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ABSTRACT In this study, we develop the n ‐fold Darboux transformation (DT) for the Kraenkel–Manna–Merle (KMM) integrable system designed to characterize the nonlinear dynamics of ultra‐short wave pulses, especially within the saturated ferromagnetic materials. A short‐wave can only propagate in the direction perpendicular to the external saturating magnetic field in some saturated ferromagnetic materials, which is called as KMM system. We achieve this by employing the gauge transformation technique between Lax pairs and derive multiple soliton solutions expressed through a determinant representation. generates novel solutions and , characterized by the ratios of two respective determinants. Furthermore, we provide the explicit forms for the nth‐order smooth positons for the KMM system by employing the degenerate DT in relation to the eigenvalues. The non‐singular solutions for the KMM system with ‐positons are derived under the specific condition where for . The dynamic characteristics of the smooth positon in the KMM system are discussed in detail, as well as the derivation of the corresponding trajectory, an approximation of the trajectory, and the concept of a “phase shift.” As an example, we report new kink‐type fronts in some types of saturated ferromagnetic materials. Finally, we find that the “phase shift” for the smooth positons is dependent on the spatial space and temporal time, while it is constant for a typical two‐soliton solution. The stronger localized shape and propagation of magnetic smooth positons implies that the novel form of microwaves is admitted and controlled in ferrites, which has important potential applications for ferrite‐loaded waveguides at microwave frequencies, rapid storage and processing of information in spintronics.
Rahman et al. (Thu,) studied this question.