Bäcklund Transformations and Explicit Solutions of (2+1)-Dimensional Barotropic and Quasi-Geostrophic Potential Vorticity Equation

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1 Commun. Theor. Phys. Beijing, China) ) pp Chinese Physial Soiety and IOP Publishing Ltd Vol. 53, No. 5, May 15, 010 Bäklund Transformations and Expliit Solutions of +1)-Dimensional Barotropi and Quasi-Geostrophi Potential Vortiity Equation HU Xiao-Rui ) 1 and CHEN Yong í ) 1,, 1 Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 0006, China Nonlinear Siene Center and Department of Mathematis, Ningbo University, Ningbo 31511, China Reeived Deember, 009; revised manusript reeived January 8, 010) Abstrat By the Bäklund transformation method, an important +1)-dimensional nonlinear barotropi and quasigeostrophi potential vortiity BQGPV) equation is investigated. Some simple speial Bäklund transformation theorems are proposed and used to get expliit solutions of the BQGPV equation. Furthermore, all solutions of a seond order linear ordinary differential equation inluding an arbitrary funtion an be used to onstrut expliit solutions of the +1)-dimensional BQGPV equation. Some figures are also given out to desribe these solutions. PACS numbers: 0.30.Jr, 0.0.Hj, 0.0.Sv, Sd Key words: Bäklund transformation, +1)-dimensional nonlinear barotropi and quasi-geostrophi potential vortiity equation, expliit solution 1 Introdution It is known that most of the models of atmospheri and oean dynamial systems are nonlinear differential equations. The expliit solutions of these differential equations an well depit and reflet fruitful phenomena in fluids and other physial fields. However, due to the high nonlinearity of the models, it is often diffiult to solve these differential equations analytially. Based on the lassial Lie symmetry approah, 1 4] Huang 5] investigated the +1)-dimensional nonlinear barotropi and quasi-geostrophi potential vortiity BQGPV) equation without foring and dissipation on a β-plane hannel and obtained its some types of expliit solutions inluding the ring solitary waves and the breaking soliton type of vortiity solutions. The BQGPV equation reads t ψ F ψ t + Jψ, ψ) + β ψ x = 0, 1) where = / x + / y denotes the two-dimensional Laplaian operator, Ja, b) = a x b y a y b x is the Jaobian operator, ψ is the dimensional steam funtion, F = L /R0 is the square of the ratio of the harateristi horizonal length sale L to the Rossby deformation radius R 0, β = β 0 L /U), and β 0 = ω 0 /R 0 )osφ 0, in whih R 0 is the Earth s radius, ω 0 is the angular frequeny of the Earth s rotation, φ 0 is the latitude, and U is the harateristi veloity sales. The subsripts x, y represent partial derivatives. The BQGPV equation 1) as one of the most important models of the atmospheri and oean dynamial systems an be widely used to study the development of turbulene and oherent strutures in atmosphere and in magnetized plasmas. 