Soliton and periodic wave structures of the NTME-II equation: analytical investigation of localized and singular solutions

This study presents a thorough method to obtain soliton solutions for the nonlinear Nurshuak–Tolkynay–Myrzakulov-II (NTME-II) equation, employing recently developed analytical techniques, including the Sardar sub-equation method and the sine-Gordon expansion method. The NTME-II equation finds broad applications in various physical contexts, such as ferromagnetic spin systems, nonlinear optics and optical fibre communications, emphasizing its importance. Different types of exact soliton solutions are explored, including single- and multi-wave patterns that reveal the equation's detailed nonlinear dynamics. The findings are depicted through 2D phase portraits and 3D surface plots, which help illustrate both the qualitative and quantitative aspects of the solutions obtained. These visualizations emphasize key soliton characteristics, including wave interactions, amplitude changes and stability features. What's more, the analysis offers a clear and detailed look into nonlinear wave propagation. The study of the interaction processes within the NTME-II equation provides valuable insights into integrable systems and the theory of nonlinear waves.