THERMAL SCIENCE, Year 05, Vol. 9, No. 4, pp. 49-435 49 KINK DEGENERACY AND ROGUE WAVE FOR POTENTIAL KADOMTSEV-PETVIASHVILI EQUATION by Hong-Ying LUO a*, Wei TAN b, Zheng-De DAI b, and Jun LIU a a College of Mathematics and Information Science, Qujing Normal University, Qujing, China b School of Mathematics and Statistics, Yunnan University, Kunming,China Original scientific paper DOI: 0.98/TSCI50449L A new method called homoclinic breather limit method is proposed to solve the Potential Kadomtsev-Petviashvili equation, breather kink-wave and periodic soliton are obtained, and kink degeneracy and new rogue wave are first found in this paper. Key words: Potential Kadomtsev-Petviashvili equation, rogue wave solution, extended homoclinic test method, kink degeneracy, exp-function method Introduction In recent years, the study of exact solutions for the non-linear evolution equations has attracted much attention [], and many non-linear phenomena were discovered [-7]. In this work, we focus on some new non-linear phenomena, i. e., the kink degeneracy and rational breather solutions, for the (+) dimensional Potential Kadomtsev-Petviashvili (PKP) equation: u + 6u u + u + u = 0 () xt x xx xxxx yy It is well-known that this equation arises in a number of remarkable non-linear problems in fluid mechanics and thermal science, and its solutions have been studied extensively. Solitary wave, soliton-like solution, and interaction among solitary waves were found [8-3]. Recently, exact periodic kink-wave and degenerative soliton solution of PKP were also elucidated [4]. In this paper, a novel approach named as homoclinic breather limit process is proposed to seek for rational breather-wave solutions. Homoclinic breather limit method In order to elucidate the new method, we consider a high dimensional non-linear evolution equation in the general form: Fu (, ut, ux, uy, uxx, uyy, ) = 0 () where u(x, y, t), and F is a polynomial of u and its derivatives. The new method takes following steps. * Corresponding author; e-mail: luohongy98@63.com
430 THERMAL SCIENCE, Year 05, Vol. 9, No. 4, pp. 49-435 Step. By Painleve analysis, a transformation: u = T( f) (3) is made for a new and unknown function f. Step. Convert eq. () into Hirota bilinear form: GD ( t, Dx, Dy; f, f ) = 0 (4) where D is the operator defined by: m k m k x t = a( x, t) b( x, t ) x= x, t= t D D a b x x t t Step 3. Solve eq. (3) using homoclinic test approach by assuming that [5]: ip( x α t) ip( x αt) Ω y+ γ Ω y+ γ 0 f( x, y, t) = + b (e + e )e + be (5) or by the extended homoclinic test approach by assuming that: p( α x+ βy+ ωt+ γ) p( αx+ βy+ ωt+ γ) 0 α β ω γ f( x, y, t) = e + b cos[ p ( x+ y+ t+ )] + be (6) where α, β, Ω, α, β, ω, p, p, b 0, b, γ, and γ are constants to be determined, or by a more general approach by the exp-function method [6]. Step 4. Substitute eq. (5) or eq. (6) into eq. (3), and equate all coefficients of ( αx+ βy+ ωt+ γ ) αx+ βy+ ωt+ γ e, e, cos( α x+ βy+ ωt + γ), sin( α x+ βy+ ωt + γ), and constant term to zero, we obtain the set of algebraic equation for α, β, ω, α, β, ω, p, p, b 0, and b. Step 5. Solve the set of algebraic equations in Step 4 using Maple and for α, β, Ω, α, β, ω, p, p, b 0, and b. Step 6. Substitute the identified values of α, β, ω, α, β, ω, p, p, b 0, and b into eq. () and eq. (5) and deduce the exact solutions of eq. (). Step 7. Using the relationship between p and p, or p and Ω of exact solution obtained in the Step 6 and let p tends to zero, we can get the rational breather-wave solution. Kink degeneracy By using Painleve test we can assume that: (ln f) x (7) where f(x, y, t) is an unknown real function to be determined. Substituting eq. (7) into eq. (), we obtain the bilinear equation: 4 x t y x Now we suppose that the solution of eq. (8) is: ( DD+ D + D) f f= 0 (8) where ξ ξ 0 η f ( x, y, t) = e + b cos( ) + be (9) ξ = px ( + βt+ ω), η = p( x βt + ω) + αy
