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1 Alied Mathematics and Comutation 7 (00) 60 Contents lists available at ScienceDirect Alied Mathematics and Comutation journal homeage: Full symmetry grous, Painlevé integrability and exact solutions of the nonisosectral BKP equation Huan-ing Zhang a, Biao Li a,c, Yong Chen a,b, * a Nonlinear Science Center, Ningbo University, Ningbo, China b Shanghai Key Laboratory of Trustworthy Comuting, East China Normal University, Shanghai 0006, China c MM Key Lab, Chinese Academy of Sciences, Beijing 00080, China article info abstract Keywords: Symmetry grou Nonisosectral BKP equation Painlevé analysis Solitons Based on the generalized symmetry grou method resented by Lou and Ma [Lou and Ma, Non-Lie symmetry grous of ( + )-dimensional nonlinear systems obtained from a simle direct method, J. Phys. A: Math. Gen. 8 (00) L9], firstly, both the Lie oint grous and the full symmetry grou of the nonisosectral BKP equation are obtained, at the same time, a relationshi is constructed between the new solutions and the old ones of equation. Secondly, the nonisosectral BKP can be roved to be Painlevé integrability by combining the standard WTC aroach with the Kruskal s simlification, some solutions are obtained by using the standard truncated Painlevé exansion. Finally, based on the relationshi by the generalized symmetry grou method and some solutions by using the standard truncated Painlevé exansion, some interesting solution are constructed. Ó 009 Elsevier Inc. All rights reserved.. Introduction The nonisosectral soliton equations are imortant hysical models, because some of them can describe the waves in a certain tye of nonuniform media [ ]. There are many methods for finding solutions of nonisosectral soliton equations, such as Darboux transformation [], IST [,,] and so on. In recent years, the study of symmetries [6 8], symmetry grous [9], symmetry reductions [0,] and grou invariant solutions of nonlinear artial differential equations (PDEs) has become one of the most exciting and extremely active areas of research [ 6]. Some owerful methods to obtain the similarity reductions of a given PDE have been develoed by mathematicians and hysicist, such as, the Lie aroach [6,9] and the direct method resented by Clarkson and Kruskal (CK) [0]. Most recently, Lou et al. develo a new symmetry grou method, named generalized symmetry grou method, in a series of aers [7 ]. By the new symmetry grou method, both the Lie oint symmetry grous and the full symmetry grou can be obtained for some PDEs []. Furthermore, the exressions of the exact finite transformations of the Lie grous are much simler than those obtained via the standard aroaches for some nonlinear PDEs. Here we use the generalized symmetry grou method and the standard WTC aroach with the Kruskal s simlification [ 7] to investigate ( + )-dimensional nonisosectral BKP equation [8]: 9u t þ yðu xxxxx þ uu xxx þ u x u xx þ u u x u xxy uu y u u u yy Þ xu y u xx u u u 6@ u y ¼ 0 where u ¼ uðx y tþ. The nonisosectral BKP equation had been researched, for examle, Deng obtained the soliton solutions for the nonisosectral BKP equation are derived through Hirota method and Pfaffian technique [9]. ðþ * Corresonding author. Address: Nonlinear Science Center, Ningbo University, Ningbo, China. address: chenyong@nbu.edu.cn (Y. Chen) /$ - see front matter Ó 009 Elsevier Inc. All rights reserved. doi:0.06/j.amc

