Journal of Applied Mathematics and Computing https://doi.org/10.1007/s12190-020-01446-0 ORIGINAL RESEARCH A qualitative study and numerical simulations for a time-delayed dispersive equation Kaïs Ammari 1 · Boumediène Chentouf 2 · Nejib Smaoui 2 Received: 13 August 2020 / Revised: 4 October 2020 / Accepted: 9 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020 Abstract This paper deals with the stability analysis of a nonlinear time-delayed dispersive equation of order four. First, we prove the well-posedness of the system and give some regularity results. Then, we show that the zero solution of the system exponentially converges to zero when the time tends to infinity provided that the time-delay is small and the damping term satisfies reasonable conditions. Lastly, an intensive numerical study is put forward and numerical illustrations of the stability result are provided. Keywords Nonlinear dispersive equation · Time-delay · Stability · Numerical simulations Mathematics Subject Classification 35L05 · 35M10 1 Introduction The qualitative and numerical analysis of nonlinear dispersive equations has attracted the attention of a huge number of authors from various disciplines. This is due to the fact that such equations describe miscellaneous physical phenomena, such as surface water waves in shallow water [22,38], turbulent states in a distributed chemical reaction system and plane flame propagation [40,52], propagation of ion-acoustic waves in plasma, and pressure waves in liquid–gas bubble mixture [27,30,37,41,65–67]. B Boumediène Chentouf boumediene.chentouf@ku.edu.kw Kaïs Ammari kais.ammari@fsm.rnu.tn Nejib Smaoui n.smaoui@ku.edu.kw 1 UR Analysis and Control of PDEs, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir, Tunisia 2 Department of Mathematics, Faculty of Science, Kuwait University, 13060 Safat, Kuwait 123