×

Infinitely many elliptic solutions to a simple equation and applications. (English) Zbl 1302.35113

The aim of the paper is to show that certain nonlinear PDE could have infinitely many solutions presented in terms of Weierstrass and Jacobi elliptic functions.
Main ingredient of the method is a nonlinear iterative formula of solutions. The authors first defined such formula for simple auxiliary differential equation \[ \left(\frac{d\eta(\xi)}{d \xi}\right)^2 = a \eta + b \eta^2 + c \eta^3. \tag{1} \] The nonlinear iterative formula for equation (1) is obtained by using Bäcklund transformation and a modified truncation approach. By using this formula the authors obtained infinitely many Weierstrass and Jacobi solutions.
The proposed method is applied to obtain infinitely many Weierstrass and Jacobi solutions to a number of special PDEs, namely \[ (u_t - 6 u u_{xx} + u_{xxx})_x + 3 u_{yy} = 0, \]
\[ \left(\frac{u_{xx}}{u}\right)_t + 2 u u_x = 0, \]
\[ \begin{aligned} i u_t + u_{xx} - u_{yy} - 2 |u|^2 u - 2 u v &= 0, \\ v_{xx} + v_{yy} + 2 \left(|u|^2\right)_{xx} &= 0. \end{aligned} \]

MSC:

35G20 Nonlinear higher-order PDEs
33E05 Elliptic functions and integrals
35A24 Methods of ordinary differential equations applied to PDEs

Software:

