Eur. Phys. J. Plus (2021) 136: 638
https://doi.org/10.1140/epjp/s13360-021-01626-2

Regular Article

Integrability, qualitative analysis and the dynamics of wave solutions for Biswas–Milovic equation

¹ Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, 31982, Al-Ahsa, Saudi Arabia
² Department of Mathematics, Faculty of Science, Mansoura University, 35516, Mansoura, Egypt

^a aelmandouh@kfu.edu.sa, adel78@mans.edu.eg

Received: 3 March 2021
Accepted: 29 May 2021
Published online: 6 June 2021

Abstract

We are interested in studying a Biswas–Milovic equation from various aspects. We prove that it does not have Painlevé property, and hence, it is non-integrable in Painlevé sense. Applying certain wave transformations, it turns to an ordinary differential equation which is equivalent to a Hamiltonian system. Based on the qualitative theorem of planar systems, we introduce the conditions that guarantee the existence of periodic, solitary and kink solutions for this equation in addition to some other singular solutions. The degeneracy of these solutions resulting from the transmission between the orbits is discussed. The 3D and 2D graphic representations for some wave solutions of this equation are presented beside the orbit relating to this solution.