THERMAL SCIENCE, Year 05, Vol. 9, No. 4, pp. 49-435 43 and β, ω, α, ω, p, p, b 0, and b are some constants to be determined later. Substituting eq. (9) into eq. (8), we have: get: 4 4 0 0 4 4 b0 p p p p p p pb0p p p p p 4 4 b0α + 8β p b+ 8b0 p + βb0 p + 3p b = 0 [ bb( p + p β + p β + p α 6 p p ) cos( η) 8 pbbp( p p)( p+ p) sin( η)]e + + [ ( + β + β + α 6 ) cos( η) + 8 ( )( + )sin( η)]e ξ Equating all coefficients of different powers of e ξ, e ξ, sin(η), and cos(η) to zero, we 4 4 p + pβ + p β + p α 6p p = 0 8 pb0p( p p)( p + p) = 0 4 4 b0α + 8β p b + 8b0 p + βb0 p + 3p b = 0 Solving the system of eqs. (0) with the aid of Maple, we get: ξ (0) b0 ( β 8 p ) α p =± p, b =, =± p β 4p 4( β + 4 p ) and β, ω, ω, p, b 0, are free. Substitute eq. () into eq. (9), we obtain the solution: () px ( + βt+ ω) px ( + βt+ ω) 0 β ω α f( x, y, t) = e + b cos[ p ( x t+ ) + y] + be () If 0 < b R, then we obtain the exact breather kink solution: 4p b sinh p( x+ β t+ ω) + ln( b) b0 sin p( x βt+ ω) + αy (3) b cosh p( x+ βt+ ω) + ln( b) + b0 cos p( x βt+ ω) + αy Substitute eq. () into eq. (3), eq. (3) can be re-written as: β 8 p b0 ( β 8 p ) ± p sinh p( x+ βt+ ω) + ln sin p 4 4( 4 ) ( x βt+ ω) + αy β+ p β+ p ux (, yt,) = (4) β 8 p b0 ( β 8 p ) ± cosh px ( + βt+ ω) + ln + cos p 4 4( 4 ) ( x βt+ ω) + α p y β+ β+ p Solution u(x, y, t) represented by eq. (4) is a breather kink-wave solution. In fact, solution u(x, y, t) is a kink wave as trajectory along the straight line x = βt ω α y p, and meanwhile evolves periodically along the straight line x = βt ω p, fig. (a). Let p tends to zero and take b 0 = ±, we can get the following rational breather solution: where 4 β(x+ ω+ ω + βy) A+ B β t + ( ωβ + β ω ) t+ βω βω 3 = + A βx ( βω βω β y) x 5 3 B = β y + ( β t β ω ) y 3 (5)
43 THERMAL SCIENCE, Year 05, Vol. 9, No. 4, pp. 49-435 The solution u(x, y, t) represented by eq. (5) is a breather solution and no longer the kinky. This shows that kink is degenerated when the period of breather wave tends to infinite in the breather kink-wave, fig. (b). this is a new non-linear phenomenon up to now. Figure. (a) the breather kink solution as p = /0, β =, b 0 = ω = ω =, y = 0, and (b) the rational breather-wave solution as α = ω = ω = 0, β = 4, y = 0 Solutions (3) is kinky periodic-wave which has speed β, the forward-direction (or backward-direction) wave shows solitary feature meanwhile takes on kinky feature with space variable x, t for PKP equation. Specially, this wave shows both kinky and periodic feature to space variable t. Such a surprising feature of weakly dispersive long-wave is first obtained. Meanwhile, notice (5) when the t tends to infinity, u tends to zero. From fig., it is observed that the kink of the solutions disappeared when the p tends to zero. More importantly, we obtained a rational breather wave solution. From periodic soliton solution to rational breather wave solution In this section, by choosing special test function in application of HTA to (+) dimensional Potential-Petviashvili equation, we obtain a periodic soliton solution and a rational breather-wave solution. Setting: ξ = x αt where α is a wave velocity. Equation () can be re-written as: αuξξ + 6uξ uξξ + uξξξξ + u yy = 0 (6) Using Painlev e analysis, we suppose that the solution of eq. () is: (ln ) (7) for some unknown real function f(x, y, t) and by substituting eq. () into eq. (), we can obtain the bilinear form: f ξ 4 ( α + D + D ) f f = 0 (8) Dξ y ξ