ATFM
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Parkes, E. J.; Duffy, B. R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Computer Physics Communications, 98, 3, 288-300 (1996) · Zbl 0948.76595 · doi:10.1016/0010-4655(96)00104-X
[2] Malfliet, W.; Hereman, W., The tanh method. I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54, 6, 563-568 (1996) · Zbl 0942.35034 · doi:10.1088/0031-8949/54/6/003
[3] Fan, E., Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 277, 4-5, 212-218 (2000) · Zbl 1167.35331 · doi:10.1016/S0375-9601(00)00725-8
[4] Wazwaz, A.-M., The tanh method for traveling wave solutions of nonlinear equations, Applied Mathematics and Computation, 154, 3, 713-723 (2004) · Zbl 1054.65106 · doi:10.1016/S0096-3003(03)00745-8
[5] Ma, W. X., Travelling wave solutions to a seventh order generalized KdV equation, Physics Letters A, 180, 3, 221-224 (1993) · doi:10.1016/0375-9601(93)90699-Z
[6] Parkes, E. J.; Zhu, Z.; Duffy, B. R.; Huang, H. C., Sech-polynomial travelling solitary-wave solutions of odd-order generalized KdV equations, Physics Letters A, 248, 2-4, 219-224 (1998)
[7] Wang, M. L., Solitary wave solutions for variant Boussinesq equations, Physics Letters A, 199, 3-4, 169-172 (1995) · Zbl 1020.35528 · doi:10.1016/0375-9601(95)00092-H
[8] Wang, M.; Zhou, Y.; Li, Z., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216, 1-5, 67-75 (1996) · Zbl 1125.35401
[9] Wazwaz, A.-M., The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Applied Mathematics and Computation, 184, 2, 1002-1014 (2007) · Zbl 1115.65106 · doi:10.1016/j.amc.2006.07.002
[10] Liu, S.; Fu, Z.; Liu, S.; Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A, 289, 1-2, 69-74 (2001) · Zbl 0972.35062 · doi:10.1016/S0375-9601(01)00580-1
[11] He, J.-H.; Wu, X.-H., Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30, 3, 700-708 (2006) · Zbl 1141.35448 · doi:10.1016/j.chaos.2006.03.020
[12] Liu, J.; Yang, K., The extended \(F\)-expansion method and exact solutions of nonlinear PDEs, Chaos, Solitons & Fractals, 22, 1, 111-121 (2004) · Zbl 1062.35105 · doi:10.1016/j.chaos.2003.12.069
[13] Ma, W.-X.; Lee, J.-H., A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation, Chaos, Solitons & Fractals, 42, 3, 1356-1363 (2009) · Zbl 1198.35231 · doi:10.1016/j.chaos.2009.03.043
[14] Ma, W.-X.; Wu, H.; He, J., Partial differential equations possessing Frobenius integrable decompositions, Physics Letters A, 364, 1, 29-32 (2007) · Zbl 1203.35059 · doi:10.1016/j.physleta.2006.11.048
[15] Ma, W.-X.; Zhu, Z., Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Applied Mathematics and Computation, 218, 24, 11871-11879 (2012) · Zbl 1280.35122 · doi:10.1016/j.amc.2012.05.049
[16] Ma, W. X., Bilinear equations, Bell polynomials and linear superposition principle, Journal of Physics: Conference Series, 411 (2013) · doi:10.1088/1742-6596/411/1/012021
[17] Ma, W. X., Generalized bilinear differential equations, Studies in Nonlinear Sciences, 2, 140-144 (2011)
[18] Wei, L., A function transformation method and exact solutions to a generalized sinh-Gordon equation, Computers & Mathematics with Applications, 60, 11, 3003-3011 (2010) · Zbl 1207.35226 · doi:10.1016/j.camwa.2010.09.062
[19] Wei, L., Exact solutions to a combined sinh-cosh-Gordon equation, Communications in Theoretical Physics, 54, 4, 599-602 (2010) · Zbl 1219.35265 · doi:10.1088/0253-6102/54/4/03
[20] Demina, M. V.; Kudryashov, N. A., From Laurent series to exact meromorphic solutions: the Kawahara equation, Physics Letters A, 374, 39, 4023-4029 (2010) · Zbl 1238.34020 · doi:10.1016/j.physleta.2010.08.013
[21] Demina, M. V.; Kudryashov, N. A., On elliptic solutions of nonlinear ordinary differential equations, Applied Mathematics and Computation, 217, 23, 9849-9853 (2011) · Zbl 1220.65090 · doi:10.1016/j.amc.2011.04.043
[22] Demina, M. V.; Kudryashov, N. A., Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations, Communications in Nonlinear Science and Numerical Simulation, 16, 3, 1127-1134 (2011) · Zbl 1221.34233 · doi:10.1016/j.cnsns.2010.06.035
[23] Kudryashov, N. A.; Sinelshchikov, D. I.; Demina, M. V., Exact solutions of the generalized Bretherton equation, Physics Letters A, 375, 7, 1074-1079 (2011) · Zbl 1242.37054 · doi:10.1016/j.physleta.2011.01.010
[24] Kudryashov, N. A.; Sinelshchikov, D. I., Exact solutions of the Swift-Hohenberg equation with dispersion, Communications in Nonlinear Science and Numerical Simulation, 17, 1, 26-34 (2012) · Zbl 1245.35095 · doi:10.1016/j.cnsns.2011.04.008
[25] Lawden, D. F., Elliptic Functions and Applications. Elliptic Functions and Applications, Applied Mathematical Sciences, 80, xiv+334 (1989), New York, NY, USA: Springer, New York, NY, USA · Zbl 0689.33001
[26] Yan, Z., An improved algebra method and its applications in nonlinear wave equations, Chaos, Solitons & Fractals, 21, 4, 1013-1021 (2004) · Zbl 1046.35103 · doi:10.1016/j.chaos.2003.12.042
[27] Kadomtsev, B. B.; Petviashvili, V. I., On the stability of solitary waves in weakyly dispersive media, Soviet Physics. Doklady, 15, 16, 539-541 (1970) · Zbl 0217.25004
[28] Qiao, Z.; Li, J., Negative-order KdV equation with both solitons and kink wave solutions, EPL, 94, 5, 1-5 (2011) · doi:10.1209/0295-5075/94/50003
[29] Hase, Y.; Satsuma, J., An \(N\)-soliton solution for the nonlinear Schrödinger equation coupled to the Boussinesq equation, Journal of the Physical Society of Japan, 57, 3, 679-682 (1988) · doi:10.1143/JPSJ.57.679
[30] Fan, E.; Zhang, J., Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters A, 305, 6, 383-392 (2002) · Zbl 1005.35063 · doi:10.1016/S0375-9601(02)01516-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.