6 9] Furthermore, Eq. 1) is a potential vortiity onservation and its solutions an indiate the movement state of free oean depending on its initial onditions. In Ref. 10], based on the standard multi-sale expansion method and the long wave approximation, one possible approximate solution to Eq. 1) was gained by Tang et al. and it has been used to explain the life yle of a bloking system without loss of the long wave property of the Rossby wave. Then Eq. 1) was restudied 11 1] by Huang and Tang et al. using the lassial Lie symmetry approah again. Tang et al. found a new symmetry and obtained two new types of similarity redution solutions as a note to the results in Ref. 5]. In this paper, Eq. 1) is reinvestigated in an alternative way, i.e. Bäklund transformation and more exat solutions of Eq. 1) are onstruted. In Ref. 13], Lou et al. have made use of this simple and essential tehnique, i.e. Bäklund transformation, to obtain some types of exat solutions of + 1)-dimensional Euler equation. Here, it is found that the same method in Ref. 13] is also valid for Eq. 1). In Se., some Bäklund transformation theorems for the BQGPV equation are given out. With the help of Bäklund transformation, some types of exat solutions are obtained and some figures are given out in Se. 3. Setion 4 is our onlusions. Bäklund Transformations of BQGPV Equation Firstly, for the onveniene of our omputation, we rewrite Eq. 1) in the following equivalent form q = ψ xx + ψ yy ψ, ) Supported by the National Natural Siene Foundation of China under Grant Nos , , and , Shanghai Leading Aademi Disipline Projet under Grant No. B41, Program for Changjiang Sholars and Innovative Researh Team in University IRT0734) yhen@sei.enu.edu.n

2 804 HU Xiao-Rui and CHEN Yong Vol. 53 q t Fψ t + Jψ, q) + βψ x = 0. 3) Then we an investigate Eqs. ) and 3) instead of Eq. 1). Theorem 1 If {q 0, ψ 0 } is a solution of Eqs. ) and 3), so is {q 1, ψ 1 } under the definition where {ω, p} is a solution of q 1 = q 0 + ω, ψ 1 = ψ 0 + p, 4) ω = p, 5) ω t Fp t + Jp, ω) + Jψ 0, ω) + Jp, q 0 ) + βp x = 0. 6) Proof Diret alulations. From Bäklund transformation 4), one an find new exat solutions {q 1, ψ 1 } from known ones {q 0, ψ 0 } when {ω, p} are solved from Eqs. 5) and 6). Furthermore, we put a onstraint between ω and p as general as ω = Qp), 7) where Qp) is an arbitrary funtion of p. Due to this ansatz, we may have a muh simplified Bäklund transformation theorem: Theorem If {q 0, ψ 0 } is a solution of Eqs. ) and 3), so is {q 1, ψ 1 } under the definition q 1 = q 0 + Qp), ψ 1 = ψ 0 + p, 8) where Qp) is an arbitrary funtion of p and p is a solution of the over-determined equation system p = Qp), 9) p t + Jψ 0, p) = 0, 10) Fp t + Jp, q 0 ) + βp x = 0. 11) Proof As for Eq. 9), it is just a simple ombination of Eqs. 5) and 7). Substituting the ansatz 7) into Eq. 6) of Theorem 1, we have p t + Jψ 0, p))q p Fp t + Jp, q 0 ) + βp x = 0, 1) where Q p dq/dp. One an see that Eq. 1) is orret for arbitrary Qp) if and only if Eqs. 10) and 11) are satisfied simultaneously. Theorem is proven. However, if the known solutions {q 0, ψ 0 } are very ompliated, it is still very diffiult to onstrut some exat solutions by solving Eqs. 9), 10), and 11). So to make sure Eqs. 10) and 11) be solved easily, we an selet the seed solution for a very speial and simple form q 0 =, 13) i.e. q 0 is an arbitrary onstant. Firstly, we should solve another seed solution ψ 0 by substituting q 0 = into Eqs. ) and 3) ψ 0xx + ψ 0yy = q 0, 14) q 0t Fψ 0t + Jψ 0, q 0 ) + βψ 0x = 0. 