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New generalized Jacobi elliptic function rational expansion method. (English) Zbl 1219.35219
Summary: A new generalized Jacobi elliptic function rational expansion method, based upon twenty-four Jacobi elliptic functions and eight double periodic Weierstrass elliptic functions, which solve the elliptic equation ϕ '2 =r+pϕ 2 +qϕ 4 , is described. As a consequence abundant new Jacobi-Weierstrass double periodic elliptic functions solutions for the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation are obtained by using this method. We show that the new method can be also used to solve other nonlinear partial differential equations (NPDEs) in mathematical physics.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35A24Methods of ordinary differential equations for PDE
35C10Series solutions of PDE
35C05Solutions of PDE in closed form
References:
[1]M.J. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge, 1991. · Zbl 0762.35001
[2]Hu, X. B.; Ma, W. X.: Application of Hirota’s bilinear formalism to the Toeplitz lattice some special soliton-like solutions, Phys. lett. A 293, 161 (2002) · Zbl 0985.35072 · doi:10.1016/S0375-9601(01)00850-7
[3]Ali, A. T.: New exact solutions of Einstein vacuum equations for rotating axially symmetric fields, Phys. scr. 79, No. 3, 035006 (2009) · Zbl 1172.83306 · doi:10.1088/0031-8949/79/03/035006
[4]Attallah, S. K.; El-Sabbagh, M. F.; Ali, A. T.: Isovector fields and similarity solutions of Einstein vacuum equations for rotating fields, Commun. nonlinear sci. Numer. simul. 12, No. 7, 1153 (2007)
[5]Mekheimer, K. S.; Husseny, S. Z.; Ali, A. T.; Abo-Elkhair, R. E.: Lie point symmetries and similarity solutions for an electrically conducting Jeffrey fluid, Phys. scr. 83, No. 1, 015017 (2011) · Zbl 1217.76088 · doi:10.1088/0031-8949/83/01/015017
[6]Suhubi, E. S.: Isovector fields and similarity solutions for general balance equations, Internat. J. Engrg. sci. 29, 133 (1991) · Zbl 0793.35018 · doi:10.1016/0020-7225(91)90083-F
[7]Reyes, E. G.; Sanchez: Explicit solutions to the Kaup–kupershmidt equation via nonlocal symmetries, Bifur. chaos 17, No. 3, 2749 (2007) · Zbl 1141.37355 · doi:10.1142/S0218127407018737
[8]Reyes, E. G.: Nonlocal symmetries and the Kaup–kupershmidt equation via nonlocal symmetries, J. math. Phys. 46, No. 7, 073507 (2005) · Zbl 1110.37052 · doi:10.1063/1.1939988
[9]Wang, M. L.; Zhou, Y. B.; Li, Z. B.: Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. lett. A 216, 67 (1996) · Zbl 1125.35401 · doi:10.1016/0375-9601(96)00283-6
[10]Ali, A. T.: A note on the exp-function method and its application to nonlinear equations, Phys. scr. 79, No. 2, 025006 (2009) · Zbl 1200.35246 · doi:10.1088/0031-8949/79/02/025006
[11]Ali, A. T.; Hassan, E. R.: General expa-function method for nonlinear evolution equations, Appl. math. Comput. 217, No. 2, 451 (2010) · Zbl 1201.65180 · doi:10.1016/j.amc.2010.06.025
[12]He, J. H.; Wu, X. H.: Exp-function method for nonlinear wave equation, Chaos solitons fractals 30, 700 (2006) · Zbl 1141.35448 · doi:10.1016/j.chaos.2006.03.020
[13]Yan, C. T.: A simple transformation for nonlinear waves, Phys. lett. A 224, 77 (1996) · Zbl 1037.35504 · doi:10.1016/S0375-9601(96)00770-0
[14]El-Wakil, S. A.; Abdou, M. A.: New exact traveling wave solutions using modified extended tanh-function method, Chaos solitons fractals 31, 840 (2007) · Zbl 1139.35388 · doi:10.1016/j.chaos.2005.10.032
[15]Fan, E. G.: Extended tanh-function method and its applications to nonlinear equations, Phys. lett. A 277, 212 (2000) · Zbl 1167.35331 · doi:10.1016/S0375-9601(00)00725-8
[16]Wu, R.; Sun, J.: Soliton-like solutions to the gKdV equation by extended mapping method, Chaos solitons fractals 31, 70 (2007) · Zbl 1138.35405 · doi:10.1016/j.chaos.2005.09.032
[17]Yomba, E.: On exact solutions of the coupled Klein–gordan–Schrödinger and the complex coupled KdV equations using mapping method, Chaos solitons fractals 21, 209 (2004) · Zbl 1046.35105 · doi:10.1016/j.chaos.2003.10.028
[18]Zhang, S.; Xie, T.: An improved generalized F-expansion method and its application to the (2+1)-dimensional KdV equations, Commun. nonlinear sci. Numer. simul. 13, 1294 (2008) · Zbl 1221.35384 · doi:10.1016/j.cnsns.2006.12.008
[19]Song, L. N.; Wang, Q.; Zheng, Y.; Zhang, H. Q.: A new extended Riccati equation rational expansion method and its application, Chaos solitons fractals 31, 548 (2007) · Zbl 1138.35403 · doi:10.1016/j.chaos.2005.10.008
