Article http://dx.doi.org/10.26855/jamc.2021.12.014

A Note on Two (3+1)-Dimensional Gardner-Type Equation with Multiple Kink Solutions


Afgan Aslanov

Computer Engineering Department, Istanbul Esenyurt University, Istanbul, Turkey.

*Corresponding author: Afgan Aslanov

Published: December 23,2021


It is difficult or impossible to find the exact solutions for many partial differential equations. In recent years, a variety of efficient and practical methods have been proposed by mathematicians. This article investigates the exact solutions of partial differential equations. The Gardner equation is well known in literature and is applicable in various branches of physics. The Gardner equation belongs to the category of non-linear partial differential equations. This equation and its generalizations are used in many areas of applications, such as hydrodynamics, plasma physics, and quantum field theory. The Gardner-type equations are the useful model to understand the propagation of negative ion acoustic plasma waves. These type of equations can be derived from the system of plasma motion equations in one dimension with arbitrarily charged cold ions and inertia neglected isothermal electrons. Numerous numerical and analytical methods have been used to study this equation. That proves the importance of this equation. In this paper, we examine the exact travelling wave solutions of the Gardner equation. The Hirota’s bilinear method and the Cole-Hopf transformation is used to obtain an elegant formula for the exact travelling wave solution. We demonstrate a correct formula for exact solutions. Mathematica software and the standart LaTex tools was used to perform the computations. The suggested approach can be used in other real-world models in science and engineering.


[1] Osborne, A. (2010). Nonlinear ocean waves and inverse scattering transform, Elsevier, Amsterdam, 2010.

[2] Konopelchenko, B. G. (1991). Inverse spectral transform for the (2+1)-dimensional Gardner equation, Inverse Problems, V. 7, 1991, pp. 739-753.

[3] Konopelchenko, B. G., Dubrovsky, V. G. (1983). On the general structure of nonlinear equations integrable by the general linear spectral problem, Phys. Lett. A, V.95, N.9, 1983, pp. 457-461.

[4] Yu, G.-F., Tam, H.-W. (2007). On the (2+1)—dimensional Gardner equation: Determinant solutions and pfaffianization, J. Math. Anal. Appl., V.330, 2007, pp. 989-1001.

[5] Wazwaz, A. M. (2007). New solitons and kink solutions for the Gardner equation, Commun. Nonlin. Sci. Numer. Simulat., V.12, N.8, 2007, pp. 1395-1404.

[6] Wazwaz, A. M. (2008). Solitons and singular solitons for the Gardner-KP equation, Appl. Math. Comput., V.204, N.1, 2008, pp. 162-169.

[7] Wazwaz, A. M. (2014). Multiple kink solutions for the (2+1)-dimensional integrable Gardner equation, Proc. Romanian Acad. A, V.15, 2014, pp. 241-246.

[8] Naz, R., Ali, Z., Naeem, I. (2013). Reductions and new exact solutions of ZK, Gardner KP, and modified KP Equations via generalized double reduction theorem, Abstract and Applied Analysis, 2013, 340564.

[9] Cai, P., Tang, J.-S., Li, Z.-B. Bifurcation of exact traveling wave solutions for Gardner and Gardner--KP equations, Intern. J. Appl. Math. Statistics, V.44, N.14, pp. 461-468.

[10] Hirota, R. (2004). The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004.

[11] Wazwaz, A. M. (2017). Two (3+1)-dimensional Gardner-type equation with multiple kink solutions, Rom. Rep. Phys., V.69, N.108, 2017.

[12] Porubov, A. V., Maugin, G. A., Andrievsky, B. R. (2011). Wave Motion, doi: 10.1016/j.wavemoti. 2011.04.012. 2011.

[13] Vassilev, V. M., Djongjorov, P. A., Hadzhilazova, M. Ts., Mladenov, I. M. (2011). Traveling Wave Solutions of the Gardner Equation and Motion of Plane Curves Governed by the mKdV Flow, AIP Conf. Proc., V.1404, N.86, 2011. doi: 10.1063/1.3659907.

How to cite this paper

A Note on Two (3+1)-Dimensional Gardner-Type Equation with Multiple Kink Solutions

How to cite this paper: Afgan Aslanov. (2021) A Note on Two (3+1)-Dimensional Gardner-Type Equation with Multiple Kink SolutionsJournal of Applied Mathematics and Computation5(4), 354-360.

DOI: http://dx.doi.org/10.26855/jamc.2021.12.014