15) For the assumption q 0 =, Eqs. 14) and 15) an be solved readily. One an see that the orresponding general solution of Eqs. 14) and 15) has the form of i 1) ψ 0 = 4 y + x + β F t ) ] + f 1 y + x + β F t ) i ] where f 1 f 1 y + + f y x + β ) ] F t i, 16) x + β ) ] F t i, f f y x + β ) ] F t i are arbitrary funtions of the indiated variables. Furthermore, ψ 0 is assured to be real on ondition that f is a omplex onjugate of f 1 and vie verse. In order to get expliit solutions of Eqs. ) and 3), here we selet f 1 + f = m 1 y + m x + β ) F t, 17) i.e. f 1 = m 1 m i y + x + β ) ] F t i, f = m 1 + m i y x + β ) ] F t i, where m 1 and m are arbitrary real onstants. Hene formula 16) beomes ψ 0 = y + x + β ) ] 4 F t + m 1 y + m x + β ) F t. 18) Then substituting the seed solution {q 0, ψ 0 } with the form of Eq. 13) and 18) into Eqs. 10) and 11), one an get p = PFx + βt) + F y ) + 4F m x + m 1 y) + 4Fβm t y)). 19) For onveniene, we denote ξ = Fx + βt) + F y ] + 4F m x + m 1 y) + 4Fβm t y). 0) Substituting p = Pξ) with Eq. 0) into Eq. 9), we an obtain m F +Fm 1 β) + ξ ] d Pξ) + dpξ) = QPξ)). The above results an be summarized as the following theorem: Theorem 3 If Pξ) P is a solution of the seond order ordinary differential equation m F +Fm 1 β) + ] d 4 ξ Pξ) dξ + dpξ) = QP),1) with ξ = Fx + βt) + F y ] + 4F m x + m 1 y) + 4Fβm t y), QP) being an arbitrary funtion of P and, m 1, m being arbitrary onstants, then ψ given by ψ = y + 4 is a solution of the BQGPV equation 1). x+ β F t ) ]+m 1 y +m x+ β F t ) +Pξ)) 3 Some Speial Expliit Solutions from Bäklund Transformations From Theorem 3, one an get fruitful expliit solutions of Eq. 1) only by solving a seond order ordinary differential equation with an arbitrary funtion QP). In this setion, we selet some speial QP) and give out the orresponding expliit solutions of Eq. 1). For the arbitrary onstant in Theorem 3, there are two irumstanes, i.e.

3 No. 5 Bäklund Transformations and Expliit Solutions of +1)-Dimensional Barotropi and 805 = 0 and 0. Then for these two ases, we will study them separately. 3.1 Speial Expliit Solutions from 0 in Theorem 3 For the parameter 0 in Eq. 1), we give out some expliit solutions of Eq. 1) for the speial seletions of k1 QP). i) For the seletion QP) = 0, Eq. 1) beomes m F +Fm 1 β) + 4 ξ ] d Pξ) dξ + 4 dpξ) 0 = 0, 3) dξ where 0 is also an arbitrary onstant. Solving Eq. 3), one an get Pξ) = 16 ] 0 m F + Fm 1 β) ) ln4m F + Fm 1 β) + ξ) ξ + k, with k 1 and k being two arbitrary onstants. Hene, one an obtain one expliit solution of Eq. 1) via ) diretly, there being ψ = 4 0 F + )x + y + β 4 F t) βf + β ) xt + m m F ) x F + m βF m 1 F ) m β y + F m βf ) k1 t + k + 16 ] 0 m F + Fm 1 β) ) ln Fx + βt) + F y ) + 4FFm x + Fm 1 β)y + βm t) + 4Fm 1 β) + 4m F ). 