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New exact solutions of the Tzitzéica type equations arising in nonlinear optics using a modified version of the improved \(\tan \left( {\varPhi \left( \xi \right)/2} \right)\)-expansion method

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Abstract

The paper deals with the Tzitzéica type nonlinear evolution equations arising in nonlinear optics and their new exact solutions. First, through the use of the Painlevé transformation and Lie symmetry method, the Tzitzéica, Dodd–Bullough–Mikhailov, and Tzitzéica–Dodd–Bullough equations are converted to nonlinear ordinary differential equations (NODEs), and then, a modified version of the improved \(\tan \left( {\varPhi \left( \xi \right)/2} \right)\)-expansion method, proposed by the authors, is adopted to generate new exact solutions of the reduced equations. The method truly recommends a reliable and capable technique to produce new exact solutions of a variety of nonlinear partial differential equations (NPDEs).

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Authors and Affiliations

  1. Department of Mathematics, Guilan Science and Research Branch, Islamic Azad University, Rasht, Iran

    K. Hosseini

  2. Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran

    K. Hosseini

  3. Department of Engineering Sciences, Faculty of Technology and Engineering, University of Guilan, East of Guilan, P. C. 44891-63157, Rudsar, Iran

    Z. Ayati

  4. Department of Mechanical Engineering, University of Guilan, P. O. Box 3756, Rasht, Iran

    R. Ansari

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  1. K. Hosseini
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  2. Z. Ayati
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Correspondence to Z. Ayati.

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Hosseini, K., Ayati, Z. & Ansari, R. New exact solutions of the Tzitzéica type equations arising in nonlinear optics using a modified version of the improved \(\tan \left( {\varPhi \left( \xi \right)/2} \right)\)-expansion method. Opt Quant Electron 49, 273 (2017). https://doi.org/10.1007/s11082-017-1094-4

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