trncated Painleve epansion 19 2 the fnction epansion method (inclding tanh epansion method sine-cosine epansion method ep-fnction method 25 and mltiple ep-fnction method 26) the bifrcation theory of planar dynamical system the F-epansion type method 29 3 G /G method and the integral bifrcation method Among these available methods for solving nonlinear PDEs some of them employed Frobenis idea directly or indirectly. Frobenis idea is aso called Frobenis integrable decompositions 37; it can redce a partial differential eqation (PDE) to an ordinary differential eqation (ODE) nder investigation for soltion. Indeed the F-epansion type methods indirectly employed Frobenis idea; crcial points of this method are to choose integrable ODE to start investigation for soltion. In fact the tanh fnction method and G /G method are special cases of sch an idea or general Frobenis idea. Direct Frobenis idea was also sed to establish the transformed rational fnction method 38 and to solve the KPP eqation 39.

2 2 Mathematical Problems in Engineering In this paper we will employ Frobenis idea together with integral bifrcation method to investigate eact traveling wave soltions of the following integrable generalization of the modified KdV eqation: t + + t +β +α α( +2 ) + 3μα μα 2 ( ) (1) 2. Direct Application of Frobenis Idea on Redcing the PDE (1) to an Integrable ODE Frobenis idea is abot changing a partial differential eqation (PDE) into an ordinary differential eqation (ODE) and then sing integrable decomposition method to investigate its eact soltions. Ths in this section we first employ the direct Frobenis idea to change(1) into an integrable ordinary differential eqation; see the following discssions. Making a traveling wave transformation ( t) = (ξ) with ξ= ct(1) can be redced to the following ordinary differential eqation (ODE): + 2 μα 2 ( )= where α β μand are constants and <α<1.themodel (1) comes from the physical and asymptotic considerations via the methodology introdced by Fokas 4in 1995;it can be regarded as a water wave model to describe the motion of waterwave.itisworthtopointotthatthespecialcaseof(1) (1 c) d dξ +(β c) d3 dξ 3 +αd dξ α(d3 dξ 3 +2d d 2 dξ dξ 2 ) + 3μα2 2 d dξ + μα 2 2 d 3 3 dξ 3 +(d dξ ) +4 d d 2 dξ dξ 2 (3) t + + t +β +α α( +2 ) + 3μα 2 2 = is also an important physical model. The above two eqations were stdied by many athors. Eqation (2) was introdced by Fchssteiner and Fokas in their previos works in The La pairs of (2) were given by Fokas in 4. New La pairs and Darbo transformation of (2) were introdcedbyyangandriin 43 recently. In44 by sing the bifrcation theory of dynamical system the eistence conditions of different kinds of traveling wave soltions of (2) were presented by Bi. In 45 by sing the same method Li and Zhang stdied(1) the eistence of solitary wave kink and antikink wave soltions ncontably infinite many smooth and nonsmooth periodic wave soltions were discssed. However eact travelling wave soltions of (1) were not obtained in 45 thogh the athors obtained some reslts of nmerical simlation for smooth and nonsmooth periodic wave soltions in this work. Moreover the investigations on eact soltions of (1) are few in the eisting literatres becase (1) is more comple than (2). Therefore in this paper employing Frobenis idea together with integral bifrcation method we will investigate different kinds of eact traveling wave soltions of (1). The rest of this paper is organized as follows. In Section 2 by sing Frobenis idea we will derive ordinary differential eqation (ODE) which is eqivalent to (1). In Section 3 by sing the integral bifrcation method combined with factoring techniqe we will investigate different kinds of eact traveling wave soltions of (1) and discss their dynamic properties when the integral constants satisfy different conditions. In Section 4 we will discss different kinds of eact traveling wave soltions of (1) nder the special case of the parameter =. (2) +μ 2 α 2 ( d d 3 2 dξ )2 dξ 3 +2d dξ (d2 dξ 2 ) = where c is wave velocity which moves along the direction of -ais and c =.Eqation(3)canberewrittenas (1 c) d dξ +(β c) d3 dξ αd( ) dξ α d dξ d2 dξ (d dξ ) +μα 2 d(3 ) dξ + μα 2 d d 2 2 dξ 2 dξ 2 +(d dξ ) +μ 2 α 2 d dξ (d dξ )2 d 2 dξ 2 =. Integrating (4)oncewe obtain (1 c) +(β c) d2 dξ α αd2 dξ (d dξ ) +μα μα 2 2 d 2 2 dξ 2 +(d dξ ) +μ 2 α 2 ( d dξ )2 d 2 dξ 2 =g where g is an integral constant. Employing direct Frobenis idea we need not change(5) into a 2-dimensional planar system as the method in we can directly integrate (5) again; see the following calcls. (4) (5)

3 Mathematical Problems in Engineering 3 Mltiplying d/dξ to the both sides of (5) yields (1 c) d dξ d d 2 +(β c) dξ dξ d 2 α2 dξ αd d 2 dξ dξ (d dξ ) +μα 2 3 d dξ + μα2 2 d d 2 3 dξ dξ 2 +(d dξ ) (6) where the coefficients r r 1 r 2 s s 1 ands 2 are defined by the following epressions: c β+μ(1 c) 1/18 (c β) ( μ) r = α 2 ( μ) μ + ( μ) r 1 = r 2 =1 μ 3α ( μ) ( μ) +μ 2 α 2 ( d dξ )3 d 2 dξ 2 =gd dξ. Integrating (6)oncewe obtain g+ 1 2 (1 c) (β c)(d dξ ) α α(d dξ ) μα μ2 α 2 ( d 4 dξ ) μα2 2 ( d =h where h is another arbitrary integral constant. When = (7)canberewrittenas 4h 4g (1 c) α μα (β c) ( d 2 dξ ) + 2α 2 3 (d dξ ) + 2μα 2 2 ( d +μ 2 α 2 ( d dξ ) 4 =. 3. Different Kinds of Eact Traveling Wave Soltions of (1) In this section by sing the integral bifrcation method combinedwithfactoringtechniqeasin36 we will investigate different kinds of eact traveling wave soltions of (1) and discss their dynamic properties via (7)and(8) Hyperbolic Fnction Soltions and Periodic Wave Soltions of (1) as the Two Integral Constants g =and h =. When ( μ)>and h = (324μ 2 (c 1) 2 + μ(648c 2 648β+ 324β c c 2 2 )+ +36(β c))/1296(μ )α 2 μ 2 = g = (18μ(c 1) + 1)/18αμ 2 =(8) canbe decomposed in the following form: r +r 1 +r 2 2 +μ 2 ( d s +s 1 +s 2 2 +α 2 ( d = (7) (8) (9) c β+μ(1 c) 1/18+ (c β) ( μ) s = μ 2 ( μ) α μ+ ( μ) s 1 = s 2 = α2 3μ 2 ( μ) μ α2 μ ( μ). (1) Ths (9) can be redced to the following two ordinary differential eqations: or r +r 1 +r 2 2 +μ 2 ( d = (11) s +s 1 +s 2 2 +α 2 ( d = (12) where the coefficients r r 1 r 2 s s 1 ands 2 are given by (1). Solving (11) nder the conditions r 2 <and q=4r r 2 r 2 1 > we obtain two hyperbolic fnction soltions of (1)as follows: r 2 1 =± 1 2r 2 q sinh ( r 2 μ ( ct)) r 1. (13) Solving (11) nder the conditions r 2 >and q=4r r 2 <weobtaintwoperiodicwavesoltionsof(1) as follows: = 1 2r 2 q sin ( r 2 μ ( ct)) ±r 1. (14) Similarly solving (12) nder the conditions s 2 <and q = 4s s 2 s 2 1 > we obtain two hyperbolic fnction soltions of (1) as follows: s 2 1 =± 1 2s 2 q sinh ( s 2 α ( ct)) s 1. (15) Solving (12) nder the conditions s 2 >and q =4s s 2 <weobtaintwoperiodicwavesoltionsof(1) as follows: = 1 2s 2 q sin ( s 2 α ( ct)) ±s 1. (16)

4 4 Mathematical Problems in Engineering 3.2. Hyperbolic Fnction Soltions and Periodic Wave Soltions of (1) as the Two Integral Constants g= and h =. When 2 μ > and g = h = (18μ 18μβ ) 2 / 1296α 2 μ 3 ( μ) = c = 1 (1/18μ)(8)canbedecomposed in the following form: μ μβ /18 α(μ + 2 μ) + μ 2 ( μ+ 2 μ) 3μ 2 2 μ = μ 2α 2 ( 2 μ) αω α(μ + 2 μ) 2 3μ sin (ω 2ξ) ± 3μ 2 μ (21) where the Ω 1 Ω 2 ω 1 ω 2 andξ are defined by + α2 2 μ (1 μ ) 2 +α 2 ( d 2 dξ ) μ μβ /18 α 2 ( μ 2 μ) + μ+ 2 μ 3α 2 μ (17) Ω 1 = 36μ (β )+ 2 μ + 36μ ( β)+ 2 μ μ( μ+ 2 μ) Ω 2 ω 1 = 2 μ μ +(μ+ μ 2 μ) 2 +μ 2 ( d 2 dξ ) =. = 36μ (β )+ 2 μ + 36μ ( β)+ 2 μ μ( μ+ 2 μ) Eqation (17) canberedcedtothefollowingtwoordinary differential eqations: or μ μβ /18 α(μ + 2 μ) + μ 2 ( μ+ 2 μ) 3μ 2 2 μ + α2 2 μ (1 μ ) 2 +α 2 ( d 2 dξ ) = μ μβ /18 α 2 ( μ 2 μ) + μ+ 2 μ 3α 2 μ +(μ+ μ 2 μ) 2 +μ 2 ( d 2 dξ ) =. (18) (19) Solving (18) we obtain two hyperbolic fnction soltions and two periodic wave soltionsof (1) as follows: μ = ± 2α 2 ( 2 μ) αω α(μ + 2 μ) 1 3μ sinh (ω 1ξ) (2) 3μ 2 2 μ ω 2 = 2 μ ξ = (1 1 μ 18μ )t. (22) Similarly solving (19) we also obtain two hyperbolic fnction soltions and two periodic wave soltions of (1) as follows: = ± 2μ ( + 2 μ) μω 3 3α sinh (ω 3ξ) μ+ 2 μ 3α 2 μ = 2μ ( + 2 μ) μω 4 3α sin (ω 4ξ) ± μ+ 2 μ 3α 2 μ where the Ω 3 Ω 4 ω 3 ω 4 andξ are defined by Ω 3 (23) (24) = 36 ( β) 2 μ +36 ( β) 2 /μ μ 2 μ ω 3 = 1 μ (μ+ μ 2 μ)

