Kink degeneracy and rogue potential solution for the (3+1)-dimensional B-type Kadomtsev Petviashvili equation

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1 Pramana J. Phys. 6) 87: DOI.7/s-6--8 c Indian Academy of Sciences Kink degeneracy and roge poenial solion for he +)-dimensional B-ype Kadomsev Peviashvili eqaion ZHENHUI XU,, HANLIN CHEN and ZHENGDE DAI Applied Technology College, Sohwes Universiy of Science and Technology, Mianyang 6, People s Repblic of China School of Science, Sohwes Universiy of Science and Technology, Mianyang 6, People s Repblic of China School of Mahemaics and Physics, Ynnan Universiy, Knming, 69, People s Repblic of China Corresponding ahor. zhenhi9@6.com MS received May ; revised 8 Sepember ; acceped 8 November ; pblished online 6 Jly 6 Absrac. In his paper, we obained he eac breaher-ype kink solion and breaher-ype periodic solion solions for he +)-dimensional B-ype Kadomsev Peviashvili BKP) eqaion sing he eended homoclinic es echniqe. Some new nonlinear phenomena, sch as kink and periodic degeneracies, are invesigaed. Using he homoclinic breaher limi mehod, some new raional breaher solions are fond as well. Meanwhile, we also obained he raional poenial solion which is fond o be js a roge wave. These resls enrich he variey of he dynamics of higher-dimensional nonlinear wave field. Keywords. B-ype Kadomsev Peviashvili eqaion; homoclinic breaher limi mehod; raional breaher solion; kink degeneracy; roge poenial solion. PACS Nos..Jr;..Jb;..Yv. Inrodcion In recen years, soliary wave solions of nonlinear evolion eqaions have begn playing imporan roles in nonlinear science fields, especially in nonlinear physical science. The soliary wave solion can provide physical informaion and more insigh ino he physical aspecs of he problem hs leading o frher applicaions []. I is well known ha here are many mehods for finding special solions of nonlinear parial differenial eqaions, sch as he inverse scaering mehod [], he homogeneos balance mehod [], he Darbo ransformaion mehod [,], he Hiroa s bilinear mehod [,6], he improved anh-mehod [7], he Lie grop mehod [8], he eended homoclinic es approach [9 ], and so on. In his work, we consider he +)-dimensional B- ype Kadomsev Peviashvili BKP) eqaion z y ) y + =, ) where : R R y R z R R. The BKP eqaion was given his name becase i is a B-ype KP eqaion []. The well-known BKP eqaion possesses many inegrable srcres sch as La formlaion and he mliple solion solions. Eac solions of he BKP eqaion have been sdied by means of some effecive approaches, sch as he comple ravelling wave solion [], periodic solions, mliple solion solions [6], Wronskian solion [7] and he Pfaffian solion [8]. However, o or bes knowledge, he beraher-ype kink and raional breaher solions o he +)-dimensional BKP eqaion ) have no ye been sdied. Therefore, in his paper, an approach of seeking raional breaher-wave solion, called he homoclinic breaher limi mehod [9,], is proposed and applied. Eac breaher kink wave and periodic breaher soliary solions are obained, kink and periodic degeneracy are invesigaed, new raional breaher solions and roge poenial solion are consrced by homoclinic breaher limi process or by Taylor epansion [,].

