The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation - Zhong Penghong; Chen Xingfa; Chen Ye (Open Mathematics)
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 HS2sH_{{{\rm{S}}}^{2}}^{s} (for the periodic solution in HS2s(T)H_{{{\rm{S}}}^{2}}^{s}\left({\mathbb{T}}) and the non-periodic solution in HS2s(R)H_{{{\rm{S}}}^{2}}^{s}\left({\mathbb{R}}), respectively). When α≠0α \ne 0, the range of the weak ill-posedness index is 1<s<321\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 32<s<72\frac{3}{2}\lt s\lt \frac{7}{2}, whereas for the non-periodic solution, the range is 1<s<21\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.