The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation

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^s2 (for the periodic solution in H_S^s2(T) and the non-periodic solution in H_S^s2(Rq), respectively). When α ≠ 0, the range of the weak ill-posedness index is 1 < s <3/2 for both periodic and non-periodic solutions. However, the periodic solution exhibits a strong ill-posedness index in the range of3/2 < s < 7/2, whereas for the non-periodic solution, the range is 1 < s < 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.