4) When 0 = 0 and k 1 = 0 in Eq. 4), we obtain a polynomial solution of Eq. 1). In other ases, the solution obtained in the form of Eq. 4) will be with singularities due to the logarithm funtion. Figures 1a) and 1b) display the struture for the solution 4) under the parameter seletions β = 1, F = 1, = 0.1, m 1 = 1, m = 0, k 1 = 10, k = 0, 5) with 0 = 1/16 and 0 = 1 at time t = 0, respetively. Both of the singularities are 0, 0). Fig. 1 The speial solution ψ given by Eq. 4) with Eq. 5) and a) 0 = 1/16, b) 0 = 1, both at time t = 0. ii) Seleting QP) = mpξ) in Eq. 1), one an obtain m F + Fm 1 β) + ] d 4 ξ Pξ) dξ + dpξ) mpξ) = 0, 6) where m is an arbitrary onstant. Solving Pξ) in Eq. 6) and substituting it into Eq. ), we get the seond type of expliit solution of Eq. 1) ψ = 4 y + x + β F t ) ] + m 1 y + m x + β F t ) + l 1 J + βfxt + 4m F x + 4Fm 1F β) { + l Y 0, 4 m F x + y + β y + 4m βf F t) + βfxt + 4m F { 0, 4 m F x + y + β F t) t + 4m F + Fm 1 β) ) ] 1/} + 4Fm 1F β) y + 4m βf t + 4m F + Fm 1 β) ) ] 1/}, 7) where l 1, l are arbitrary onstants, J and Y are the Bessel funtions of the first and the seond kind, respetively. Then we give out Fig. to display the property of the solution 7). The parameter seletions in Figs. a) and b) x

4 806 HU Xiao-Rui and CHEN Yong Vol. 53 are respetively at time t = 0. β = 1, F = 1, = 0.1, m 1 = 1, m = 0, m = 1/16, l 1 = 15, l = 0, 8) β = 1, F = 1, = 0.1, m 1 = 0, m = 1, m = 1/16, l 1 = 10, l = 0, 9) Fig. a) and b) desribe the steam funtion ψ given by Eq. 7) with Eqs. 8) and 9) separately at time t = 0. iii) Seleting QP) = expp), Eq. 1) beomes m F + Fm 1 β) + ] d 4 ξ Pξ) dξ + dpξ) expp) = 0. 30) The expliit solution of Eq. 30) is a1 )seh 1/) Pξ) = ln a 1 /)ln4m F + 4m 1 F β) + ξ) a )) ] 84m F + 4m 1 F β), 31) + ξ) where a 1 and a are two arbitrary onstants. Hene from Pξ) in Eq. 31) with ξ in Eq. 0), one an gain the expliit solution of Eq. 1) from Eq. ). Beause the expression of speial solution is so ompliated that we do not write it here. iv) For the QP) = ξ sinξ), Eq. 1) beomes m F + Fm 1 β) + ] d 4 ξ Pξ) dξ + dpξ) ξ sinξ) = 0. 3) By solving Pξ) in Eq. 3), one an readily hek that ψ = 1 {Ci ζ + k)8k sink) + 4k 8)osk)] + Si ζ + k)4k 8)sink) 8k osk)] + 4k 1)ζ sinζ) + k 3)osζ)] + b 1 lnζ + k)) + 4F ζ + 4Fβy + 4b F )} 33) is the expliit solution of Eq. 1). In Eq. 33), Si, Ci are respetively the Sine integral and the Cosine integral and b 1, b are two arbitrary onstants. Here for onveniene, we denote ζ = F x + y + β F t) + βfxt + 4FF β)y, k = 4β F). 34) 3. Speial Exat Solutions from = 0 in Theorem 3 In this setion, we will give out some another expliit solutions of Eq. 1) for the parameter = 0 in Eq. 1). In this ase, theorem 3 will be more speial and simplified. Here we rewrite it as follows. Theorem 4 If Pξ) P is a solution of the two-order ordinary differential equation m F + Fm 1 β) ) d Pξ) dξ = QP), 35) with ξ = Fm x + m 1 y) + βm t y), QP) being an arbitrary funtion of P and m 1, m being arbitrary onstants, then ψ given by ψ = m 1 y + m x + β F t ) + Pξ) 36) is a solution of the BQGPV equation 1). For QP) in Eq. 35), we an not list all of them and Eq. 35) is not solved easily for every QP). Here we only list some speial seletions whih are of interest to us. We will selet QP) as P, P 3, expp), sinp), tanhξ), ξ sinξ), and ξp. Substituting them into Eq. 35), the orresponding solutions an be solved readily. We list them in the following.

