Painlevé Test for the Certain (2+1)-Dimensional Nonlinear Evolution Equations. Abstract

Transcription

1 Painlevé Test for the Certain (2+1)-Dimensional Nonlinear Evolution Equations T. Alagesan and Y. Chung Department of Information and Communications, Kwangju Institute of Science and Technology, 1 Oryong-dong, Buk-gu, Gwangju , Korea K. Nakkeeran Photonics Research Center and Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (Dated: February 24, 2004) Abstract We present the explicit Painlevé test to the certain (2+1)-dimensional nonlinear evolution equations such as uncoupled breaking soliton equation, the Infeld-Rowlands equation and Potential Kadomstev-Petviashvili equation. The associated Bäcklund transformations and bilinear form are also obtained for all these system of equations directly from the Painlevé test. Keywords: Painlevé (P-) test, uncoupled breaking soliton (UBS) equation, Infeld-Rowlands (IR) equation, potential Kadomstev-Petviashvili (PKP) equation, associated Bäcklund transformation, bilinear form. Electronic address: alagesan@infcom.kjist.ac.kr Electronic address: ychung@kjist.ac.kr Electronic address: ennaks@polyu.edu.hk 1

2 I. INTRODUCTION The role of higher dimensional nonlinear evolution equations (HDNEE) has been increased enormously in the major fields of science and engineering due to their potential applications. To test the integrability of these HDNEE, Painlevé analysis (or P-test) is a well known technique in all the branchs of nonlinear science. The (2+1)-dimensional breaking soliton equation has been used to describe the interaction of Riemann wave propagation using the long wave propagation. The typical breaking soliton equation has been obtained first by Calogero and Degasperis in It can be reduced to the well know KdV equation by selecting suitable transformations and the self-dual Yang-Mills equation is also found to be a class of breaking soliton equation. New families of overtuning soliton solutions and soliton-like solutions for the typical breaking soliton equation have been derived through computer-algebra-based method and symbolic computation respectively [1, 2]. The singularity structure analysis of the two coupled breaking soliton equation has been carried out and proved that system equation possesses the P-property for the appropriate parametric condition [3]. They have also generated its bilinear form through the P-test and reported the possibilities of exponentially localized structures called the dromion and multidromians. The Infeld-Rowlands (IR) equation [4] has been studied and reported that it does not belong to the class of integrable in the three dimensional case. In reality, some of its reductions to ordinary differential equations do not have the P-property. The authors have analyzed the symmetry group of IR equation and obtained a class of its one- and twodimensional subgroups. Its symmetry group has shown to use these subgroups to perform symmetry reduction and to obtain some group invariant solutions. Moreover, a study of the stability of the Landau-Ginzburg equation has led to the present system (IR equation) after the suitable rescaling. The perturbative approach [5] to solitons in higher dimensions has been systematically implemented by Infeld and Rowlands. The numerical soliton-like solutions of the potential Kadomstev-Petviashvili (PKP) equation has been studied by Adomian s decomposition method [6]. They have obtained the exact and numerical solitary-wave solutions of the PKP equation for certain initial conditions. Recently, the studies on PKP equation using the homogeneous balance method and symbolic computation obtained the travelling wave solutions and soliton-like solutions respectively 2

3 [7, 8]. The conditional integrability of the Kadomstev-Petviashvili (KP) hierarchy has been discussed in detail [9]. Here, we consider certain physically important (2+1)-dimensional nonlinear evolution equations namely uncoupled breaking soliton (UBS) equation, Infeld-Rowlands (IR)equation and potential Kadomtsev-Petviashvili (PKP) equation which pass the P-test and derive the associated Bäcklund transformation and bilinear form directly from the P-test. II. (2+1)-DIMENSIONAL UBS EQUATION A. Painlevé test In this paper, first we present the P-test of the (2+1)-Dimensional UBS equation. The (2+1)-Dimensional UBS equation takes the following form [2]: u xt 4u x u xy 2u y u xx + u xxxy =0. (1) The leading order of the solution of Eq. (1) is assumed as u u 0 φ α. (2) On substituting Eq. (2) into Eq. (1) and equating the most dominant terms, the following results are obtained: α = 1; u 0 = 2φ x. (3) For finding the resonances, the full Laurent series u = u 0 φ 1 + u j φ j 1 (4) is substituted into Eq. (1) and by equating the coefficients of (φ j 5 ), the polynomial equation in j is derived as j=1 Using Eq. (5), the resonances are found to be j 4 10j 3 +23j 2 +10j 24 = 0. (5) j = 1, 1, 4, 6. (6) 3