5 Mathematical Problems in Engineering 5 Ω 4 where the Ω 5 Ω 6 ω 5 ω 6 andξ are defined by = 36 ( β) 2 μ +36 ( β) 2 /μ μ 2 μ ω 4 = 1 μ μ+ μ 2 μ ξ= (1 1 18μ )t. (25) Ω 5 = 72μ ( β) (32 μ) + 72μ (β )+3 2 μ μ( μ+ 2 μ) ω 5 = 2 μ μ Ω Hyperbolic Fnction Soltions and Periodic Wave Soltions of (1) as the Two Integral Constants g = and h=. When 2 μ >and g=(+18μ(β ))/18αμ 2 2 μ = h = c = ( +18(β μ) μ)/18( μ+ 2 μ)(8)canbedecomposedinthefollowingform: 18μ ( β) α(μ + 2 μ) + 9μ 2 ( μ+ 2 μ) 3μ 2 2 μ α 2 ( 2 μ) + 2 +α 2 ( d 2 μ dξ ) μ+ 2 μ +(1+ μ 3α 2 μ 2 μ) 2 +μ 2 ( d 2 dξ ) =. (26) Similarly solving (26) we obtain for hyperbolic fnction soltionsandforperiodic wavesoltionsof (1) as follows: = ± μ 2α 2 ( 2 μ) = 72μ ( β) (32 μ) + 72μ (β )+3 2 μ μ( μ+ 2 μ) ω 6 = 2 μ μ ξ= +18 (β μ)+18 2 μ t 18 ( μ+ 2 μ) 2 μ + 2 μ = ± 6αμ ( + 2 μ) 2 μ = sinh ( + 2 μ ξ) 1 μ 2 μ + 2 μ 6αμ ( + 2 μ) 2 μ sin ( + 2 μ ξ) ± 1 μ (29) (3) (31) where ξ= (( +18(β μ) μ)/18( μ+ 2 μ))t. = αω α( μ 2 μ) 5 3μ sinh (ω 5ξ) ± 3μ 2 2 μ μ 2α 2 ( 2 μ) (27) 3.4. Hyperbolic Fnction Soltions Periodic Wave Soltions and Rational Fnction Soltion of (1) as the Two Integral Constants g=h=. (i) When μ= β> and g=h= c=β/(8)canbedecomposedinthefollowingform: α 3 2 ( (β )) + α2 2 +α 2 ( d αω α( μ 2 μ) 6 3μ sin (ω 6ξ) 3μ 2 2 μ (28) (β ) + 2 +μ 2 ( d 2 3α dξ ) =. (32)

6 6 Mathematical Problems in Engineering Solving (32) we obtain for hyperbolic fnction soltions and for periodic wave soltions of (1) as follows: = 6(β ) cos 2 ( 1 2μ B 6( β) ξ) (4) =± (β ) 6α sinh ( ξ ) 1 ( <<β) (33) ( < β < B>) = 6(β ) cosh 2 ( 1 2μ B 6(β ) ξ) (β > >B>) (41) = (β ) 6α = (β ) 6α sin ( ξ )±1 (β> >) (34) sinh ( ξ μ )±1 ( <<β) (35) where ξ = (β/)t and A B have been given above. (iii)when μ = and g = h = c = β/ (8) canbe decomposed in the following form: 3α ( β) 2 + α2 2 +α 2 ( d 2 dξ ) 2 3α +μ2 ( d =. (42) =± (β ) 6α sin ( ξ μ ) 1 (β > >) (36) Solving (42) we obtain a hyperbolic fnction soltion a periodic wave soltion and a rational fnction soltion as follows: = 3(β ) cos 2 ( 1 ξ) ( >) (43) α 2 where ξ = (β/)t. (ii)when μ= /18(β ) β> and g=h= c=β/ (8)canbedecomposedinthefollowingform: A αa 6(β ) 2 +α 2 ( d 2 dξ ) B B 6(β ) 2 +μ 2 ( d 2 dξ ) = (37) where A = 6α(β )18(β ) 3 36(β ) 2 + 2(β )/ 2 B = (18(β )+3 36(β ) 2 + 2(β ))/54α(β ). Solving (37) we obtain two hyperbolic fnction soltions and two periodic wave soltions of (1) as follows: = 6(β ) α = 6(β ) α cos 2 ( 1 2 A 6α ( β) ξ) ( < β < A>) cosh 2 ( 1 2 A 6α (β ) ξ) ( <<βa<) (38) (39) = 3(β ) cosh 2 ( 1 ξ) ( <) (44) α 2 = 1 6αμ 2 ξ2 (45) where ξ = (β/)t and A B have been given above. All the above eact soltions which were obtained by s are smooth travelling wave soltions inclding smooth periodic wave soltions and smooth hyperbolic fnction soltions. In order to show the dynamical profiles of periodic wave soltions as eamples we plot the graphs of soltions (14) and(38) forα =.5 β =.8 μ = 1.2 t =.1 which are shown in Figres 1(a) and 1(b) Peakon Soltions nder Some Special Parametric Condition. The epression in the right side of (8) cannotberedced to a form of a 2 b(d/dξ) 2 2 =bysing the factoring techniqe becase this eqation contains the terms (4h/) (4g/) + (2α/3) 3 + (2α/3)(d/dξ) 2.Thsthepeakon soltions of (1)schasCammasa-Holm sformre δ ξ cannot be obtained by direct integral method together with factoring techniqe as in 36. However the research works given by Li et al. in 45 show that the peakon soltions of (1) eist thogh they did not obtain eact peakon soltions of this eqation. Indeed the eistence of peakon soltion of (1)isprovedbyLi viaanalysisofphaseportraitsinthispaper.wenoticethatthe terms 3 and (d/dξ) 2 are kindred terms when =re δ ξ. Ths we assme that (1) has peakon soltions of the form =re δ ξ or =s+re δ ξ.

