TY - JOUR
AU - Allami, Mohammed
AU - Mutashar, A. K.
AU - Rashid, A. S.
PY - 2019/09/30
Y2 - 2022/09/29
TI - On solutions of the combined KdV-nKdV equation
JF - Al-Mustansiriyah Journal of Science
JA - MJS
VL - 30
IS - 2
SE - Mathematics
DO - 10.23851/mjs.v30i2.482
UR - https://mjs.uomustansiriyah.edu.iq/index.php/MJS/article/view/482
SP - 33-45
AB - 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.
ER -