The higher-order modified Korteweg-de Vries equation: Its soliton, breather and approximate solutions

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In this study, the fifth-order modified Korteweg-de Vries (F-MKdV) equation is first addressed using Hirota's bilinear method. Thereafter, the exact and approximative solutions of the generalized form of the F-MKdV equation are investigated using the modified Kudryashov method, the Riccati equation and its Backlund transformation method, the solitary wave ansatz method, and the homotopy perturbation transform method (HPTM). As a result, solitons, breather, and solitary wave solutions are derived from these methods. In particular, we obtain some new solutions such as the dark soliton, bright soliton, singular soliton, periodic trigonometric, exponential, hyperbolic, and rational solutions. The constraint conditions associated with the resulting solutions are also discussed in detail. The HPTM is employed to construct approximate solutions to the aforementioned generalized model due to its strong nonlinear terms and only a few terms are required to obtain accurate solutions. These findings may help to understand complex nonlinear phenomena.