By utilizing Eq.(3) into the Eq.(1) with \(h_{1}=0,\) we achieve
$$\begin{aligned} \left\{ \begin{array}{c} \omega M^{^{\prime \prime }}-MN^{^{\prime }}=0, \\ \omega N^{^{\prime \prime }}+MM^{^{\prime }}=0. \end{array} \right. \end{aligned}$$
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Exploration of soliton solutions of the nonlinear Kraenkel-Manna-Merle system using innovative methods in ferromagnetic materials
