ABSTRACT

Novel approximate and analytical solutions to the fifth-order Duffing equation (FODE) are reported. These solutions are expressed in terms of the Jacobi elliptic and trigonometric functions. As a possible realization in the nature of the FODE, we study some nonlinear differential equations (NLDEs) that describe many physical problems. The numerical solution to the FODE is compared with the theoretical results. Moreover, the proposed method shows a new vision, which could be of great interest in the solution of the family of higher-order nonlinear Schrödinger equation such as the cubic-quintic NLSE (CQNLSE) which is used for interpreting the high Langmuir fields energy in magnetoplasmas. This method extends to find the solution of the derivative NLSE (DNLSE), which is used to describe the weakly modulated Alfvén waves propagation in magnetoplasmas. Furthermore, the equation of motion for the general pendulum could be solved using the techniques under consideration. All mentioned equations could be reduced to the FODE via a suitable transformation. Finally, this study is very interesting in describing several natural phenomena and solving many physical NLDEs that were difficult to solve them before. Moreover, the analytical bright soliton solution to the higher-order NLSE with incorporating cubic-quintic nonlinearity is obtained.

摘要

报告了五阶杜芬方程 (FODE) 的新近似解和解析解。这些解用雅可比椭圆函数和三角函数表示。作为 FODE 本质的一种可能实现，我们研究了一些描述许多物理问题的非线性微分方程 (NLDE)。FODE 的数值解与理论结果进行了比较。此外，所提出的方法展示了一个新的愿景，它可能对高阶非线性薛定谔方程族的解有很大的兴趣，例如用于解释高朗缪尔场能量的三次-五次NLSE（CQNLSE）。磁浆体。该方法扩展到找到导数 NLSE (DNLSE) 的解，该解用于描述磁等离子体中弱调制的阿尔文波传播。此外，可以使用所考虑的技术求解一般摆的运动方程。所有提到的方程都可以通过适当的变换简化为 FODE。最后，这项研究在描述几种自然现象和解决许多以前难以解决的物理 NLDE 方面非常有趣。此外，获得了包含三次五次非线性的高阶NLSE的解析亮孤子解。