In this paper, we present a comprehensive study on the $(4+1)$-Fokas equation that includes new solutions and conservation laws of the equation. Specifically, for the first time, new shrunken-periodic and fascinating interactions on the bright soliton background have been investigated. Furthermore, a bright, dark, dipole soliton, tripole soliton, multipole soliton, periodic, doubly-periodic, damped-periodic, kink, lump, rogue wave, breather and their interactions have been attained and graphically illustrated. In addition, a significant interaction between four parabolic-periodic solitary waves is first detected and emphasized by harnessing 3D and contour plots. On the other hand, the latter has been compared to an experimentally-induced Moiré effect from a monitor screen camera capture. These outstanding results are achieved using the Lie symmetry approach. The multiple reduction process of the $(4+1)$-Fokas equation through the Lie algebra spanned elements, and the corresponding optimal system guarantees 23 solutions in general forms. To determine the physical meaning of the latter, an appropriate ansatz for the arbitrary functions and parameters has been proposed and listed in the corresponding tables, which leads to the various types of solitary wave solutions. Finally, the conserved vectors for the underlying equation are derived using the multiplier method.

在本文中，我们对$(4+1)$-Fokas 方程，包括方程的新解和守恒定律。具体来说，首次研究了明亮孤子背景上新的收缩周期和迷人的相互作用。此外，还获得了明亮的、黑暗的、偶极子孤子、三极子孤子、多极子孤子、周期性、双周期性、阻尼周期性、扭结、团块、流氓波、呼吸及其相互作用，并以图形方式说明了它们。此外，首先通过利用 3D 和等高线图检测和强调四个抛物线周期孤立波之间的显着相互作用。另一方面，后者已与监视器屏幕摄像机捕获的实验诱导的莫尔效应进行了比较。这些突出的结果是使用李对称方法实现的。多重还原过程$(4+1)$-Fokas方程通过李代数跨越元素，对应的最优系统保证23个一般形式的解。为了确定后者的物理意义，针对任意函数和参数提出了适当的 ansatz 并列在相应的表格中，这导致了各种类型的孤立波解。最后，使用乘法器方法导出基础方程的守恒向量。