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**Exact solutions of \(\phi^4\) equation using Lie symmetry approach along with the simplest equation and exp-function methods.**
*(English)*
Zbl 1257.35075

Summary: This paper obtains the exact solutions of the \(\phi^4\) equation. The Lie symmetry approach along with the simplest equation method and the Exp-function method are used to obtain these solutions. As a simplest equation we have used the equation of Riccati in the simplest equation method. Exact solutions obtained are travelling wave solutions.

### MSC:

35G20 | Nonlinear higher-order PDEs |

35C07 | Traveling wave solutions |

35B06 | Symmetries, invariants, etc. in context of PDEs |

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\textit{H. Jafari} et al., Abstr. Appl. Anal. 2012, Article ID 350287, 7 p. (2012; Zbl 1257.35075)

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### References:

[1] | M. Duranda and D. Langevin, “Physicochemical approach to the theory of foam drainage,” The European Physical Journal E, vol. 7, pp. 35-44, 2002. |

[2] | E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212-218, 2000. · Zbl 1167.35331 |

[3] | L. A. Ostrovsky, “Nonlinear internal waves in a rotating ocean,” Oceanology, vol. 18, pp. 119-125, 1978. |

[4] | A.-M. Wazwaz, “The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1467-1475, 2007. · Zbl 1119.65100 |

[5] | A.-M. Wazwaz, “The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations,” Chaos, Solitons & Fractals, vol. 25, no. 1, pp. 55-63, 2005. · Zbl 1070.35076 |

[6] | A.-M. Wazwaz, “The sine-cosine method for obtaining solutions with compact and noncompact structures,” Applied Mathematics and Computation, vol. 159, no. 2, pp. 559-576, 2004. · Zbl 1061.35121 |

[7] | R. Abazari, “Application of (G\(^{\prime}\)/G)-expansion method to travelling wave solutions of three nonlinear evolution equation,” Computers & Fluids, vol. 39, no. 10, pp. 1957-1963, 2010. · Zbl 1245.76096 |

[8] | E. Salehpour, H. Jafari, and N. Kadkhoda, “Application of, (G\(^{\prime}\)/G)-expansion method to nonlinear Lienard equation,” Indian Journal of Science and Technology, vol. 5, pp. 2554-2556, 2012. |

[9] | H. Jafari, N. Kadkhoda, and C. M. Khalique, “Travelling wave solutions of nonlinear evolution equations using the simplest equation method,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 2084-2088, 2012. · Zbl 1268.35107 |

[10] | N. A. Kudryashov, “Simplest equation method to look for exact solutions of nonlinear differential equations,” Chaos, Solitons & Fractals, vol. 24, no. 5, pp. 1217-1231, 2005. · Zbl 1069.35018 |

[11] | N. A. Kudryashov and N. B. Loguinova, “Extended simplest equation method for nonlinear differential equations,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 396-402, 2008. · Zbl 1168.34003 |

[12] | N. K. Vitanov, “Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 2050-2060, 2010. · Zbl 1222.35062 |

[13] | J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700-708, 2006. · Zbl 1141.35448 |

[14] | X.-H. Wu and J.-H. He, “EXP-function method and its application to nonlinear equations,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 903-910, 2008. · Zbl 1153.35384 |

[15] | X. W. Zhou, Y. X. Wen, and J. H. He, “Exp-function method to solve the non-linear dispersive K(m,n) equations,” International Journal of Nonlinear Science and Numerical Simulation, vol. 9, pp. 301-306, 2008. · Zbl 06942352 |

[16] | G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. · Zbl 0698.35001 |

[17] | N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1-3, CRC Press, Boca Raton, Fla, USA, 19941996. · Zbl 0864.35001 |

[18] | A. G. Johnpillai and C. M. Khalique, “Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1207-1215, 2011. · Zbl 1221.35338 |

[19] | P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 2nd edition, 1993. · Zbl 0785.58003 |

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