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Exact travelling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVB equation using the new approach of generalized \(\left (\boldsymbol { {G^{\prime }/G}} \right )\)-expansion method

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Abstract

In this paper, the new generalized (\(G^{\prime } /G)\)-expansion method is executed to find the travelling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVB equation. The efficiency of this method for finding exact and travelling wave solutions has been demonstrated. It is shown that the new approach of generalized (\({G}^{\prime }/G)\)-expansion method is a straightforward and effective mathematical tool for solving nonlinear evolution equations in applied mathematics, mathematical physics and engineering. Moreover, this procedure reduces the large volume of calculations.

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Authors and Affiliations

  1. Department of Mathematics, Pabna University of Science and Technology, Pabna, Bangladesh

    MD NUR ALAM & M FAZLUL HOQUE

  2. Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh

    M ALI AKBAR

Authors
  1. MD NUR ALAM
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  2. M ALI AKBAR
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  3. M FAZLUL HOQUE
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Correspondence to MD NUR ALAM.

Appendix. Zayed solutions

Appendix. Zayed solutions

Zayed [30] examined the exact solutions of the nonlinear (3+1)-dimensional mKdV-ZK equation by using the \((G^{\prime } /G)\)-expansion method. He found the following five solutions of the form:

$$ u(\xi )=-3\sqrt{\frac{-2}{\alpha }}i\sec h\xi , $$
(A.1)
$$ u(\xi )=3\sqrt{\frac{-2}{\alpha }}i\sec \xi , $$
(A.2)
$$ u(\xi )=\pm 3\sqrt{\frac{-2}{\alpha }}(\coth \xi -\tanh \xi ), $$
(A.3)
$$u(\xi )=\pm 3\sqrt{\frac{-2}{\alpha }}(\cot \xi +\tan \xi ), $$
(A.4)
$$ u(\xi )=\pm 3\sqrt{\frac{-2}{\alpha }}\left({\frac{B}{B\xi +c_{1} }} \right). $$
(A.5)

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ALAM, M.N., AKBAR, M.A. & HOQUE, M.F. Exact travelling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVB equation using the new approach of generalized \(\left (\boldsymbol { {G^{\prime }/G}} \right )\)-expansion method. Pramana - J Phys 83, 317–329 (2014). https://doi.org/10.1007/s12043-014-0776-8

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  • DOI: https://doi.org/10.1007/s12043-014-0776-8

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