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...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-02-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2024-0103 |
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| Summary: | 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 HS2s{H}_{{{\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≠0\alpha \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. |
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| ISSN: | 2391-5455 |