In this work, we investigate the Myrzakulov–Lakshmanan equation, which is an integrable evolution equation in the Lax representation sense and plays an important role in applications of magnetism and nano-physics. This equation can also be regarded as a \((2+1)\) -dimensional extension of the nonlinear Schrödinger equation and appears in several physical contexts related to ferromagnetic spin dynamics and nonlinear wave propagation. Motivated by the importance of exact wave structures in nonlinear physical systems, we employ the extended hyperbolic function method to construct new analytical solutions of the model. In contrast to many existing studies that focus on limited solution structures, this work provides a broader class of exact solutions including dark, bright, dark–bright, exponential, and periodic soliton solutions. The proposed technique is direct, concise, and capable of generating rich families of solutions for nonlinear partial differential equations. The validity and accuracy of the obtained solutions are verified by direct substitution into the original system. Furthermore, the model reduces to the well-known Schrödinger equation for \(\alpha = 1\) and \(\beta = 0\) , and to the Zakharov–Strachan equation for \(\alpha = 0\) and \(\beta = 1\) ; therefore, the derived solutions also satisfy these important limiting cases. In addition, we investigate the modulation instability of the system by reducing the coupled system to a single evolution equation. The modulation instability gain spectrum is obtained and the influence of different parameters on the instability growth rate is analyzed. To illustrate the dynamical behavior of the obtained solutions, several profiles are presented through 3D, contour, and 2D plots using Mathematica software.The obtained results provide new analytical wave structures for the Myrzakulov–Lakshmanan equation and contribute to the understanding of nonlinear wave propagation in ferromagnetism and nano-magnetic media.
