An extended time-dependent KdV6 equation

Abdul-Majid Wazwaz (Department of Mathematics, Saint Xavier University, Chicago, Illinois, USA)
Gui-Qiong Xu (Shanghai University, Shanghai, China)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Publication date: 24 June 2019



The purpose of this paper is to develop a new time-dependent KdV6 equation. The authors derive multiple soliton solutions and multiple complex soliton solutions for a time-dependent equation.


The newly developed time-dependent model has been handled by using the Hirota’s direct method. The authors also use the complex Hirota’s criteria for deriving multiple complex soliton solutions.


The examined extension of the KdV6 model exhibits complete integrability for any analytic time-dependent coefficient.

Research limitations/implications

The paper presents a new efficient algorithm for constructing extended models which give a variety of multiple real and complex soliton solutions.

Practical implications

The paper introduced a new time-dependent KdV6 equation, where integrability is emphasized for any analytic time-dependent function.

Social implications

The findings are new and promising. Multiple real and multiple complex soliton solutions were formally derived.


This is an entirely new work where a new time-dependent KdV6 equation is established. This is the first time that the KdV6 equation is examined as a time-dependent equation. Moreover, the complete integrability of this newly developed equation is emphasized via using Painlevé test.



Wazwaz, A. and Xu, G. (2019), "An extended time-dependent KdV6 equation", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. ahead-of-print No. ahead-of-print.

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