The physics of the (4 + 1)-dimensional Fokas equation follows necessarily from the physical nature of the Kadomtsev–Petviashvili and Davey-Stewartson equations in wave theory. In this article, we consider the nonlinear $(4+1)$-dimensional Fokas partial differential equation. Two recognizable methods namely improved F-expansion and generalized exp$(-\varphi (\xi ))$-expansion methods are proposed to investigate the new wave structures of (4 + 1)-dimensional Fokas dynamical model which have never been constructed before. As a result, new solutions are in different solitons such as dark soliton, bright soliton, combined dark-bright soliton solutions, periodic, and solitary wave solutions. The new generalized solitary wave solutions can be constructed by assigning the specific values to those parameters involved in these methods. The achieved results are compared with existing results in the literature and we found that some of results are not available in literature. The derived results are explained graphically, to understand the phenomena of the proposed model. The constructed results rendering that the consider methods in this article are simple, effective, and easy to find the solution of many other nonlinear higher-dimensional models which arise in several areas of science and engineering.

（4 +1）维Fokas方程的物理性质必然源自波浪理论中Kadomtsev–Petviashvili和Davey-Stewartson方程的物理性质。在本文中，我们考虑了非线性$（4+\mathrm{1个}）$维Fokas偏微分方程。两种公认的方法，即改进的F展开和广义的exp$（--\varphi （\xi ））$提出了用扩展方法研究（4 +1）维Fokas动力学模型的新波形结构，这种结构以前从未建立过。结果，新的解决方案出现在不同的孤子中，例如暗孤子，亮孤子，组合的暗亮孤子解决方案，周期和孤立波解决方案。通过将特定值分配给这些方法中涉及的那些参数，可以构造新的广义孤立波解。将获得的结果与文献中的现有结果进行比较，我们发现某些结果在文献中不可用。用图形方式解释得出的结果，以了解所提出模型的现象。构造结果表明本文中的审议方法简单，有效，