Myrzakulov-Lakshmanan-XXXII-Equation: Painlevé Integrability, New Variety Soliton Solution Types

The objective of this study is to investigate the Myrzakulov-Lakshmanan-XXXII (ML-XXXII) equation, which represents the generalized Heisenberg ferromagnetic equation recently introduced at the Eurasian International Centre for Theoretical Physics, Kazakhstan. The proposed model is a generalized modified form of the Heisenberg ferromagnetic spin-chain equation with bilinear and anisotropic interactions in the semi classical limit. It has significant importance in describing various magnetic phenomena, including nonlinear long-range ferromagnetic ordering, the magnetic properties of many insulating crystals, and spin interactions. For the first time, the resonance associated with the arbitrariness of the singular manifold of the proposed model is investigated by applying the Painlevé analysis. Furthermore, three different semi-analytical techniques, namely the Extended Simple Equation Algorithm (ESEA), the Paul-Painlevé Approach Algorithm (PPAA), and the Riccati-Bernoulli Sub-Ordinary Differential Equation Algorithm (RBSODA), are employed to derive new semi-analytical solutions for the proposed model. To the best of our knowledge, this is the first attempt to obtain such semi-analytical solutions for the ML-XXXII equation. The obtained results yield several types of rational soliton solutions with different dynamical behaviors, including hyperbolic periodic wave solutions, singular soliton solutions representing repulsive interaction forces, and W-shaped soliton solutions, which can be interpreted as a combination of bright and dark solitons. A comparison with previously published solutions demonstrates that the obtained semi-analytical solutions are novel and have not been reported previously. Furthermore, two-dimensional and three-dimensional plots are presented to illustrate the dynamic properties and propagation behaviors of the newly derived semi-analytical solutions.