7 Mathematical Problems in Engineering (a) Periodic wave defined by (14) (b) Periodic wave defined by (38) Figre 1: The graphs of profiles for the smooth periodic wave soltions defined by (14). (a) When g =and h =wespposethat(8) hasa peakon soltion as the following form: =s+re δ ξ (46) where the parameters s randδ can be determined frther in the below discssions. We define s+re δ ξ =s+rep( δ ξ 2 ) when ξ =;particlarly=s+r=(constant) when ξ= and obviosly =s+ris a constant soltion which satisfies (1). When ξ =sbstitting(46)(i.e.=s+rep( δ ξ 2 )) into (8)weobtain C +C 1 ep ( δ ξ 2 )+C 2 ep ( 2δ ξ 2 ) +C 3 ep ( 3δ ξ 2 )+C 4 ep ( 4δ ξ 2 )= where coefficients C C 1 C 2 C 3 and C 4 satisfy C = 4h 4gs + 2αs3 2 (1 c) s2 + 3 C 2 = C 1 = 4 (1 c) rs 2 (1 c) r2 + 2αrs + 2αr2 s + μα2 s 4 4gr + 4μα2 rs 3 + 6μα2 r 2 s 2 + 2μα 2 r 2 s 2 δ 2 + 2(β c)r2 δ 2 + 2αr2 sδ 2 3 (47) C 3 = μα2 r 4 + 2μα 2 r 4 δ 2 +μ 2 α 2 r 4 δ 4 C 4 = μα2 r 4 + 2μα 2 r 4 δ 2 +μ 2 α 2 r 4 δ 4. (48) Let the coefficients of every terms of ep-fnction (inclding the term of constant) in (47)aszero;itfollows C = C 1 = C 2 = C 3 = C 4 =. (49) Solving the above grop of eqations yields h g c= (2 3μ + 36μ 2 36βμ)+( 36βμ) 2 μ 36μ (μ 2 μ) = (2 +μ 36μ βμ)+( + 36βμ 36μ) 2 μ 216μ 2 (μ 2 μ) = (32 +μ 72μ βμ)+(3 + 72βμ 72μ) 2 μ 5184μ 3 α 2 (μ 2 μ) r=r(free parameter) s= 1 6αμ δ= 2 μ μ (5)

8 8 Mathematical Problems in Engineering (a) Peakon wave defined by (51) (b) Peakon wave defined by (57) Figre 2: The graphs of profiles for the nonsmooth peakon soltions defined by (51) and(57) nder different parametric vales: (a) = 1.4; (b) = 1.4. where <<μ.thswhentheconstantsc gand h satisfy the above conditions (1) has a peakon soltion as follows: = 1 2 6αμ +rep ( μ ct ) (51) μ where r is an arbitrary nonzero constant and c is given above. (b)when g = h = and β can be regarded as a free parameter we sppose that (8) hasapeakonsoltionasthe following form: = re δ ξ (52) where the parameters r δ can be determined frther in the below discssions. When ξ =sbstitting(52) (i.e.= re δ ξ 2 )into(8) we obtain A ep ( 2δ ξ 2 )+Bep ( 3δ ξ 2 ) +C ep ( 4δ ξ 2 )= where coefficients A BandC satisfy A= 2 (1 c) r2 + 2(β c) r2 δ2 B= 2α r α r3 δ2 C= μα2 r 4 + 2μα 2 r 4 δ2 +μ 2 α 2 r 4 δ4. (53) (54) Let the coefficients of every terms of ep-fnction in (53) as zero; it follows A = B = C =. (55) Solving the above grop of eqations yields 1 β=μ= < δ = r = r c = c. (56) Ths when the constants g=h=and β=μ= δ = 1/ (1) has a peakon soltion as follows: = r ep ( 1 ct ) (57) where r and c are arbitrary nonzero constants. In order to show the dynamical profiles of peakon soltions (51) and(57) we plot the graphs of them for α =.5 β =.8 μ =.2 r = 5 c = 4 t =.1 whichare shown in Figres 2(a) and 2(b). 4. Different Kinds of Eact Soltions nder the Special Case = Under the special case of parameter =(7)canberewritten as h g (1 c) α μα β(d dξ ) =. (58) Solving (58) in different kinds of parametric conditions we obtain different kinds of eact traveling wave soltions inclding solitary wave soltions and kink wave soltions; see the below discssions.

