Spatial analyticity of solutions of a nonlocal perturbation of the KdV equation
Let $\mathcal{H}$ denote the Hilbert transform and $\eta \ge 0$. We show that if the initial data of the following problems $ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \, u(\cdot , 0) = \phi (\cdot)$ and $ v_t + \frac{1}{2} (v_x)^2 + v_{xxx} + \eta(\mathcal{H} v_x...
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| Format: | Article |
| Language: | English |
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University of Szeged
2005-11-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=230 |
| _version_ | 1827951446724706304 |
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| author | B. Alvarez Samaniego |
| author_facet | B. Alvarez Samaniego |
| author_sort | B. Alvarez Samaniego |
| collection | DOAJ |
| description | Let $\mathcal{H}$ denote the Hilbert transform and $\eta \ge 0$. We show that if the initial data of the following problems
$ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \,
u(\cdot , 0) = \phi (\cdot)$ and
$ v_t + \frac{1}{2} (v_x)^2 + v_{xxx} + \eta(\mathcal{H} v_x + \mathcal{H} v_{xxx}) = 0, \,
v(\cdot , 0) = \psi (\cdot)$
has an analytic continuation to a strip containing the real axis, then the solution has the same property, although the width of the strip might diminish with time. When $\eta>0$ and the initial data is complex-valued we prove local well-posedness of the two problems above in spaces of analytic functions, which implies the constancy over time of the radius of the strip of analyticity in the complex plane around the real axis. |
| first_indexed | 2024-04-09T13:41:56Z |
| format | Article |
| id | doaj.art-aec6aa04fdc844b98a4977fc8f09c7f5 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:41:56Z |
| publishDate | 2005-11-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-aec6aa04fdc844b98a4977fc8f09c7f52023-05-09T07:52:57ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752005-11-0120052012110.14232/ejqtde.2005.1.20230Spatial analyticity of solutions of a nonlocal perturbation of the KdV equationB. Alvarez Samaniego0IMECC-UNICAMP, Campinas, BrasilLet $\mathcal{H}$ denote the Hilbert transform and $\eta \ge 0$. We show that if the initial data of the following problems $ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \, u(\cdot , 0) = \phi (\cdot)$ and $ v_t + \frac{1}{2} (v_x)^2 + v_{xxx} + \eta(\mathcal{H} v_x + \mathcal{H} v_{xxx}) = 0, \, v(\cdot , 0) = \psi (\cdot)$ has an analytic continuation to a strip containing the real axis, then the solution has the same property, although the width of the strip might diminish with time. When $\eta>0$ and the initial data is complex-valued we prove local well-posedness of the two problems above in spaces of analytic functions, which implies the constancy over time of the radius of the strip of analyticity in the complex plane around the real axis.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=230 |
| spellingShingle | B. Alvarez Samaniego Spatial analyticity of solutions of a nonlocal perturbation of the KdV equation Electronic Journal of Qualitative Theory of Differential Equations |
| title | Spatial analyticity of solutions of a nonlocal perturbation of the KdV equation |
| title_full | Spatial analyticity of solutions of a nonlocal perturbation of the KdV equation |
| title_fullStr | Spatial analyticity of solutions of a nonlocal perturbation of the KdV equation |
| title_full_unstemmed | Spatial analyticity of solutions of a nonlocal perturbation of the KdV equation |
| title_short | Spatial analyticity of solutions of a nonlocal perturbation of the KdV equation |
| title_sort | spatial analyticity of solutions of a nonlocal perturbation of the kdv equation |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=230 |
| work_keys_str_mv | AT balvarezsamaniego spatialanalyticityofsolutionsofanonlocalperturbationofthekdvequation |