THERMAL SCIENCE, Year 05, Vol. 9, No. 4, pp. 49-435 433 With regard to eq. (3), using the homoclinic test technique [4], we are going to seek the solution of the form: ipξ ipξ Ω y+ γ Ω y+ γ f( ξ, y) = + b (e + e )e + b e (9) where p, Ω, γ, b, and b are all real to be determined below. Substituting eq. (4) into eq. (3) yields the exact solution of eq. () in the form: Ω y+ bp e sin( pξ ) Ω y+ γ ( Ω y+ γ) ξ (0) + b cos( p )e + b e ξ f, y Computing D f D f f, and 4 Dξ f f, we obtain: Ω y+ γ Ω y+ γ Ω y+ Dξ f f = 4bp e [cos( pξ)e b + cos( pξ) + be ] Ω y+ γ Ω y+ γ Ω y+ D y f f = 4 Ω e [cos( pξ) be b + cos( pξ) b + be ] 4 4 Ω y+ γ Ω y+ γ Ω y+ Dξ f f = 4 bp e [cos( pξ)e b + cos( pξ) + 8be ] Substituting eqs. (6) into eq. (3) we get: 4 Ω+ y γ 4 [(4bpαb + 4Ω bb + 4 bpb) cos( pξ)e + (4bpα + 4Ω b+ 4 bp ) cos( pξ) + () 4 α Ω+ y γ Ω+ y γ + (8b p + 8Ω b + 3 b p )e ]e = 0 Ω+ y γ Equating all coefficients of different powers of cos( pξ )e, to zero, we get: 4 4bp αb + 4Ω bb + 4bpb = 0 4 4bp α + 4Ω b + 4bp = 0 4 8b p α + 8Ω b + 3b p = 0 Solving above equations we get: 4 ( α + 4 p ) b α p p, b Ω = = α + p cos( pξ), and e Ω+ y γ () (3) Substituting ξ = x αt into eq. (5), eq. (5) can be rewritten: bp sin[ px ( αt)] (4) b cosh[( Ω y+ ) + ln( b)] + b cos[ p( x αt)] Substituting eq. (8) into eq. (9), we get exact periodic soliton solution of eq. (): bpsin[ px ( αt)] ( α + 4 p ) b ( α + 4 p ) b cosh py p ln b α + γ + cos[ p( x αt)] α p + + α + p (5)
434 THERMAL SCIENCE, Year 05, Vol. 9, No. 4, pp. 49-435 Let p tends to zero and γ = 0, b =, we can get the following rational breather-wave solution: 4 α( x αt) (6) 3 αx xα t + α t y α + 3 Figure. (a) the periodic soliton solution as p = b =, α = 00, γ = 0, y = x, and (b) the rational breather-wave solution as α = 6, y = x Solutions (5) is periodic solitary wave which have speed α, the forward-direction (or backward-direction) wave shows solitary feature meanwhile takes on periodical feature with space variable x, t for PKP equation. In particular, this wave shows both solitary and periodic feature to space variable t. This is a strange and interesting physical phenomenon to the evolution of -D flow of shallow-water waves having small amplitudes. We will study this interesting phenomenon in further work. From fig., it is observed that the kink of the solutions disappeared when the p tends to zero. More importantly, we obtained a rational breather wave solution Conclusions In summary, the aid of extended homoclinic test method, we successfully applied this method to the (+) dimensional PKP equation. Some exact periodic soliton solution, exact breather cross-kink solutions and rational solutions are obtained. More importantly, it is an effective method for many equations seeking rational solutions. Our results enrich the variety of the dynamics of high-dimensional systems. This method is simple and straightforward. We will investigate other types of non-linear evolution equations and non-integrable systems. Acknowledgments The work was supported by Chinese Natural Science Foundation Grant No. 36048, Qujing Normal University Natural Science Foundation Grant No. 0QN06. References [] Debtnath, L., Non-Linear Partial Differential Equations for Scientist and Engineers, Birkhauser, Boston, USA, 997 [] Hong T., et al., Bogoliubov Quasiparticles Carried by Dark Solitonic Excitations in Nonuniform Bose- Einstein Condensates, Chinese Phys. Lett., 5 (998), 3, pp. 550-55
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