9 Mathematical Problems in Engineering Different Kinds of Eact Traveling Wave Soltions for the Constants g=h=. When two integral constants g h are both zero (i.e. g=h=) (58)canberedcedto ( d = c 1 β 2 α 3β 3 μα2 2β 4. (59) Under different parametric conditions solving (59) we obtain a series of eact traveling wave soltions as follows: = 6 (c 1) sech 2 ((1/2) (c 1) /βξ) 2α + 9αμ (c 1) 1 ± tanh ((1/2) (c 1)/βξ) 2 where c>1 β>andξ= ct: = (6) 6 (c 1) α1 18μ (1 c) 1sinh ( (c 1) /βξ) (61) where c>1 β> μ < 1/18(c 1)andξ= ct: = 6 (c 1) α1+ 18μ (c 1) +1sin ( (1 c) /βξ) (62) where c<1 β> μ < 1/18(c 1)andξ= ct: = 6 (c 1) α1 18μ (c 1) +1cosh ( (c 1) /βξ) (63) where c>1 β> μ > 1/18(c 1)andξ= ct: = 6 (c 1) α1 18μ (c 1) +1cos ( (1 c) /βξ) (64) where c<1 β> μ > 1/18(c 1)andξ= ct: = 3(c 1) (αcosh 2 ( 1 2 c 1 β ξ) μ (1 c) 3 2 where c>1 β> μ<and ξ= ct: = 3(c 1) (αcos 2 ( 1 2 c 1 β ξ) μ (c 1) +3 2 sinh ( c 1 1 β ξ)) sin ( 1 c 1 β ξ)) (65) (66) where c < 1 β > μ < and ξ= ct: or = 1 6αμ 1 ± tanh ( ξ) (67) βμ = 1 6αμ 1±coth ( ξ) (68) βμ where β>μ<andξ = (1 (1/18μ))t. Among these eact travelling wave soltions obtained in Section 4.1 it is worth pointing that the soltions (6) and (63) describe smooth solitary waves and the soltion (67) describes kink wave and antikink wave (Figre 4). In order to show the dynamic profiles of soltions (6) and(63) we plot their graphs for α =.5 β =.8 c = 5 and t =.1 which are shown in Figres 3(a) and 3(b). In order to show the dynamic profiles of soltion (67) we plot its graphs for α =.5 β =.8 μ=.2and t=.1 which are shown in Figres 3(a) and 3(b) Eact Traveling Wave Soltions for the Constants g = and h=. When g =and h=(58)canberedcedto g (1 c) α μα β(d dξ ) =. (69) If βμ < then(69)canberewrittenas d 3 + (2/3αμ) 2 (2(c 1) /μα 2 ) (4g/μα 2 ) =±α μ 2β dξ. (7) By sing factoring method (7) can be rewritten in the following two forms: d ( ) ( z )( z 1 )( z 1 ) or d ( z ) ( )( z 1 )( z 1 ) =±α μ 2β dξ (z <) =±α μ 2β dξ (z >) (71) (72) where z z 1 and z 1 arerootsoftheeqation 3 +(2/3αμ) 2 (2(c 1)/μα 2 ) (4g/μα 2 ) = and the z is real root; the z 1 z 1 are conjgate comple roots. z z 1 and z 1 can be epressed by the parameters α μ c and g andherewe

10 1 Mathematical Problems in Engineering (a) Bright solitary wave defined by (6) 4 (b) Dark solitary wave defined by (63) Figre 3: The graphs of profiles for smooth solitary wave soltions defined by (6)and(63) nder different parametric vales: (a) μ =.7; (b) μ= (a) Kink wave (b) Antikink wave Figre 4: The graphs of profiles for kink wave and antikink wave soltions defined by (67)with + and. omit their epressions becase they are very comple. Bt we can always obtain their vales by sing compter once the parameters α μ c and g arefiedonconcretevales.for eample taking α =.8 c = 5 μ = 5 g= 4wegetz = z 1 = iand z 1 = i; takingα=.8 c=5μ= 5 g = 4 wegetz = z 1 = iand z 1 = i.

11 Mathematical Problems in Engineering 11 Solving (71) and (72) we obtain two periodic wave soltionsof (1) as follows: =z z 1 cn ( ω 1ξ m 1 ) 1 where m 1 ( z 1 cn ( ω 1ξ m 1 ) 1 + z 2 2z Re (z 1 )+ z cn ( ω 1 ξ m 1 )±1) (73) ω 1 =α μ z 1 z 2 2z Re (z 1 )+ z 1 2 = z 1 2 2z Re (z 1 )+2 z 1 z 2 2z Re (z 1 )+ z 1 2. z 1 z 2 2z Re (z 1 )+ z 1 2 (74) If βμ > then(69)canberewrittenas d 3 + (2/3αμ) 2 (2(c 1) /μα 2 ) (4g/μα 2 ) =±α μ 2β dξ. 2β (75) By sing factoring method (75) can be rewritten in the following two forms: d ( ) ( z )( z 1 )( z 1 ) or d (z )( ) ( z 1 )( z 1 ) =±α μ 2β dξ (z <) =±α μ 2β dξ (z >) (76) (77) where z z 1 and z 1 are same as the above case. Solving (76) and (77) we obtain two periodic wave soltionsof (1) as follows: =z z cn ( ω 2ξ m 2 ) ( z cn ( ω 2ξ m 2 ) + z 2 2z Re (z 1 )+ z cn ( ω 2 ξ m 2 )) (78) where m 2 = z z 2 2z Re (z 1 )+ z cn ( ω 2 ξ m 2 ) ( z 1 1 cn ( ω 2ξ m 2 ) + z 2 2z Re (z 1 )+ z cn ( ω 2 ξ m 2 )) 1 ω 2 =α μ z 1 z 2 2z Re (z 1 )+ z 1 2 = 1 2 2z Re (z 1 ) 2 z z 1 z 2 2z Re (z 1 )+ z 1 2 2β (79) z 1 z 2 2z Re (z 1 )+ z 1 2. (8) 4.3. Eact Traveling Wave Soltions for the Constants g = and h =. When g =and h =(58)canberewrittenas d ( μα2 2β μα 3 2 (c 1) μα 2 2 4g 1/2 4h μα2 μα 2 ) =±dξ. (81) The types of soltions of (81) contain many