2 Page of 8 Pramana J. Phys. 6) 87:. Homoclinic breaher limi mehod Consider a high-dimensional nonlinear evolion eqaion of he general form P,,, y, z,, yy, zz,...) =, ) where =,y,z,) and P is a polynomial of and is derivaives. The basic idea of he homoclinic breaher limi mehod can be epressed in he following five seps: Sep By Painlevé analysis [], a ransformaion = Tf), ) is made for some new nknown fncion f. Sep By sing he ransformaion in Sep, he original eqaion can be convered ino Hiroa s bilinear form GD,D,D y,d z ; f, f ) =, ) where he D-operaor [] is defined by QD,D y,d z,d,...)f,y,z,,...) G,y,z,,...) = Q, y y, z z,,...) F, y, z,,...)g, y, z,,...) =,y =y,z =z, =,...,) where Q is a polynomial of D,D y,d,... Sep As we know, he breaher of inegrable PDE is sally in he form of a raional fncion as he nmeraor and denominaor are he combinaion of fncions of cos, sin, cosh, sinh, and so f can be conjecred as a combinaion of cos and cosh or sin and sinh). Then, sbsie his rial form o he bilinear eqaion, eq. ), o ge a se of algebraic eqaion for some parameers, solve he above se of eqaion o obain homoclinic breaher wave solion, which was called he eended homoclinic es approach EHTA)in []. Sep Le he period of periodic wave go o infinie in homoclinic breaher wave solion. We can hen obain a raional breaher wave solion. Sep Solving he poenial of breaher wave solion in Sep and leing p ends o zero, we can obain a raional homoclinic heeroclinic) wave and his wave is js a roge wave [7].. Applicaions. Kink degeneracy and new raional breaher solion By sing Painlevé es, we can assme ha,y,z,) = ln f), 6) where f,y,z,) is an nknown real fncion. Sbsiing eq. 6) ino eq. ), we obain he following bilinear form: D z D D D y + D )f f =, 7) where D z D f f = ff z f z f ), D D yf f = f y f f f y +f f y f f y ). Wih regard o eq. 7), we can seek he solion in he form f = e pξ + δ cospη) + δ e pξ, 8) where ξ = + a y + b z + c,η = + a y + b z + c,a,b,c,a,b,c,p,p,δ,δ are real consans o be deermined. Sbsiing eq. 8) ino eq. 7) and eqaing all he coefficiens of differen powers of e ξ, e ξ, sinη), cosη) and he consan erm o zero, we can obain a se of algebraic eqaions for p, p,a i,b i,c i,δ i i =, ) as follows: a p a p + a p + c b + a p +c b + 6 = a p p a p p + a p p + p +b c p c b p + a p = δ p a δ p b c δ p δ p 6a δ p + b c δ p =. 9) Solving eq. 9) wih he aid of Maple, we ge he following resls: p a δ p p + p ) a δ p + δ 6δ )) +p a = δ p p + p, ) c = p a p + a p ) p + a p) b p + p ), b a δ p p +p) +p p δ δ) a δ )) p +δ +p b = δ p p + p) a p + a p + ), p p c = ) a p + a p + ) b p + p ), )

3 Pramana J. Phys. 6) 87: Page of 8 where a,b,δ,δ,p,p are some free real consans. There are differen choices for δ,δ and p in ). Here, we specially ake δ i,i =, andp sch as δ = p +,δ = p +,p = p in eq. ), so ha i is more easy o ge he form of /asp, in order o obain raional breaher solion. In his case, eq. ) can be rewrien as a = a p + a p, + b = a p b p + p + b + ) + 8p + 6 a p + )p, + ) c = a p +, b c = a p +. ) b Sbsiing eq. ) ino eq. 8), we have f,y,z,) = p +coshp + H y + K z + L ) + lnp + )) p + where cosp + a y + b z L )), ) H = a p + a p, + K = b a p + a p + p + ) a p + )p, + ) L = a p +. b Sbsiing eq. ) ino eq. 6) yields he solion of he +)-D BKP eqaion as follows:,y,z,) = p p + sinhp + H y + K z + L ) + lnp + )) + p + sinp + a y + b z L ))) p + coshp + H y + K z + L ) + lnp + )). ) p + cosp + a y + b z L )) The solion,y,z,) represened by eq. ) is a breaher-ype kink solion. I is generaed by he ineracion beween he solion wih variable X = p + H y + K z + L )+ lnp + ) and he periodic wave wih variable Y = p + a y + b z L ). If p in eq. ), we can ge he raional breaher solion as follows:,y,z,) = b + y z + b z) b + a y + b z) + ) + b + y z) a b y + )) + b. ) The solion,y,z,) represened by eq. ) is a new raional breaher solion. Noe ha ends o zero in eq. ), when ±, and so i is no longer kinky. Sch a srprising feare of weakly dispersive long wave is firs obained. Meanwhile, his shows ha kink is degeneraed when he period of breaher wave ends o infiniy in he breaher kink wave. Figres,, and ehibi he evolion breaher kink wave and raional breaher wave in he, ) and, y) planes, respecively. This is a new nonlinear phenomenon ill now. By choosing he special es fncion, we obained a kinky periodic-wave solion and a new raional breaher solion.. Kinky periodic degradaion and new raional breaher solion In his secion, we apply he homoclinic breaher limi mehod o he +)-dimensional BKP eqaion. Figre. The breaher-ype kink solion solion when a =,b =,p =,y = z =.