4 As usual, the resonance at j = 1 corresponds to the arbitrariness of the singular manifold φ(x, y, t) = 0. In order to check the existence of sufficient number of arbitrary functions at the other resonance values, the full Laurent expansion (4) is substituted in Eq. (1). From the coefficient of (φ 5 ), the explicit value of u 0 is obtained as given in Eq. (3). Collecting the coefficient of (φ 4 ), the following equation is obtained 72φ 2 x φ yφ xx +48φ 3 x φ xy 32φ 2 x φ yφ xx 16φ 2 x φ yφ xx 32φ 3 x φ xy 16φ 3 x φ xy 24φ 2 x φ yφ xx =0. (7) Absence of u 1 in Eq. (7) proves that u 1 is arbitrary. This corresponds to the resonance value at j = 1. Then, collecting the coefficient of (φ 3 ), the explicit value of u 2 is obtained as u 2 = 1 24φ 3 x φ y [ 4φ 2 x φ t 56φ x φ xx φ xy 16u 1x φ 2 xφ y 8u 1y φ 3 x +8φ x φ y φ xx +12φ 2 xxφ y +48φ xx φ xy +8φ 2 xφ xxy ]. (8) Proceeding further to the coefficient of (φ 2 ), the value of u 3 can be obtained as u 3 = 1 24φ 3 x φ y [ 16φxx φ xxy 2φ xx φ t 4φ x φ xt +8u 1xy φ 2 x +24u 2φ 2 x φ xy +8u 1x φ y φ xx +16u 1x φ x φ xy +24u 2 φ x φ xx φ y +8φ xy φ xxx 12φ xxy +12u 1y φ x φ xx 2φ xxxx φ y 8φ x φ xxxy 8φ xxx φ xy +4u 1xx φ x φ y ]. (9) Similarly, collecting the coefficient of (φ 1 ), the result is obtained as 8u 1xy φ xx +12u 2 φ xx φ xy 8u 2x φ x φ xy 8u 3y φ 3 x 40u 3φ 2 x φ xy +8u 1x φ xy 2φ xxt 8u 2xy φ 2 x +8u 2 φ x φ xxy +4u 1xx φ xy 4u 2xx φ x φ y 16u 3x φ 2 xφ y +4u 1y φ xxx 4u 2y φ x φ xx +4u 2 φ y φ xxx 32u 3 φ x φ y φ xx 2φ xxxxy =0. (10) Absence of u 4 in Eq. (10) proves that u 4 is arbitrary. This corresponds to the resonance value at j = 4. From the coefficient of (φ 0 ), the value of u 5 can be obtained as 4

