On solutions of the combined KdV-nKdV equation

• Published in: Al-Mustansiriyah journal of science Vol. 30; no. 2; pp. 33 - 45
• Main Authors: Allami, Mohammed, Mutashar, A. K., Rashid, A. S.
• Format: Journal Article
• Language: English; Arabic
• Published: Al-Mustansiriyah University 30.09.2019
• ISSN: 1814-635X; 2521-3520
• DOI: 10.23851/mjs.v30i2.482

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.