4 Page of 8 Pramana J. Phys. 6) 87: e p+b z+d ), sinp y + b z + c + d )), cosp y + b z+c +d )) and consan erm o zero, we can obain a se of algebraic eqaions for c, bi, δi i =, ). Solving he sysem wih he aid of Maple, we ge he following resls: b = Figre. The raional breaher solion when a =, b =, y = z =. Sppose ha he solion of eq. 7) is f, y, z, ) = e p+b z+d ) + δ cosp y + b z + c + d )) + δ ep+b z+d ), ) where b, b, c, d, d, δ, δ, p, p are free real consans. Sbsiing eq. ) ino eqs 7), and eqaing all he coefficiens of differen powers of ep+b z+d ), p, c p, b c p = δ = δ. 6) Sbsiing eq. 6) ino eq. ) and aking b c >, we have p f, y, z, ) = δ cosh p + z + d c δ + ln +δ cos py + bz + c + d). b c 7) Sbsiing eq. 7) ino eq. 6) yields he kinky periodic solion solion of he +)-D BKP eqaion as follows: p δ sinh p + pc z + d + ln δ, y, z, ) =. δ cosh p + pc z + d + ln δ + δ cos py + b z + c + d ) b c The solion, y, z, ) represened by eq. 8) can be considered as a kink solion of he variable X=p + p z + d + ln δ c spread along he direcion of variable Y = see figre ). Especially, for he same reason as dealing wih eq. ), we choose δ = in eq. 8), while p, we can ge he raional breaher solion as follows:, y, z, ) = py + b z + c + d ) b c bc bc + d) + d) + y + bz + c + d). 9) 8) y Figre. The breaher-ype kink solion solion when a =, b =, p =, = z =. y Figre. The raional breaher solion when a =, b =, = z =.

5 Pramana J. Phys. 6) 87: Page of 8 U U Figre. The kinky periodic solion solion when b = /,c =,δ =,p =,d = d = y = z =. The solion,y,z,) represened by eq. 9) is a breaher wave which no longer has periodic kink feare. Here, periodic kink degeneracy occrs when he period of he periodic wave ends o infiniy. I was observed ha he periodic kink feare of he solion disappeared when p ends o zero. More imporanly, we obained a new raional breaher wave solion see figre 6).. Periodic degeneracy and new raional breaher solion In his secion, we obained a breaher-ype periodic solion solion and a raional breaher solion by choosing anoher special es fncion. Sppose ha he solion of eq. 7) is f,y,z,) = e py+b z+c+d ) +δ cosp + b z + d )) +δ e py+b z+c+d ), ) where b,b,c,δ,δ,p,p are free real consans.,y,z,) =..... Figre 6. The raional breaher solion when b = /, c =,d = d = y = z =. Sbsiing eq. ) ino eqs 7), and eqaing all he coefficiens of differen powers of e py+b z+c+d ), e py+b z+c+d ), sinp + b z + d )), cosp + b z+d )) and consan erm o zero, we can obain a se of algebraic eqaions for c, b i,δ i i =, ). Solving he sysem wih he aid of Maple, we ge he following resls: b = b p b c, p = p, δ = δ. ) Sbsiing eq. ) ino eq. ) and aking b c>, we have f,y,z,) = δ cosh py + b z + c + d ) )) + ln δ b c +δ cos p ) b p z + d, ) Sbsiing eq. ) ino eq. 6), we obain a breaherype periodic solion solion of BKP eqaion as follows: ) ) b cpδ sin b c p b p z + d δ cosh py + b z + c + d ) + ln δ )) )). ) + δ cos b c p b p z + d. The solion,y,z,) represenedbyeq. ) can be considered as a solion of variable ) X = py + b z + c + d ) + ln δ spread along he direcion of variable b c Y = p ) b p z + d see figre 7). Similar o he way we deal wih eq. ), here we choose δ = in eq. ), when p, and we can ge he raional breaher solion as follows figre 8):,y,z,) = b c + d ) y + b z + c + d ) + b c + d ). )

6 Page 6 of 8 Pramana J. Phys. 6) 87: Solion,y,z,) represened by eq. ) is a breaher wave which no longer has periodic feare. Here, periodic degeneracy occrs when he period of he periodic wave ends infiniy. This is a srange and ineresing physical phenomenon which cases he evolion of shallow waer waves having small amplides. I is observed ha he periodic feare of he solion disappeared when p ends o zero. More imporanly, we obained a new raional breaher wave solionsee figre ).. Roge poenial solion In his secion, we solve he poenial of eq. ) and le p end o zero. We hen obain a raional homoclinic heeroclinic) wave and his wave is js a roge wave. Solving he poenial of eq. ), we have φ =,y,z,)) = p p + p + p + sinhp + H y + K z + L ) + lnp + )) sinp + a y + b z L ))) p + coshp + H y + K z + L ) + lnp + )) p + cosp + a y + b z L ))) ) where H = a p + a p, + K = b a p + a p + p + ) a p + )p, + ) L = a p + b and φ is a breaher-ype periodic solion see figre 9). Le p anda = in eq. ). By comping, we obain he raional breaher wave, and i is js a roge wave as follows see figre ): U roge wave = 8b 6b z + b ) + b + y + b z) b + b z) + b ) b + y + b z) + ) + b z ) + ) + b ). 6) U conains wo waves wih differen velociies and direcions. I is easy o verify ha U roge wave is a raional breaher-ype wave. In fac, U roge wave conains wo waves wih differen velociies and direcions. From figre, we can see ha U roge wave has one pper dominan peak and wo small holes. The spaial srcre of he fncion U roge wave is similar o he srcre of he roge waves which has been a poin of ho discssion in recen years. In fac, U for fied as y, z and ±. So, U roge wave is no only a raional breaher wave b also a roge wave solion, he amplide of which is hree imes higher han is srronding waves and U roge wave generally forms in a shor ime. Remark. By sing he same mehodology as for eq. ), we can solve he poenial of solions of eqs 8) and U U Figre 7. The breaher-ype periodic solion solion when b =,p =,c =,δ =,d = d = y = z = Figre 8. The raional breaher solion when b =, c =,d = d = y = z =...

