This research is based on three main pillars which are studying the computational solutions of the modified Benjamin–Bona–Mahony (BBM) equation via the modified Khater method then investigating the stability property of the obtained solutions through the Hamiltonian system’s features, evaluating the initial and boundary conditions that allow applying the B-spline collection schemes (cubic, quantic, septic) to find the numerical solutions of the suggested model and to explain the matching between our obtained types of solutions. This model is used in the optical illusions field by describing the propagation of long waves in the nonlinear dispersive media in a visual illusion. An optical illusion is an illusion caused by the visual system and characterized. Four distinct types of sketches (2D, 3D, contour, and Stream plots) have been employed for better understanding our obtained solutions and their convergence.

这项研究基于三个主要支柱，即通过改进的Khater方法研究改进的本杰明-博纳-马洪尼（BBM）方程的计算解，然后通过哈密顿系统的特征研究获得的解的稳定性，评估初始值和初始值。边界条件，允许应用B样条收集方案（三次，定量，化粪池）来找到建议模型的数值解，并解释我们获得的解的类型之间的匹配。通过描述长波在视觉幻觉中的非线性色散介质中的传播，将该模型用于光学幻觉领域。视觉上的错觉是由视觉系统引起的错觉并具有特征。四种不同类型的草图（2D，3D，轮廓，