5 1 u 5 = 24φ 3 xφ y [u 2x φ t + u 2xxx φ y + u 2t φ x + u 2 φ xt +2u 3 φ x φ t +8φ xx u 2xy +14u 3x φ xx φ y +10u 3 φ xxx φ y +10u 3y φ x φ xx +3u 2xxy φ x +18u 4 φ x φ xx φ y +30u 3 φ xx φ xy 2u 3xy φ 2 x 10u 4yφ 3 x 48u 4φ 2 x φ xy 14u 4x φ 2 x φ y +3u 3 φ xxx φ y 4u 1x u 1xy 4u 1x u 2x φ y 4u 1x u 2y φ x 4u 1x u 2 φ xy 8u 3 u 1x φ x φ y + u 1xt +11u 2x φ xxy +u 2y φ xxx + u 2 φ xxxy 4u 1xy u 2 φ x 8u 2 u 2x φ x φ y 4u 2 u 2y φ 2 x 4u2 2 φ xφ xy 12u 2 u 3 φ 2 xφ y + u 1xxxy +2u 3xx φ x φ y +22u 3 φ x φ xxy 24u 4 φ x φ xx φ y +7u 2xx φ xy +18u 4 φ 2 x φ y 2u 1y u 1xx 4u 1y u 2x φ x 2u 1y u 2 φ xx 4u 1y u 3 φ 2 x +4u 2y φ xxx 2u 1xx u 2 φ y 2u 2 2 φ ] xxφ y +12u 3x φ xy. (11) Collecting the coefficient of (φ 1 ), the result is obtained as u 2xt +2u 3x φ t +2u 3t φ x +2u 3 φ xt +6u 4 φ x φ t +14u 3xy φ xx +30u 4y φ x φ xx +54u 4 φ xx φ xy +u 2xxxy +2u 3xx φ y +34u 4x φ xx φ y 2u 1y u 2xx 4u 1x u 2xy 8u 1x u 3x φ y 8u 1x u 3y φ x +6u 3xxy φ x +36u 4x φ xy 24u 1x u 4 φ x φ y 4u 2x u 1xy 8u 2x u 2y φ x 4u 2 u 2x φ xy 2u 1xx u 2y 14u 4xx φ x φ y 4u 2 u 2xy φ x 16u 2 u 3x φ x φ y 8u 2 u 3y φ 2 x 16u 2u 3 φ x φ xy 36u 2 u 4 φ 2 x φ y 4u 2 2x φ y +2u 3y φ xxx 8u 1xy u 3 φ x 12u 2y u 3 φ 2 x 24u2 3 φ2 x φ y +42u 4 φ x φ xxy +10u 3xx φ xy +72u 5 φ x φ y φ xx +24u 5 φ 2 x φ xy 8u 1y u 3x φ x 4u 3 u 1y φ xx 12u 1y u 4 φ 2 x +2u 3φ xxxy 2u 2 u 2y φ xx 2u 2 u 2xx φ y 8u 2 u 3 φ xx φ y +4u 3y φ xxx 4u 3 u 1xx φ y +18u 4 φ xx φ y +10u 4xy φ 2 x 8u 1xu 3 φ xy 24u 2x u 3 φ x φ y +14u 3x φ xxy +8u 4x φ x φ xy +24u 5x φ 2 x φ y =0. (12) Absence of u 6 in Eq. (12) proves that u 6 is arbitrary. This corresponds to the resonance value at j =6. As Eq. (1) admits sufficient number of arbitrary functions, it is concluded that the Eq. (1) passes the P-test and hence it is expected to be integrable. B. Associated Bäcklund transformation To construct the Bäcklund transformation of Eq. (1), let us truncate the Laurent series at the constant level term to give u = u 0 φ 1 + u 1, (13) 5

6 where the pair of function (u, u 1 ) satisfy Eq. (1) and hence Eq. (13) may be treated as the associated Bäcklund transformation of Eq. (1). C. Bilinear form To construct the bilinear form, we take the vacuum solution u 1 = 0 in Eq. (13) which leads to u = u 0 φ 1 = 2 (ln φ). x (14) With the help of the above dependent variable transformation in Eq. (1), the Hirota s bilinear form is found to be ( Dx D t + Dx 3 D ) y φ φ =0. (15) Having obtained the bilinear form, one can construct the soliton solution by expanding the dependent variables in terms of power series. III. (2+1)-DIMENSIONAL IR EQUATION A. Painlevé test Next we perform the P-test of the (2+1)-Dimensional IR equation. takes the following form [4]: The IR equation u t +2u x u xx + u xxxx + u xy =0. (16) For the P-test, the leading order of the solution of Eq. (16) is assumed as u u 0 φ α, (17) where u 0 and φ are analytic functions of x, y and t. α is a negative integer to be determined. Substituting Eq. (17) into Eq. (16) and balancing the most dominant terms, the results obtained as follows: α = 1; u 0 =6φ x. (18) 6