[20]Chen, Y.; Yan, Z.: The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations, Chaos solitons fractals 29, 948 (2006) · Zbl 1142.35603 · doi:10.1016/j.chaos.2005.08.071
[21]Li, Z.; Dong, H.: Abundant new travelling wave solutions for the (2+1)-dimensional sine-Gordon equation, Chaos solitons fractals 37, 547 (2008) · Zbl 1143.35357 · doi:10.1016/j.chaos.2006.09.030
[22]Chen, H. T.; Zhang, H. Q.: Improved Jacobian elliptic function method and its applications, Chaos solitons fractals 15, 585 (2003) · Zbl 1037.35027 · doi:10.1016/S0960-0779(02)00147-9
[23]Chen, H. T.; Yin, H.: A note on the elliptic equation method, Commun. nonlinear sci. Numer. simul. 13, 547 (2008) · Zbl 1131.35069 · doi:10.1016/j.cnsns.2006.06.007
[24]Chen, Y.; Wang, Q.; Li, B.: Elliptic equation rational expansion method and new traveling solutions for Whitham–Broer–Kaup equations, Chaos solitons fractals 26, 231 (2005) · Zbl 1080.35080 · doi:10.1016/j.chaos.2004.12.020
[25]El-Sabbagh, M. F.; Ali, A. T.: New generalized Jacobi elliptic function expansion method, Commun. nonlinear sci. Numer. simul. 13, 1758 (2008) · Zbl 1221.35331 · doi:10.1016/j.cnsns.2007.04.014
[26]Fu, Z. T.; Liu, S. K.; Liu, S. D.; Zhao, Q.: New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. lett. A 290, 72 (2001) · Zbl 0977.35094 · doi:10.1016/S0375-9601(01)00644-2
[27]Liu, S. K.; Fu, Z. T.; Liu, S. D.; Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. lett. A 289, 69 (2001) · Zbl 0972.35062 · doi:10.1016/S0375-9601(01)00580-1
[28]Chen, Y.; Wang, Q.; Li, B.: Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly-periodic solutions of nonlinear evolution equations, Z. nat.forsch. A 59, No. 9, 536 (2004)
[29]Chen, Y.; Wang, Q.: A new Riccati equation rational expansion method and its application, Z. nat.forsch. A 60, No. 1–2, 1 (2005)
[30]Chen, Y.; Wang, Q.: A new elliptic equation rational expansion method and its application to the shallow long wave approximate equations, Appl. math. Comput. 173, 1163 (2006) · Zbl 1088.65087 · doi:10.1016/j.amc.2005.04.061
[31]Wang, Q.; Chen, Y.; Zhang, H.: A new Jacobi elliptic function rational expansion method and its application to (1+1)-dimensional dispersive long wave equation, Chaos solitons fractals 23, 477 (2005) · Zbl 1072.35510 · doi:10.1016/j.chaos.2004.04.029
[32]Wang, Q.; Zhang, H.: A new Riccati equation rational expansion method and its application to (2+1)-dimensional Burgers equation, Chaos solitons fractals 25, 1019 (2005) · Zbl 1070.35073 · doi:10.1016/j.chaos.2005.01.039
[33]Chen, Y.; Yan, Z.; Zhang, H.: New explicit solitary wave solutions for (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation, Phys. lett. A 307, 107 (2003) · Zbl 1006.35083 · doi:10.1016/S0375-9601(02)01668-7
[34]El-Sabbagh, M. F.; Ali, A. T.: New exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized (2+1)-dimensional Boussinesq equation, Int. J. Nonlinear sci. Numer. simul. 6, No. 2, 151 (2005)
[35]Xie, F.; Zhang, Y.; Lü, Z.: Symbolic computation in non-linear evolution equation: application to (3+1)-dimensional KP equation, Chaos solitons fractals 24, 257 (2005) · Zbl 1067.35095 · doi:10.1016/j.chaos.2004.09.019
[36]Ablowitz, M. J.; Kaup, D.; Newell, A.; Seger, H. J.: The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. appl. Math. 35, 249 (1975)
[37]Ablowitz, M. J.; Seger, H. J.: On the evolution of packets of water waves, J. fluid. Mech. 92, 539 (1979) · Zbl 0413.76009 · doi:10.1017/S0022112079000835
[38]Ablowitz, M. J.; Yaacov, D. B.; Fokas, A. S.: On the inverse scattering transform for the Kadomtsev–Petviashvili equation, Stud. appl. Math. 69, 135 (1983) · Zbl 0527.35080
[39]Sirendaoreji, S. J.: Auxiliary equation method for solving nonlinear partial differential equations, Phys. lett. A 309, 387 (2003) · Zbl 1011.35035 · doi:10.1016/S0375-9601(03)00196-8
[40]Elgarayhi, A.: New periodic wave solutions the shallow water equations and the generalized Klein–Gordon equation, Commun. nonlinear sci. Numer. simul. 13, 877 (2008) · Zbl 1221.35332 · doi:10.1016/j.cnsns.2006.07.013
[41]El-Sabbagh, M. F.; Ali, A. T.; El-Ganaini, S.: New abundant exact solutions for the system of (2+1)-dimensional Burgers equations, Appl. math. Inf. sci. 2, No. 1, 31 (2008) · Zbl 1146.35078
[42]Zhang, S.: Application of exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer–Kaup–kupershmidt equations, Phys. lett. A 2, No. 1, 31 (2008)
[43]Khani, F.; Hamedi-Nezhad, S.; Darvishi, M. T.; Ryu, S. W.: New solitary wave and periodic solutions of the foam drainage equation using exp-function method, Nonlinear anal. RWA 10, No. 3, 1904 (2009) · Zbl 1168.35302 · doi:10.1016/j.nonrwa.2008.02.030