cases de to the roots of the qartic eqation 4 + (2/3μα) 3 (2(c 1)/μα 2 ) 2 (4g/μα 2 ) (4h/μα 2 ) = which is in (81) and have many possibilities bt all soltions of (1) nder different kinds of cases are periodic soltions of Jacobian elliptic fnction type sch as the forms of soltions (73) (78) and (79). For convenience to discss here we only consider one case which this qartic eqation has for real roots. Obviosly we can always obtain the real roots of this qartic eqation by sing compter once the parameters α β μ c g and h arefiedonconcretevales.spposing φ 1 φ 2 φ 3 and φ 4 are for real roots of this qartic eqation we will obtain different kinds of periodic wave soltions of Jacobian elliptic fnction type for (1); see the following discssions. Case 1. Under βμ < respectively taking different root of the φ 1 φ 2 φ 3 and φ 4 as initial vale to integrate (81) yields d =±α μ ξ φ 1 ( φ 1 )( φ 2 )( φ 3 )( φ 4 ) 2β d ξ (82) where >φ 1 >φ 2 >φ 3 >φ 4 and φ 2 d =±α μ ξ (φ 1 ) (φ 2 ) ( φ 3 )( φ 4 ) 2β d ξ (83)

12 12 Mathematical Problems in Engineering where φ 1 >φ 2 > φ 3 >φ 4 and φ 3 d =±α μ ξ (φ 1 ) (φ 2 ) ( φ 3 )( φ 4 ) 2β d ξ (84) where φ 1 >φ 2 >φ 3 >φ 4 and φ 4 d =±α μ ξ (φ 1 ) (φ 2 ) (φ 3 ) (φ 4 ) 2β d ξ (85) where φ 1 >φ 2 >>φ 3 >φ 4 >. Respectively completing the above integrals in (82) (83) (84) and (85) we obtain for periodic wave soltions of Jacobian elliptic fnction type of (1) as follows: = (φ 1φ 2 φ 2 φ 4 ) sn 2 (Ω 1 ξ k 1 )+(φ 1 φ 4 φ 1 φ 2 ) (φ 1 φ 4 ) sn 2 (Ω 1 ξ k 1 )+(φ 4 φ 2 ) = (φ 1φ 2 φ 1 φ 3 ) sn 2 (Ω 1 ξ k 1 )+(φ 2 φ 3 φ 1 φ 2 ) (φ 2 φ 3 ) sn 2 (Ω 1 ξ k 1 )+(φ 3 φ 1 ) = (φ 1φ 2 φ 2 φ 4 ) sn 2 (Ω 1 ξ k 1 )+(φ 1 φ 4 φ 1 φ 2 ) (φ 1 φ 4 ) sn 2 (Ω 1 ξ k 1 )+(φ 4 φ 2 ) = (φ 2φ 4 φ 3 φ 4 ) sn 2 (Ω 1 ξ k 1 )+(φ 3 φ 4 φ 2 φ 3 ) (φ 2 φ 3 ) sn 2 (Ω 1 ξ k 1 )+(φ 4 φ 2 ) = (φ 1φ 2 φ 2 φ 4 ) sn 2 (Ω 1 ξ k 1 )+(φ 1 φ 4 φ 1 φ 2 ) (φ 1 φ 4 ) sn 2 (Ω 1 ξ k 1 )+(φ 4 φ 2 ) = (φ 1φ 3 φ 3 φ 4 ) sn 2 (Ω 1 ξ k 1 )+(φ 3 φ 4 φ 1 φ 4 ) (φ 1 φ 4 ) sn 2 (Ω 1 ξ k 1 )+(φ 3 φ 1 ) (86) where Ω 1 = ±(α/2) μ(φ 1 φ 3 )(φ 2 φ 4 )/2β and k 1 = (φ 2 φ 3 )(φ 1 φ 4 )/(φ 1 φ 3 )(φ 2 φ 4 ). Case 2. Under βμ > respectively taking different root of the φ 1 φ 2 φ 3 and φ 4 as initial vale to integrate (81) yields φ 1 d =±α μ ξ (φ 1 ) ( φ 2 )( φ 3 )( φ 4 ) 2β d ξ (87) where φ 1 > φ 2 >φ 3 >φ 4 and φ 2 d =±α μ ξ (φ 1 ) ( φ 2 )( φ 3 )( φ 4 ) 2β d ξ (88) where φ 1 >φ 2 >φ 3 >φ 4 and φ 3 d =±α μ ξ (φ 1 ) (φ 2 ) (φ 3 ) ( φ 4 ) 2β d ξ (89) where φ 1 >φ 2 >φ 3 > φ 4 and φ 4 d =±α μ ξ (φ 1 ) (φ 2 ) (φ 3 ) ( φ 4 ) 2β d ξ (9) where φ 1 >φ 2 >φ 3 >φ 4. Similarly completing the above integrals in (87) (88) (89) and (9) we obtain another for periodic wave soltions of Jacobian elliptic fnction type of (1) as follows: = (φ 1φ 4 φ 2 φ 4 ) sn 2 (Ω 2 ξ k 2 )+(φ 1 φ 2 φ 1 φ 4 ) (φ 1 φ 2 ) sn 2 (Ω 2 ξ k 2 )+(φ 2 φ 4 ) = (φ 1φ 3 φ 2 φ 3 ) sn 2 (Ω 2 ξ k 2 )+(φ 2 φ 3 φ 1 φ 2 ) (φ 1 φ 2 ) sn 2 (Ω 2 ξ k 2 )+(φ 3 φ 1 ) = (φ 2φ 3 φ 2 φ 4 ) sn 2 (Ω 2 ξ k 2 )+(φ 3 φ 4 φ 2 φ 3 ) (φ 3 φ 4 ) sn 2 (Ω 2 ξ k 2 )+(φ 4 φ 2 ) (91) = (φ 1φ 3 φ 1 φ 4 ) sn 2 (Ω 2 ξ k 2 )+(φ 1 φ 4 φ 3 φ 4 ) (92) (φ 3 φ 4 ) sn 2 (Ω 2 ξ k 2 ) + (φ 3 φ 1 ) where Ω 2 = ±(α/2) μ(φ 1 φ 3 )(φ 2 φ 4 )/2β and k 2 = (φ 1 φ 2 )(φ 3 φ 4 )/(φ 1 φ 3 )(φ 2 φ 4 ). In Sections 4.2 and 4.3 all the eact soltions obtained by s are periodic soltions of Jacobian elliptic fnction types. As an eample we plot the graphs of profiles of the soltions (78)and(79)forα =.3 β =.8 c = 1 μ = 2and t=.1 which are shown in Figres 5(a) and 5(b). 5. Conclsions Thogh Frobenis idea is a well-known general method it can solve some very comple PDE models with highly nonlinear terms and high order terms sch as (1) whenitcombines with the integral bifrcation method. In this work by sing Frobenis idea together with integral bifrcation method we stdy the third order nonlinear water wave model (1). Under different kinds of parametric conditions we obtain eight types of eact travelling wave soltions inclding the smooth solitary wave soltions (6) (63) and (65) the nonsmooth peakon wave soltions (51) and(57) the kink wave and antikink wave soltions (67) and(68) the smooth periodic wave soltions of trigonometric