7 To find the resonances, the full Laurent expansion of the solution is substituted as u = u 0 φ u j φ j 1 + (19) in Eq. (16) and equating the coefficients of (φ j 5 ) by using Eq. (18), the polynomial equation in j is derived as j 4 10j 3 +23j 2 +10j 24 = 0. (20) From the Eq. (20), the resonance values are found to be j = 1, 1, 4, 6. (21) As usual, the resonance value at j = 1 represents the arbitrariness of the singular manifold φ(x, y, t) = 0. From the coefficients of (φ 5 ), the value of u 0 is obtained explicitly as given in Eq. (18). In order to prove the existence of arbitrary functions at the other resonance values, the full Laurent expansion Eq. (19) is substituted into Eq. (16). Collecting the coefficient of (φ 4 ), the following equation is obtained 8u 0 u 0x φ 2 x +2u2 0 φ xφ xx 24u 0x φ 3 x 36u 0φ 2 x φ xx =0. (22) Absence of u 1 in Eq. (22) proves that u 1 is arbitrary. This corresponds to the resonance value at j = 1. Then, collecting the coefficients of (φ 3 ), the following result is obtained u 2 = 1 [ 2φ 2 2φ 4 x φ xxx 2φ 3 x u 1x 3φ x φ 2 xx φ2 x φ ] y. (23) x Next, collecting the coefficients of (φ 2 ), the following result is obtained u 3 = 1 4φ 4 x [ φx φ t 2φ xx φ xxx +8u 2 φ 2 xφ xx +6u 1x φ x φ xx +2u 1xx φ 2 x + φ xxx +4φ x φ xxxx + φ xx φ y +2φ x φ xy ]. (24) Collecting the coefficient of (φ 1 ), the following equation is obtained φ xt + φ xxy + φ xxxxx +2u 1xx φ xx +2u 2 φ 2 xx 4u 3x φ 3 x 2u 2xx φ 2 x +2u 1xφ xxx 2u 2x φ x φ xx +2u 2 φ x φ xxx 12u 3 φ 2 x φ xx =0. (25) 7

8 Absence of u 4 in Eq. (25) proves that u 4 is arbitrary. This corresponds to the resonance value at j = 4. Collecting the coefficients of (φ 0 ), the following result is obtained u 5 = 1 24φ 4 x [u 1t + u 2 φ t + u 1xy + u 2x φ y + u 2y φ x + u 2 φ xy +2u 3 φ x φ y +18u 2xx φ xx +u 1xxxx +36u 3x φ x φ xx +30u 3 φ 2 xx 24u 4xφ 3 x +2u 1xu 1xx +4u 1x u 2x φ x +4u 1x u 3 φ 2 x +16u 2x φ xxx +2u 1xx u 2 φ x +4u 2 u 2x φ 2 x +2u 2 2φ x φ xx ] +2u 1x u 2 φ xx +4u 2xxx φ x + u 2 φ xxxx +32u 3 φ x φ xxx 36u 4 φ 2 xφ xx. (26) Collecting the coefficient of (φ 1 ), the following equation is obtained as u 2t +2u 3 φ t + u 2xy +2u 3x φ y +2u 3y φ x +2u 3 φ xy +24u 3xx φ xx + 108u 4x φ x φ xx +54u 4 φ 2 xx +24u 5 φ 2 x φ xx +2u 1x u 2xx +8u 1x u 3x φ x +4u 1x u 3 φ xx +2u 1xx u 2x +4u 2 2x φ x +2u 2 u 2x φ xx +2u 2 u 2xx φ x +8u 2 u 3 φ x φ xx +20u 3x φ xxx 24u 3x φ x φ xx +4u 1xx u 3 φ x +12u 2x u 3 φ 2 x +60u 4 φ x φ xxx +24u 5x φ 3 x + u 2xxxx +8u 3xxx φ x +2u 3 φ xxx +24u 4xx φ 2 x +24u 5 φ 4 x =0. (27) Absence of u 6 in Eq. (27) proves that u 6 is arbitrary. This corresponds to the resonance value at j = 6. As the system of Eq. (16) admits sufficient number of arbitrary functions, it is concluded that Eq. (16) passes the P-test and hence it is expected to be integrable. B. Associated Bäcklund transformation To construct the Bäcklund transformation of Eq. (16), let us truncate the Laurent series at the constant level term as u = u 0 φ 1 + u 1, (28) where the pair of function (u, u 1 ) satisfy Eq. (16) and hence Eq. (28) may be treated as the associated Bäcklund transformation of Eq. (16). C. Bilinear form To construct the bilinear form, we take the vacuum solution u 1 = 0 in Eq. (28) which leads to u = u 0 φ 1 =6 (ln φ). (29) x 8

