On solutions of the combined KdV-nKdV equation

Authors

  • Mohammed Allami Mathematics Department, college of education, Misan university
  • A. K. Mutashar Mathematics Department, college of education, Misan university
  • A. S. Rashid Mathematics Department, college of education, Misan university

DOI:

https://doi.org/10.23851/mjs.v30i2.482

Abstract

The aim of this work is to deal with a new integrable nonlinear equation of wave propagation, the combined of the Korteweg-de vries equation and the negative order Korteweg-de vries equation (combined KdV-nKdV) equation, which was more recently proposed by Wazwaz. Upon using wave reduction variable, it turns out that the reduced combined KdV-nKdV equation is alike the reduced (3+1)-dimensional Jimbo Miwa (JM) equation, the reduced (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (PYTSF) equation and the reduced (3 + 1)¬dimensional generalized shallow water (GSW) equation in the trav¬elling wave. In fact, the four transformed equations belong to the same class of ordinary differential equation. With the benefit of a well known general solutions for the reduced equation, we show that sub¬jects to some scaling and change of parameters, a variety of families of solutions are constructed for the combined KdV-nKdV equation which can be expressed in terms of rational functions, exponential functions and periodic solutions of trigonometric functions and hyperbolic func¬tions. In addition to that the equation admits solitary waves, and double periodic waves in terms of special functions such as Jacobian elliptic functions and Weierstrass elliptic functions.

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References

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Published

30-09-2019

How to Cite

[1]
M. Allami, A. K. Mutashar, and A. S. Rashid, “On solutions of the combined KdV-nKdV equation”, Al-Mustansiriyah Journal of Science, vol. 30, no. 2, pp. 33–45, Sep. 2019.

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Section

Mathematics