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.

We consider some nonlinear partial differential equations in higher dimensions, the negative order of the Calogero-Bogoyavelnskii-Schiff (nCBS) equationin (2+1) dimensions, the combined of the Calogero-Bogoyavelnskii-Schiff equation and the negative order of the Calogero-Bogoyavelnskii-Schiff equation (CBS-nCBS) in (2+1) dimensions, and two models of the negative order Korteweg de Vries (nKdV) equations in (3+1) dimensions. We show that these equations can be reduced to the same class of ordinary differential equations via wave reduction variable. Solutions in terms of symmetrical Fibonacci and Lucas functions are presented by implementation of the modified Kudryashov method.

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