fnction type (14) (16) (21) (24) (28) (31) (34) (36) (38) (4) and (43) the nonsmooth periodic wave soltions of trigonometric fnctiontype (62) (64) and (66) the periodic wave soltions of Jacobian elliptic fnction type (78) (86) (91) and (92) the hyperbolic fnction soltions (13) (15) (2) (23) (27) (3) (33) (35) (44) and (61) and the rational fnction soltion (45). Thogh the types of these soltions obtained in this work are not new and they are familiar types the reslts of (1) obtained by s in this paper did not appear in any eisting literatres. Particlarly compared with reference 45 all reslts obtained in this paper are new. Among these soltions obtained in this paper some of them have direct

13 Mathematical Problems in Engineering (a) Periodic wave defined by (78) (b) Periodic wave defined by (79) Figre 5: The graphs of profiles for periodic wave soltions of Jacobin elliptic fnction types defined by (78) and(79) nder different parametric vales: (a) g=4;(b) g= 4. physical applications. For eample sing the smooth solitary wave soltions nonsmooth peakon wave soltions and kink and antikink wave soltions we can eplain lots of motion phenomena for water wave; indeed (1)is jst a very important water wave model. Conflict of Interests The athor declares that there is no conflict of interests regarding the pblication of this paper. Acknowledgments This work is spported by the Natral Science Fondation of China nder Grant no the Natral Science Fondation of Scientific and Technical Committee of Chongqing City nder Grant no. cstc214jcyja14 the Natral Science Fondation of Chongqing Normal University nder Grant no. 13XLR2 and the Program Fondation of Chongqing Innovation Team Project in University nder Grant no. KJTD2138. References 1 C. S. Gardner J. M. Greene M. D. Krskal and R. M. Mira Method for solving the Korteweg-deVries eqation Physical Review Lettersvol.19no.19pp C.S.GardnerJ.M.GreeneM.D.KrskalandR.M.Mira Korteweg-deVries eqation and generalization. VI. Methods for eact soltion Commnications on Pre and Applied Mathematicsvol.27pp V.B.MatveevandM.A.SalleDarbo Transformations and Solitons Springer Berlin Germany C.H.GB.L.GoY.S.Lietal.Soliton Theory and Its Applications Springer New York NY USA C. Rogers and W. K. Schief Bäcklnd and Darbo Transformations Cambridge Tets in Applied Mathematics Cambridge University Press Cambridge UK M.R.MirraBäcklnd Transformation Springer Berlin Germany M. Wadati H. Sanki and K. Konno Relationships among inverse method Bäcklnd transformation and an infinite nmber of conservation laws Progress of Theoretical Physicsvol.53 no. 2 pp I.T.Khabibllin TheBäcklnd transformation and integrable initial bondary vale problems Mathematical Notes of the Academy of Sciences of the USSRvol.49no.4pp R. Hirota Eact soltion of the korteweg-de vries eqation for mltiple collisions of solitons Physical Review Letters vol. 27 no.18pp R. Hirota The Direct Method in Soliton Theoryeditedandtranslated by A. Nimmo C. Gilson Cambridge University Press Cambridge UK G. W. Blman and S. Kmei Symmetries and Differential Eqation vol. 154ofApplied Mathematica Sciences Springer New York NY USA P. J. Olver Applications of Lie Grops to Differential Eqations SpringerNewYorkNYUSA P. A. Clarkson and M. D. Krskal New similarity redctions of the Bossinesq eqation JornalofMathematicalPhysicsvol. 3 no. 1 pp P. A. Clarkson Nonclassical symmetry redctions of the Bossinesq eqation Chaos Solitons and Fractalsvol.5no.12pp P. A. Clarkson and E. L. Mansfield Symmetry redctions and eact soltions of shallow water wave eqations Acta Applicandae Mathematicaevol.39no.1 3pp