5 No. 5 Bäklund Transformations and Expliit Solutions of +1)-Dimensional Barotropi and 807 P 1 ξ) = 6R P ξ + 1, 0, ), P 3 ξ) = ln) + ln P 4 ξ) = artan n ξ ] P ξ) = 3 sn ), i, R 5 seh 5 ξ + 6 ) R 7 R + ξ + 8 ), R P 5 ξ) = 1 R ξ + dilog 1 + expξ))] + P 6 ξ) = 1 R ξ sinξ) + osξ)] + 11ξ + 1, )], ), sn 7 R + 9 ln R ) ξ + 10, 7 R + ξ + 8 ), R )], 7 R + P 7 ξ) = 13 Ai R /3 ξ) + 14 Bi R /3 ξ). Here, dilog x) = x 1 lnz 1 z dz, at time t = 1. Ai and Bi are Airy wave funtion, R = m F + Fm 1 β), i 1 and k k = 1,...,14) are arbitrary onstants. Then the orresponding expliit solutions ψ k of Eq. 1) are obtained by Eq. 36), i.e. ψ k = m 1 y +m x+ β F t )+P k ξ), k = 1,...,7), 37) with ξ = Fm x + m 1 y) + βm t y). Some figures are given out to depit these solutions ψ k k =, 3, 4, 6, 7). Figure 3 shows the real part of omplex solution ψ and the orresponding parameters are with t = 1. m = 1, 3 = 1, 4 = 1, 38) Fig. 4 ψ 3 in Eq. 37) with Eq. 39) at t = 1. Figure 5 desribes the solution ψ 4 in Eq. 37). The parameter seletions of ψ 4 in Figs. 5a) and 5b) are respetively m = 0, 7 = 1, 8 = 1, 40) m = 1, 7 = 1, 8 = 1, 41) both at time t = 1. Figure 6 desribes the solution ψ 6 in Eq. 37) at t = 0. The parameter seletions in ψ 6 are Fig. 3 The real part of ψ given by Eq. 37) with Eq. 38) at t = 1. β = 1, F = 1, m 1 = 0, m = 1, 11 = 0.1, 1 = 0. 4) Figure 7 depits ψ 7 in Eq. 37) with the parameters Figure 4 depits ψ 3 with the parameters m = 1, 5 = 5, 6 = 1, 39) at t = 0. β = 0.1, F =, m 1 = 1, m = 1, 13 = 10, 14 = 0, 43)

6 808 HU Xiao-Rui and CHEN Yong Vol. 53 Fig. 7 ψ 7 in Eq. 37) with Eq. 43) at t = 0. Fig. 5 a) ψ 4 given by Eq. 37) with Eq. 40) and b) ψ 4 given by Eq. 37) with Eq. 41) both at t = 1. Fig. 6 ψ 6 in Eq. 37) with Eq. 4) at t = 0. 4 Conlusions For the BQGPV equation 1) being one of the most important models of the atmospheri and oean dynamial systems, to obtain its expliit solutions is very meaningful. Lou et al. have gained some types of expliit solutions using the lassial symmetry. But solving some of the redued equations of Lie group is still diffiult. To gain more expliit solutions of Eq. 1), the Bäklund transformation method is used in this paper. Firstly, some simple speial Bäklund transformation theorems are proposed. It is shown that all solutions of a seond order linear ordinary differential equation an be used to onstrut exat solutions of + 1)-dimensional BQGPV equation. For eleven speial irumstanes of the ordinary differential equation, we give out their solutions. Based that the orresponding expliit solutions of Eq. 1), whih are expressed by speial funtions, are obtained. Furthermore, some figures are given out to depit these solutions. These figures show rih strutures of solutions of BQGPV equation, whih may be interesting in studying atmospheri and oean dynamial systems. However, some expliit solutions we obtained have singularities. How to explain these singularities and to searh for more exat solutions with no singularities of the BQGPV equation are left to be investigated later. Referenes 1] S. Lie, Arh. Math ) 38. ] L.V. Ovsiannikov, Group Analysis of Differential Equations, Aademi, New York 198). 3] P.J. Olver, Appliations of Lie Groups to Differential Equations, Springer, Verlag 1993). 4] G.W. Bluman and S.C. Ano, Symmetry and Integration Methods for Differential Equations, Springer, New York 00). 5] F. Huang and S.Y. Lou, Phys. Lett. A ) 48. 6] J. Pedlosky, Geophysial Fluid Dynamis, Springer, New York 1979). 7] W. Horton and A. Hasegawa, Chaos ) 7. 8] P.K. Shukla, G.T. Birk, and R. Bingham, Geophys. Res. Lett. 1995) ] D Luo Wlave Motion ) 315; ibid ) ] X.Y. Tang, F. Huang, and S.Y. Lou, Chin. Phys. Lett ) ] F. Huang and S.Y. Lou, Commun. Theor. Phys ) ] X.Y. Tang and K.S. Padma, Commun. Theor. Phys ) 9. 13] S.Y. Lou, M. Jia, F. Huang, and X.Y. Tang, Int. J. Theor. Phys ) 08.