2 6 H.-. Zhang et al. / Alied Mathematics and Comutation 7 (00) 60 This aer is arranged as follows: In Section, both the Lie oint grous and the full symmetry grou of the nonisosectral BKP equation are obtained, at the same time, a relationshi is constructed between the new solutions and the old ones of equation. In Section, we give the roof of the Painlevé integrability by combining the standard WTC aroach with the Kruskal s simlification [ 7] and to obtain some exact solutions by using Painlevé exansion. Based on the relationshi by the generalized symmetry grou method and some solutions by using the standard truncated Painlevé exansion, some interesting solution are constructed. In Section, we give the conclusion of the article.. The full symmetry grou of a ( + )-dimensional nonisosectral BKP equation In order to obtain the full symmetry grou of the the nonisosectral BKP equation, firstly, we let u ¼ v x : Substituting Eqs. () into (), then Eq. () becomes: ðv x v xxxx þ v xx v xxx þ v xv xx þ v xxxxxx v x v xy v xx v y v yy v xxxy Þy þ 9v xt xv xy v xxx v x v xxv 6v y ¼ 0 where v ¼ vðx y tþ. Let v ¼ a þ bvðn g sþ where a b n g and s are functions of {x, y, t}. Restricting Vðn g sþ V, and satisfies the same form as the nonisosectral BKP equations () but with new indeendent variables, i.e., ðv n V nnnn þ V nn V nnn þ V n V nn þ V nnnnnn V n V ng V nn V g V gg V nnng Þg þ 9V ns nv ng V nnn V n V nnv 6V g ¼ 0: Substituting Eqs. () into (), then eliminate V nnnnnn by using Eq. (), from that, the remained determining equations of the functions n g s a b can be got by vanishing the coefficients of V and its derivatives, then we find out the general solution of the determining equations by tedious calculations. The result reads ds s ¼ s 0 g ¼ðd s y þ g0 Þ n ¼ b ¼ d y a ¼ 0 d s y þ g0 s x g 0 d s y þ g 0 8 þ >< Z y 6 y >: y d s y þ g0 x þ n ðy tþ ð7þ þ y x s 6 d s y þ g 0 y n yðy tþd þ s d s y þ g 0 y 8 d y s t s þ s 0 y t g 0 þ g y s þ d g g 0 þ d n t ðy tþ d s y þ g 0 7 dy þ c 0 where n 0 n 0 ðtþ g 0 g 0 ðtþ s 0 s 0 ðtþ c 0 c 0 ðtþ are arbitrary functions of time t and n ðy tþ ¼ ð9ds 8s t y þ 0y g d þ 8n 0 s Þ d s y þ g0 t while the constants d ossess discrete values determined by þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i 0 þ d ¼ þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 0 þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i 0 In summary, the following theorem holds: Theorem. IfV Vðx y tþ is a solution of the Eq. (), then so is v ¼ a þ bvðn g sþ y 9 >= d n 0 s 8 ðþ ðþ ðþ ðþ ð6þ ð8þ > þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 0 þ : ð9þ where a b n g s are given by Eqs. (6) (8), discrete value of the d are given by Eq. (9). The relationshi is constructed between the new solutions and the old ones of Eq. (). Thus from the theorem of Eq. () with Eq. (), we also can obtain the relationshi between the new solutions and the old ones of Eq. (). ð0þ

3 H.-. Zhang et al. / Alied Mathematics and Comutation 7 (00) 60 7 From the symmetry grou theorem, for the nonisosectral BKP equation, the symmetry grou is divided into five sectors which corresond to d ¼ d ¼ þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i 0 þ d ¼ þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 0 d ¼ þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i 0 d ¼ þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 0 þ of theorem resectively. That is to say, the full symmetry grou, G CBKP, exressed by theorem for the comlex the nonisosectral BKP equation is the roduct of the usual Lie oint symmetry grou Sðd ¼ Þ and the discrete grou D G CBKP ¼D S D fi R R R R g where I is the identity transformation, and R : vðx y tþ! þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 0 þ v þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i 0 þ! x y t R : vðx y tþ! þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i 0 þ v þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 0 þ! x y t R : vðx y tþ! þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 0 v þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i 0! x y t R : vðx y tþ! þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i 0 v þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 0! x y t : From theorem, by restricting ðf f ðtþ g gðtþ h hðtþþ ðtþþ s ¼ t þ f n 0 ¼ g g 0 ¼ h c 0 ¼ then Eq. (0) can be written as v ¼ V þ rðvþ rðvþ ¼V t g þ! y h þ yg t V y þ 9 8 yg tt þ y f þ xg t þ xh þ y ht V x þ 9 6 yg ttt þ Vg t þ x h þ xh t 0 y 7 y þ y ft þ 6 y htt þ y y Vh y ðþ ðþ þ 8 xg tt þ xf y The equivalent vector exression of the above symmetry is (! C ¼ y xf 9 y y ft 8 yg tt xg þ yg ttt þ Vg t 8 xg (!! ) xh þ y y y Vh x h þ xh t y 0 y 7 y 6 r ðf Þþr ðgþþr ðhþþr ðþ which is exactly the same as that we obtained by the standard Lie aroach. The commutation relations for the Kac Moody Virasoro algebra among r ðf Þ r ðgþ r ðhþ and r ðþ are as follows: ½r ðf Þ r ðf ÞŠ ¼ 0 ½r ðf Þ r ðþš ¼ 0 ½r ðhþ r ðþš ¼ 0 ½r ð Þ r ð ÞŠ ¼ 0 ½r ðf Þ r ðgþš ¼ r gf t g tf ½r ðf Þ r ðhþš ¼ r fh t f th ½r ðg Þ r ðg ÞŠ ¼ r ðg g t g g t Þ ½r ðgþ r ðhþš ¼ r gh t g th ½r ðgþ r ðþš ¼ r g t þ g t ½r ðh Þ r ðh ÞŠ ¼ r ðh h t h h t Þ:

4 8 H.-. Zhang et al. / Alied Mathematics and Comutation 7 (00) 60 We can use the relationshi form this theorem to obtain new solutions from old solutions. So in the next section, we rove the nonisosectral BKP equation is Painlevé integrability and obtain a solution by using the standard truncated Painlevé exansion.. Painlevé integrability of the ( + )-dimensional the nonisosectral BKP equation and some solution from the truncated Painlevé exansion Painlevé analysis is one of the most owerful method to rove the integrability of a model develoed by WTC (Weiss Tabor Canvela) [ 7]. If one needs only to rove the Painlevé roerty of a model, one may use the Kruskal s simlification for WTC method. Furthermore, the Painlevé analysis can also be used to find some exact solutions no matter whether the model is integrable or not. At first, the Painlevé exansion may have the form: v ¼ X j¼0 v j f j a ðþ where the arbitrary function f f ðx y tþ may have different forms in different aroaches, v j v j ðx y tþ ðj ¼ 0...Þ. Using any one ossible form, the final conclusion will be exactly the same. In order to give out a comlete treatment, it is convenient by using the Kruskal s simlification, i.e., f ¼ x þ wðy tþ with wðy tþ w being an arbitrary function of y and t. By substituting v ¼ v 0 f a into Eq. (), comaring the leading order terms for f! 0, we get two ossible branch: a ¼ v 0 ¼ f x ðþ ðþ ð6þ or v 0 ¼ f x : For the first branch, by equating the coefficients of f j, the olynomial equation in j is derived as: ð7þ j 6 j þ j 7j þ j þ 96j 60 ¼ 0: Using Eq. (8), the resonances are found to be j ¼ 6 0: For j ¼ and, consequently v v and v are arbitrary functions. For j ¼ and, we obtain v ¼ 90y xw y 9w t þ yw y v ¼ 60y ðyw yy þ w y Þ: ð9þ For j ¼ 6 v 6 is arbitrary function. For j ¼ 7 8 and 9, we get v 7 ¼ xw 860y y 9w t þ yw y þ y w y w yy ðþ v 8 ¼ 70yw 90700y y x 60yw y w t þ 600y w y 80yw yy w y 96xw t w y þ9w t þ 6x w y 00y w yyy ðþ v 9 ¼ 7y w 60800y y w yy 08yxwyt 60y w y w yt þ 6yw tt 09xw y y þ 97w yw t y w y y 8x w y þ xw t þ 8yx w yy þ 80y w yy w t þ 60y xw y w yy : ðþ Then for j ¼ 0 v 0 is arbitrary function. U to now, we rove that all the resonance conditions are satisfied for the first branch. It is concluded that the Eq. () asses the P-test in the first branch. For the second branch, we can get the olynomial equation in j is ð8þ ð0þ j 6 j þ j j 86j þ 6j þ 70 ¼ 0 then we get the resonances ðþ

