The Riccati Equation Mapping Method for Solving Nonlinear Partial Differential Equations in Mathematical Physics

Elsayed Mohamed Elsayed ZAYED, Hoda Ibrahim Sayed AHMED


In this article, many new exact solutions of the (2+1)-dimensional nonlinear Boussinesq-Kadomtsev-Petviashvili equation and the (1+1)-dimensional nonlinear heat conduction equation are constructed using the Riccati equation mapping method. By means of this method, many new exact solutions are successfully obtained. This method can be applied to many other nonlinear evolution equations in mathematical physics.



The Riccati equation mapping method, the (2+1)-dimensional nonlinear Boussinesq-Kadomtsev-Petviashvili equation, the (1+1)-dimensional nonlinear heat conduction equation, exact solutions

Full Text:



WM Zhang and LX Tian. An extended tanh-method and its application to the soliton breaking equation. J. Phys. Conf. Ser. 2008; 96, 012069.

CQ Dai and YZ Ni. Novel interactions between semi-foldons of the (2+1)-dimensional Boiti-Leon-Pempinelli equation. Phys. Scr. 2006; 74, 584.

CL Zheng, LQ Chen and JF Zheng. Peakon, compacton and loop excitatations with periodic behavior in Kdv type models related to Schrödinger system. Phys. Lett. A 2005; 340, 397-402.

S Tsuchiya, F Dalfovo and L Pitaevskii. Solitons in two-dimensional Bose-Einstein condensates. Phys. Rev. A 2008; 77, 045601.

JP Gollub and MC Cross. Nonlinear dynamics: Chaos in space and time. Nature 2000; 404, 710-1.

CL Zheng, GP Cai and JY Qiang. Chaos, solitons and fractals in (2+1)-dimensional KdV system derived from a periodic wave solution. Chaos Soliton. Fract. 2007; 34, 1575-83.

JP Fang, CL Zheng and JM Zhu. New variable separation excitations, rectangle like solitons and fractal solitons in the Boiti-Leon-Pempinelli system. Acta Phys. Sin. 2005; 54, 2990-3006.

CL Zheng, JP Fang and LQ Chen. Bell-like and peak-like loop solitons in (2+1)-dimensional Boiti-Leon-Pempinelli system. Acta Phys. Sin. 2005; 54, 1468-508.

SD Zhu. The generalizing Riccati equation mapping method in nonlinear evolution equation: application to (2+1)-dimensional Boiti-Leon-Permpinelle equation. Chaos Soliton. Fract. 2008; 37, 1335-42.

HY Ruan and YX Chen. Study on solitons interaction in the (2+1)-dimensional Nizhnik-Novikov-Veselov equation. Acta Phys. Sin. 2003; 52, 1313-406.

ZY Ma and CL Zheng. Two classes of fractal structures for the (2+1)-dimensional dispersive long wave equation. Chin. Phys. 2006; 15, 45-108.

CQ Dai and YZ Ni. Novel interactions between solitons of the (2+1)-dimensional dispersive long wave equation. Chaos, Soliton. Fract. 2008; 37, 269-77.

SH Ma, QB Ren, JP Fang and CL Zheng. Special soliton structures and the phenomena of fission and annihilation of solitons for the (2+1)-dimensional Broer-Kaup system with variable coefficients. Acta Phys. Sin. 2007; 57, 6777-807.

JF Zhang, WH Ruang and CL Zheng. Coherent soliton structures of a new (2+1)-dimensional evolution equation. Acta Phys. Sin. 2002; 52, 2676-707.

SH Ma, JP Fang and QB Ren. New mapping solutions and localized structures for the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system. Acta Phys. Sin. 2007; 56, 6784-807.

SH Ma, XH Wu, JP Fang and CL Zheng. New exact solutions and special soliton structures for the (3+1)-dimensional Burgers system. Acta Phys. Sin. 2008; 57, 11-7.

SH Ma, JP Fang and HP Zhun. Dromion soliton waves and the their evolution in the background of Jacobi sine waves. Acta Phys. Sin. 2007; 56, 4319-407.

ZY Ma. The projective Riccati equation expansion method and variable-separation solutions for the nonlinear physical differential equation in physics. Chin. Phys. 2007; 16, 1848-54.

A Huber. A note on a class of solitary -like solutions of the Tzitzéica equation generated by a similarity reduction. Phys. D: Nonlinear Phenom. 2008; 237, 1079-87.

CL Bai, XQ Liu and H Zhao. New localized excitations in a (2+1)-dimensional Broer-Kaup system, Chin. Phys. 2005; 14, 285-92.

BG Konopelcheno and VG Dubrovsky. Some new integrable nonlinear evolution equation in (2+1)-dimensions. Phys. Lett. A 1984; 102, 15-7.

A Maccart. A new integrable Davey-Stewartson -type equation. J. Math. Phys.1999; 40, 3971-7.

Z Jiang and RK Sullough. Combined ã and Riemann-Hilbert inverse methods for integrable nonlinear evolution equations in (2+1)-dimensions. J. Phys. A: Math. Gen. 1987; 20, L429-L435.

J Lin, SY Lou and KL Wang. Multi-soliton solutions of the Konopelchenko-Dubrovsky equation. Chin. Phys. Lett. 2001; 18, 1173-5.

A Bekir. Applications of the extended tanh-method for coupled nonlinear evolution equations. Commun. Nonlinear Sci. Numer. Simulat. 2008; 13, 1748-57.

DS Wang and HQ Zhang. Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equation. Chaos Soliton. Fract. 2005; 25, 601-10.

LN Song and HQ Zhang. New exact solutions for Konopelchenko-Dubrovsky equation using an extended Riccati equation rational expansion method and symbolic computation. Appl. Math. Comput. 2007; 187, 1373-88.

AM Wazwaz. New kinks and solitons solutions to the (2+1)-dimensional Konopelchenko-Dubrovsky equation. Math. Comput. Model 2007; 45, 473-9.

S Zhang. Symbolic computation and new families of exact non-traveling wave solutions of (2+1)-dimensional Konopelchenko-Dubrovsky equations. Chaos Soliton. Fract. 2007; 31, 951-9.

S Zhang. The periodic wave solutions for the (2+1)-dimensional Konopelchenko-Dubrovsky equations. Chaos Soliton. Fract. 2006; 36, 1213-20.

TC Xia, ZS Lü and HQ Zhang. Symbolic computation and new families of exact soliton-like solutions of Konopelchenko-Dubrovsky equations. Chaos Soliton. Fract. 2004; 20, 561-6.

B Zheng. Exact solutions for two nonlinear equations. WSEAS Trans. Math. 2010; 9, 458-67.


  • There are currently no refbacks.

Online ISSN: 2228-835X

Last updated: 26 October 2017