14 14 Mathematical Problems in Engineering 16 W. Hereman P. P. Banerjee A. Korpel G. Assanto A. van Immerzeele and A. Meerpoel Eact solitary wave soltions of nonlinear evoltion and wave eqations sing a direct algebraic method Jornal of Physics A: Mathematical and General vol. 19 no. 5 pp P. W. Doyle Separation of variables for scalar evoltion eqations in one space dimension Jornal of Physics A: Mathematical and Generalvol.29no.23pp P. W. Doyle and P. J. Vassilio Separation of variables for the 1-dimensional non-linear diffsion eqation International Jornal of Non-Linear Mechanics vol.33no.2pp J. Weiss M. Tabor and G. Carnevale The Painlevépropertyfor partial differential eqations Jornal of Mathematical Physics vol. 24 no. 3 pp R. Conte Invariant painlevé analysis of partial differential eqations Physics Letters Avol.14no.7-8pp W. Malfliet and W. Hereman The tanh method. I. Eact soltions of nonlinear evoltion and wave eqations Physica Scriptavol.54no.6pp A.-M. Wazwaz The tanh method for traveling wave soltions of nonlinear eqations Applied Mathematics and Comptation vol. 154 no. 3 pp A. M. Wazwaz A sine-cosine method for handling nonlinear wave eqations Mathematical and Compter Modelling vol. 4 no. 5-6 pp A. Wazwaz The sine-cosine method for obtaining soltions with compact and noncompact strctres Applied Mathematics and Comptationvol.159no.2pp J. He and X. W Ep-fnction method for nonlinear wave eqations Chaos Solitons and Fractalsvol.3no.3pp W. X. Ma T. Hang and Y. Zhang A mltiple ep-fnction method for nonlinear differential eqations and its application Physica Scriptavol.82no.6ArticleID J. Li and Z. Li Smooth and non-smooth traveling waves in a nonlinearly dispersive eqation Applied Mathematical Modellingvol.25no.1pp J. Li and Z. Li Traveling wave soltions for a class of nonlinear dispersive eqations Chinese Annals of Mathematics vol.23 no. 3 pp Y. Zho M. Wang and Y. Wang Periodic wave soltions to a copled KdV eqations with variable coefficients Physics Letters Avol.38no.1pp J. Li and K. Yang The etended F-epansion method and eact soltions of nonlinear PDEs Chaos Solitons and Fractals vol. 22 no. 1 pp M. Wang X. Li and J. Zhang The (G /G)-epansion method and travelling wave soltions of nonlinear evoltion eqations in mathematical physics Physics Letters A vol.372no.4pp E. M. E. Zayed and K. A. Gepreel The (G /G)-epansion method for finding traveling wave soltions of nonlinear partial differential eqations in mathematical physics Jornal of Mathematical Physicsvol.5no.1ArticleID W. Ri B. He Y. Long and C. Chen The integral bifrcation method and its application for solving a family of thirdorder dispersive PDEs Nonlinear Analysis: Theory Methods & Applicationsvol.69no.4pp R. Weigo L. Yao H. Bin and L. Zhenyang Integral bifrcation method combined with compter for solving a higher order wave eqation of KdV type International Jornal of Compter Mathematicsvol.87no.1pp W. Ri B. He S. Xie and Y. Long Application of the integral bifrcation method for solving modified Camassa-Holm and Degasperis-Procesi eqations Nonlinear Analysis: Theory Methods & Applicationsvol.71no.7-8pp W. Ri The integral bifrcation method combined with factoring techniqe for investigating eact soltions and their dynamical properties of a generalized Gardner eqation Nonlinear Dynamicsvol.76no.2pp W.X.MaH.WandJ.He Partialdifferentialeqationspossessing Frobenis integrable decompositions Physics Letters A vol. 364 no. 1 pp W. X. Ma and J. H. Lee A transformed rational fnction method and eact soltions to the 3+1dimensional Jimbo- Miwa eqation Chaos Solitons & Fractals vol. 42 no. 3 pp W. X. Ma and B. Fchssteiner Eplicit and eact soltions to a Kolmogorov-Petrovskii-PISknov eqation International Jornal of Non-Linear Mechanics vol.31no.3pp A. S. Fokas On a class of physically important integrable eqations Physica D: Nonlinear Phenomenavol.87no.1 4pp B. Fchssteiner and A. S. Fokas Symplectic strctres their Bäcklnd transformations and hereditary symmetries Physica Dvol.4no.1pp / B. Fchssteiner The Lie algebra strctre of nonlinear evoltion eqations admitting infinite-dimensional abelian symmetry grops Progress of Theoretical Physicsvol.65no.3pp H.YangandW.Ri NewLapairsandDarbotransformation and its application to a shallow water wave model of generalized KdV type Mathematical Problems in Engineering vol. 213 Article ID pages Q. Bi Wave patterns associated with a singlar line for a bi- Hamiltonian system Physics Letter Avol.369no.5-6pp J. Li and J. Zhang Bifrcations of travelling wave soltions for the generalization form of the modified KdV eqation Chaos Solitons & Fractalsvol.21no.4pp

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