5 j ¼ 6 H.-. Zhang et al. / Alied Mathematics and Comutation 7 (00) 60 9 we can also rove that all the resonance conditions are satisfied for the second branch. In all, it is concluded that the Eq. () asses the P-test and hence it is exected to be integrable whether in the first branch or second branch. Then we use Painlevé exansion to obtain some solutions of Eq. (). Because in the first branch, when j ¼ v is a resonance oint, let v ¼ f x ¼ ½lnðf ÞŠ f x f ¼ aðtþþexððtþx þ qðtþy þ wðtþþ: ðþ ð6þ Substituting Eqs. () and (6) into Eq. (), collecting the coefficient of exððtþx þ qðtþy þ wðtþþ x and y, then we can get qðtþ ¼ t ðtþ wðtþ ¼ln½aðtÞŠ þ Z t ðtþdt ð7þ aðtþ are arbitrary function of t, ðtþ should be satisfied t ðtþ 7ðtÞ ttðtþ 6 ðtþþ ðtþ t ðtþ ¼0: ð8þ So we get the solution of Eq. (), v ¼ t ðtþxþ tðtþyþ ðtþe R ðtþdt R t ðtþdt þ e ðtþxþ tðtþyþ : Then we also obtain the solution of Eq. (), u ¼ ðtþsech xðtþþ t ðtþy þ Z ðtþdt : ð9þ If we let ðtþ ¼ k ðtþþq ðtþ k ðtþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ðtþ ¼ 9c 6t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9c 6t then Eq. (8) become one-soliton solution of the nonisosectral BKP, which had been given in [9]. If we give out the aroriate form of ðtþ, we can obtain more abundant solutions of Eq. (). By theorem, we also can obtain the new solution of Eq. (), s ðsþnþ sðsþgþ ðsþe R ðsþds v ¼ a þ b R s ð0þ þ e ððsþnþ sðsþgþ ðsþdsþ where a b n g and s are determined by Eqs. (6) (8), ðsþ is satisfied s ðsþ 7ðsÞ ssðsþ 6 ðsþþ ðsþ s ðsþ ¼0: In the second branch, if we let f have the same form as Eq. (), we cannot obtain a solution of Eq. (), which contains three variables x y and t. So we let f ¼ xðtþ þ qðy tþ: Substituting Eqs. (7) and () into Eq. (), collecting the coefficient of x and y, then we get ðþ qðy tþ ¼ 7 y 7 F ðtþþ 7 tðtþy þ F ðtþ ðþ where ðtþ F ðtþ and F ðtþ is arbitrary function of t, then we can obtain the solution of Eq. () ðtþ v ¼ xðtþþ7y 7F ðtþþ7 t ðtþy þ F ðtþ then the solution of Eq. () is 6ðtÞ u ¼ : xðtþþ7y 7F ðtþþ7 t ðtþy þ F ðtþ We can also get a new solution of Eq. () by theorem.

6 60 H.-. Zhang et al. / Alied Mathematics and Comutation 7 (00) 60. Conclusions In summary, making use of the generalized symmetry grou method and symbolic comutation, the fully symmetry transformation grous for the nonisosectral BKP equation are given. The full symmetry grou of the nonisosectral BKP equation is a roduct of one discrete grou ðd Þ and one infinite dimensional Kac Moody Virasoro tye Lie grou with four arbitrary functions. The relationshi is constructed between the new solutions and the old ones of equation, which is a relationshi of grou invariant solution. If we have obtained a solution of the nonisosectral BKP equation by other method, we can use this relationshi to obtain another solution. Then we rove the nonisosectral BKP can ass the Painlevé test and hence it is exected to be integrable. So we obtain a solution of the nonisosectral BKP by the standard truncated Painlevé exansion, then we get new general solution by the relationshi, which is the known one-soliton solution [9]. It is necessary to oint out that the general solution contain all its grou invariant solutions, so it is enough to submit it to the relationshi for one time only. Acknowledgements We would like to thank Prof. Senyue Lou, Dr. Yuqi Li, Zhongzhou Dong, Xiaorui Hu, Jia Wang and Wangchuan Ye for their enthusiastic guidance and helful discussions. The work is suorted by the National Natural Science Foundation of China (Grant Nos. 0700, 077 and 90780), Shanghai Leading Academic Disciline Project (No. B), Program for Changjiang Scholars and Innovative Research Team in University (IRT07) Zhejiang Provincial Natural Science Foundations of China (Grant No. 6008), Ningbo Natural Science Foundation (Grant Nos. 007A6009 and 006A6009) and K.C. Wong Magna Fund in Ningbo University. References [] R. Hirota, J. 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