On the approximate and analytical solutions to the fifth-order Duffing oscillator and its physical applications

Abstract

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.