In this article, we derive various exact solutions and patterns for the complex modified Korteweg–De Vries system of equation (cmKdV) with a generalized innovative extended direct algebra method. The Korteweg-De Vries system exhibits the scientific dynamics of water particles at the surface and beyond the surface level. The system also has applications in ferromagnetic materials, nonlinear optics, and solitons theory. The innovative direct algebra method is applied to obtain dark, multiple, singular, breather and bright wave patterns. This method also provides staggering wave solutions for the complex modified Kortweg-De Vries system in the form of hyperbolic and trigonometric functions. These recovered solutions for the considered model and are more efficient, concise and general than the extant ones. The wave patterns are properly explained with 2-D and 3-D graphs to elucidate wave behaviour for some selected solutions derived for the system. Lastly, the solutions in this work will greatly advance various fields of application of the Kortweg-De Vries equation like optical fibres, ferromagnetic materials, nonlinear optics, signal processing, water waves, plasma physics, soliton theory, string theory and other contemporary sciences.

Akinyemi, L., Senol, M., & Iyiola, O. S. (2021). Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method. Mathematics and Computers in Simulation, 182, 211-233. http://dx.doi.org/10.1016/j.matcom.2020.10.017 Ibrahim, I. S., Sabi’u, J., Gambo, Y. Y. U., Rezapour, S., & Inc, M. (2024). Dynamic soliton solutions for the modified complex Korteweg-de Vries system. Optical and Quantum Electronics, 56(6), 954. https://doi.org/10.1007/s11082-024-06821-w Khater, M. M., Akinyemi, L., Elagan, S. K., El-Shorbagy, M. A., Alfalqi, S. H., Alzaidi, J. F., & Alshehri, N. A. (2021). Bright–dark soliton waves’ dynamics in pseudo spherical surfaces through the nonlinear Kaup–Kupershmidt equation. Symmetry, 13(6), 963. http://dx.doi.org/10.3390/sym13060963 Muhammad, U. A., Sabi’u, J., Salahshour, S., & Rezazadeh, H. (2024). Soliton solutions of (2+ 1) complex modified Korteweg–de Vries system using improved Sardar method. Optical and Quantum Electronics, 56(5), 802. https://doi.org/10.1007/s11082-024-06591-5 Rahman, M., Karaca, Y., Sun, M., Baleanu, D., & Alfwzan, W. F. (2024). Complex behaviors and various soliton profiles of (2+ 1)-dimensional complex modified Korteweg-de-Vries Equation. Optical and Quantum Electronics, 56(5), 878. Rezazadeh, H. New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity, Optik, 167(2018), 218-227. Sabi’u, J., Das, P. K., Pashrashid, A., & Rezazadeh, H. (2022). Exact solitary optical wave solutions and modulational instability of the truncated Ω-fractional Lakshamanan–Porsezian–Daniel model with Kerr, parabolic, and anti-cubic nonlinear laws. Optical and Quantum Electronics, 54(5), 269. Shaikhova, G., Kutum, B., &Myrzakulov, R. (2022). Periodic traveling wave, bright and dark soliton solutions of the (2+ 1)-dimensional complex modified Korteweg-de Vries system of equations by using three different methods. AIMS Mathematics, 7(10), 18948-18970. Yuan, F. (2021). The order-n breather and degenerate breather solutions of the (2+ 1)-dimensional cmKdVequations. International Journal of Modern Physics B, 35(04).