9 With the help of the above dependent variable transformation in Eq. (16), the Hirota s bilinear form is found to be ( D 4 x + D x D y ) φ φ =0. (30) Having obtained the bilinear form, one can construct the soliton solution by expanding the dependent variables in terms of power series. IV. (2+1)-DIMENSIONAL PKP EQUATION A. Painlevé test Finally, we present the P-test of the (2+1)-Dimensional PKP equation. The (2+1)- Dimensional PKP equation takes the following form [6]: u xt u xu xx u xxxx u yy =0. (31) The leading order of the solution of Eq. (31) is assumed as u u 0 φ α. (32) On substituting Eq. (32) into Eq. (31) and equating the most dominant terms, the following results are obtained: α = 1; u 0 =2φ x. (33) For finding the resonances, the full Laurent series u = u 0 φ 1 + u j φ j 1 (34) is substituted in Eq. (31) and by equating the coefficients of (φ j 5 ), the polynomial equation in j is derived as j=1 Using Eq. (35), the resonances are found to be j 4 10j 3 +23j 2 +10j 24 = 0. (35) j = 1, 1, 4, 6. (36) 9

10 As usual, the resonance at j = 1 corresponds to the arbitrariness of the singular manifold φ(x, y, t) = 0. In order to check the existence of sufficient number of arbitrary functions at the other resonance values, the full Laurent expansion (34) is substituted in Eq. (31). From the coefficient of (φ 5 ), the explicit value of u 0 is obtained as given in Eq. (33). Collecting the coefficient of (φ 4 ), the following equation is obtained 6u 0 u 0x φ 2 x +3u2 0 φ xφ x 18u 0 φ 2 x φ xx 12u 0x φ 3 x =0. (37) Absence of u 1 in Eq. (37) proves that u 1 is arbitrary. This corresponds to the resonance value at j = 1. Then, collecting the coefficient of (φ 3 ), the explicit value of u 2 is obtained as u 2 = 1 [ 9φx φ 2 6φ 4 xx 3φ xφ 2 y 4φ2 x φ t 10φ 2 x φ ] xxx 6u 1x φ 3 x. (38) x Proceeding further to the coefficient of (φ 2 ), the value of u 3 can be obtained as u 3 = 1 12φ 4 x [ 4φxx φ t +8φ x φ xt 2φ xx φ xxx +6u 1xx φ 2 x +24u 2 φ 2 xφ xx +18u 1x φ x φ xx +4φ x φ xxxx + φ x φ xxx +6φ xy φ y +3φ x φ yy ]. (39) Collecting the coefficient of (φ 1 ), the following equation is obtained 4φ xxt +3φ xyy + φ xxxxx +6u 1xx φ x +12u 2x φ 2 xφ xx +6u 2 φ 2 xx 6u 2xx φ 2 x 12u 3x φ 3 x 36u 3φ 2 x φ xx +6u 1x φ xxx 18u 2x φ x φ xx +6u 2 φ x φ xxx =0. (40) Absence of u 4 in Eq. (40) proves that u 4 is arbitrary. This corresponds to the resonance value at j = 4. From the coefficient of (φ 0 ), the value of u 5 can be obtained as u 5 = 1 24φ 4 x [ u1xxxx +4u 1xt +4u 2x φ t +4u 2t φ x +4u 2 φ xt +8u 3 φ x φ t +6u 3 φ 2 y +3u 1yy +6u 2y φ y +3u 2 φ yy +30u 3 φ 2 xx 24u 4x φ 3 x 72u 4 φ 2 xφ xx +6u 1x u 1xx +12u 3 u 1x φ 2 x +16u 2xφ xxx +6u 1xx u 2 φ x +12u 2 u 2x φ 2 x +6u2 2 φ xφ xx +32u 3 φ x φ xxx +18u 2xx φ xx + u 2 φ xxxx +12u 3xx φ xx +36u 4 φ 4 x φ xx 2436u 5 φ 4 x ] +48u 3x φ 2 xφ xx +12u 1x u 2x φ x 12u 3x φ x φ xx +4u 2xxx φ x +6u 1x u 2 φ xx. (41) 10

