LAUSR

Abstract Based on an equivalent derivative non-linear Schrödinger equation, we derive some periodic and non-periodic two-parameter solutions of the deformed continuous Heisenberg spin (DCHS) equation. The ill-posedness of these solutions is demonstrated through Fourier integral estimates in the Sobolev space H S 2 s {H}_{{{\rm{S}}}^{2}}^{s} (for the periodic solution in H S 2 s ( T ) {H}_{{{\rm{S}}}^{2}}^{s}\left({\mathbb{T}}) and the non-periodic solution in H S 2 s ( R ) {H}_{{{\rm{S}}}^{2}}^{s}\left({\mathbb{R}}) , respectively). When α ≠ 0 \alpha \ne 0 , the range of the weak ill-posedness index is 1 < s < 3 2 1\lt s\lt \frac{3}{2} for both periodic and non-periodic solutions. However, the periodic solution exhibits a strong ill-posedness index in the range of 3 2 < s < 7 2 \frac{3}{2}\lt s\lt \frac{7}{2} , whereas for the non-periodic solution, the range is 1 < s < 2 1\lt s\lt 2 . These findings extend our previous work on the DCHS model to include the case of periodic solutions and explore a different fractional Sobolev space.