7 Pramana J. Phys. 6) 87: Page 7 of Figre 9. The breaher-ype periodic solion φ when a =,b =,p =,y = z =. U Figre. U roge wave when b =,y = z =. ) in. and. respecively when p, o ge roge poenial solions.. Conclsion In smmary, by sccessflly applying he eended homoclinic es echniqe o he +)-dimensional B-ype Kadomsev Peviashvili eqaion, we obained eac kink breaher, kinky periodic and periodically breaher soliary solions. By sing he homoclinic breaher limi mehod proposed in his work, we obained some new raional breaher solions. Frhermore, we invesigaed wo new physical phenomena, kink and periodic degeneracy. Or resls show differen dynamics of high-dimensional sysems. Meanwhile, we also obained he raional poenial solion which is js a roge wave. This mehod is simple and sraighforward. In he fre, we shall invesigae oher ypes of nonlinear evolion eqaions and non-inegrable sysems. Acknowledgemens The ahors hank he reviewer for valable sggesions and help. 6 This work was sppored by he Chinese Naral Science Fondaion Gran Nos 68, 9769), Sichan Edcaional Science Fondaion Gran No. ZB) and Sohwes Universiy of Science and Technology Fondaion Gran No. z8). References [] M J Ablowiz and P A Clarkson, Solions, nonlinear evolion eqaions and inverse scaering Cambridge Universiy Press, 99) [] M L Wang, Phys. Le. A, ) [] C H G, H S H and Z X Zho, Darbor ransformaion in solion heory and geomeric applicaions Shanghai Science and Technology Press, Shanghai, 999) [] V B Maveev and M A Salle, Darbo ransformaion and solions Springer, 99) [] R Hiroa, Fndamenal properies of he binary operaors in solion heory and heir generalizaion, in: Dynamical problem in solion sysems edied by S Takeno, Springer Series in Synergeics, Vol. Springer, Berlin, 98) [6] R Hiroa, The direc mehod in solion heory Cambridge Universiy Press, Cambridge, ) [7] S A El-Wakil and M A Abdo, Nonlin. Anal. 68, 8) [8] P J Olver, Applicaion of Lie grops o differenial eqaions Springer, New York, 98) [9] ZDDai,JLi,XPZengandZJLi,Phys. Le. A 7, 98 8) [] L A Dickey, Solion eqaions and Hamilonian sysems, nd edn World Scienific, Singapore, ) [] Z D Dai and D Q Xian, Commn. Nonlinear Sci. Nmer. Simla., 9 9) [] Z H X, D Q Xian and H L Chen, Appl. Mah. Comp., 9 ) [] M Jimbo and T Miwa, Pbl. Res. Ins. Mah. Sci. 9, 9 98) [] H F Shen and M H T, J. Mah. Phys., 7 ) [] K L Tian, J P Cheng and Y Cheng, Science China- Mahemaics, 7 ) [6] H C Ma, Y Wang and Z Y Qin, Appl. Mah. Comp. 8, 6 9) [7] A M Wazwaz, Comp. Flids 86, 7 ) [8] Y L Kang, Y Zhang and L G Jin, Appl. Mah. Comp., ) [9] M G Asaad and W X Ma, Appl. Mah. Comp. 8, ) [] Z H X, H L Chen and Z D Dai, Appl. Mah. Le. 7, ) [] N Akhmediev, A Ankiewicz and M Taki, Phys. Le. A 7, 67 9) [] H L Chen, Z H X and Z D Dai, Abs. Appl. Anal. 7, 7867 ) [] Z D Dai, J Li and D L Li, Appl. Mah. Comp. 7, 6 9) [] Z D Dai, J Li, X P Zeng and Z J Li, Phys. Le. A 7, 98 8) [] A Ankiewicz, J M Soo-Crespo and N Akhmediev, Phys. Rev. E 8, 66 )

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