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A powerful mathematical method for front instability analysis that wasrecently developed in the field of nonlinear dynamics is applied to the 1+1(spatial and time) dimensional Landau-Lifshitz-Gilbert (LLG) equation. From theessential spectrum of the LLG equation, it is shown that the famous Walkerrigid body propagating domain wall (DW) is not stable against the spin waveemission. In the low field region only stern spin waves are emitted while bothstern and bow waves are generated under high fields. By using the properties ofthe absolute spectrum of the LLG equation, it is concluded that in a highenough field, but below the Walker breakdown field, the Walker solution couldbe convective/absolute unstable if the transverse magnetic anisotropy is largerthan a critical value, corresponding to a significant modification of the DWprofile and DW propagating speed. Since the Walker solution of 1+1 dimensionalLLG equation can be realized in experiments, our results could be also used totest the mathematical method in a controlled manner.

On the Instabilities of the Walker Propagating Domain Wall Solution