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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De Gruyter
2025-02-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2024-0103 |
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| author | Zhong Penghong Chen Xingfa Chen Ye |
| author_facet | Zhong Penghong Chen Xingfa Chen Ye |
| author_sort | Zhong Penghong |
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| description | 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. |
| format | Article |
| id | doaj-art-4813fa86cc4446b49be54d94b0240e96 |
| institution | Kabale University |
| issn | 2391-5455 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | De Gruyter |
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| spelling | doaj-art-4813fa86cc4446b49be54d94b0240e962025-02-10T13:24:35ZengDe GruyterOpen Mathematics2391-54552025-02-0123119119410.1515/math-2024-0103The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equationZhong Penghong0Chen Xingfa1Chen Ye2Department of Applied Mathematics, Guangdong University of Education, Guangzhou 510640, P. R. ChinaDepartment of Applied Mathematics, Guangdong University of Education, Guangzhou 510640, P. R. ChinaDepartment of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011, United States of AmericaBased 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.https://doi.org/10.1515/math-2024-0103heisenberg spinsolitonill-posednessfourier integral35q6035b35 |
| spellingShingle | Zhong Penghong Chen Xingfa Chen Ye The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation Open Mathematics heisenberg spin soliton ill-posedness fourier integral 35q60 35b35 |
| title | The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation |
| title_full | The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation |
| title_fullStr | The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation |
| title_full_unstemmed | The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation |
| title_short | The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation |
| title_sort | ill posedness of the non periodic traveling wave solution for the deformed continuous heisenberg spin equation |
| topic | heisenberg spin soliton ill-posedness fourier integral 35q60 35b35 |
| url | https://doi.org/10.1515/math-2024-0103 |
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