11 Finally, collecting the coefficient of (φ 1 ), the following equation is obtained 4u 2xt +8u 3x φ t +8u 3t φ x +8u 3 φ xt +24u 4 φ x φ t +3u 2yy +12u 3y φ y +6u 3 φ yy +108u 4x φ x φ xx +54u 4 φ 2 xx +6u 1x u 2xx +24u 1x φ 3x φ x +12u 1x u 3 φ xx +24u 5 φ 4 x +36u 2x u 3 φ 2 x +6u 2u 2xx φ x +24u 2 u 3x φ 2 x +24u 2u 3 φ x φ xx +20u 3x φ xxx + u 2xxxx +6u 2 u 2x φ xx +8u 3xxx φ x +2u 3 φ xxxx +36u 4xx φ 2 x +24u 5xφ 3 x +24u 5φ 2 x φ xx +18u 4 φ 2 y +24u 3xxφ xx +6u 1xx u 2x +12u 2 2x φ x +12u 1xx u 3 φ x +60u 4 φ x φ xxx =0. (42) Absence of u 6 in Eq. (42) proves that u 6 is arbitrary. This corresponds to the resonance value at j = 6. As Eq. (31) admits sufficient number of arbitrary functions, it is concluded that the Eq. (31) passes the P-test and hence it is expected to be integrable. B. Associated Bäcklund transformation To construct the Bäcklund transformation of Eq. (31), let us truncate the Laurent series at the constant level term to give u = u 0 φ 1 + u 1, (43) where the pair of function (u, u 1 ) satisfy Eq. (31) and hence Eq. (43) may be treated as the associated Bäcklund transformation of Eq. (31). C. Bilinear form To construct the bilinear form, we take the vacuum solution u 1 = 0 in Eq. (43) which leads to u = u 0 φ 1 =2 (ln φ). (44) x With the help of the above dependent variable transformation in Eq. (31), the Hirota s bilinear form is found to be ( D x D t D4 x + 3 ) 4 D2 y φ φ =0. (45) Having obtained the bilinear form, one can construct the soliton solution by expanding the dependent variables in terms of power series. 11

12 V. CONCLUSION To conclude, the P-test for the (2+1)-Dimensional PKP, UBS and IR equations has been presented. All of these system of equations have the same polynomial equation. Finally, we have also derived its associated Bäcklund transformation and bilinear form directly from the P-test. Acknowledgments This work was performed under the partial support from the Brain Korea-21(BK-21) project, Ministry of Education and in part supported by IT-Professorship Program, IITA, MIC, Korea. K.N acknowledges the support of The Hong Kong Polytechnic University (Project No. G-YW85). K.N is also very grateful to P.K.A. Wai for all kinds of help. [1] Y. T. Gao and B. Tian, Computers Math. Applic., 30, 97 (1995). [2] Z. Y. Yan and H. Q. Zhang, Computers Math. Applic., 44, 1439 (2002). [3] R. Radha and M. Lakshmanan, Phys. Lett. A., 197, 7 (1995). [4] M. Faucher and P. Winternitz, Phys. Rev. E., 48, 3066 (1993). [5] E. Infeld and G. Rowlands, Phys. Rev. A., 43, 4537 (1991). [6] D. Kaya and S. M. El-Sayed, Phys. Lett. A., 320, 192 (2003). [7] M. Senthilvelan, Appl. Math. Comput., 123, 381 (2001). [8] D. S. Li and H. Q. Zhang, Appl. Math. Comput., 146, 381 (2003). [9] B. Dorizzi, B. Grammaticos, A. RamaniandP. Winternitz, J. Math. Phys., 27, 2848 (1986). 12