Influence of damping effects on the propagation of magnetic waves in ferrites

•We investigate solution of the dissipative KMM-system•We propose indeed an integrable system with its associated Lax-pairs, construct its soliton solutions using the Wadati–Konno–Ichikawa scheme•We show that by means of ultra-fast process assumption, the solution of the proposed-system approximates very well the solution of the dissipative KMM-system of our interest.•For consistency, analytical results, we proceed to some numerical simulations, that confirm the results obtained analytically.