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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.16973 |
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| _version_ | 1866910598694961152 |
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| author | Bejenaru, Ioan Pillai, Mohandas Tataru, Daniel |
| author_facet | Bejenaru, Ioan Pillai, Mohandas Tataru, Daniel |
| contents | We consider equivariant solutions for the Schrödinger Map equation in $2+1$ dimensions, with values into $\mathbb{S}^2$. Within each equivariance class $m \in \mathbb{Z}$ this admits a lowest energy nontrivial steady state $Q^m$, which extends to a two dimensional family of steady states by scaling and rotation. If $|m| \geq 3$ then these ground states are known to be stable in the energy space $\dot H^1$, whereas instability and even finite time blow-up along the ground state family may occur if $|m| = 1$. In this article we consider the most delicate case $|m| = 2$. Our main result asserts that small $\dot H^1$ perturbations of the ground state $Q^2$ yield global in time solutions, which satisfy global dispersive bounds. Unlike the higher equivariance classes, here we expect solutions to move arbitrarily far along the soliton family; however, we are able to provide a time dependent bound on the growth of the scale modulation parameter. We also show that within the equivariant class the ground state is stable in a slightly stronger topology $X \subset \dot H^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16973 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Near soliton evolution for $2$-equivariant Schrödinger Maps in two space dimensions Bejenaru, Ioan Pillai, Mohandas Tataru, Daniel Analysis of PDEs We consider equivariant solutions for the Schrödinger Map equation in $2+1$ dimensions, with values into $\mathbb{S}^2$. Within each equivariance class $m \in \mathbb{Z}$ this admits a lowest energy nontrivial steady state $Q^m$, which extends to a two dimensional family of steady states by scaling and rotation. If $|m| \geq 3$ then these ground states are known to be stable in the energy space $\dot H^1$, whereas instability and even finite time blow-up along the ground state family may occur if $|m| = 1$. In this article we consider the most delicate case $|m| = 2$. Our main result asserts that small $\dot H^1$ perturbations of the ground state $Q^2$ yield global in time solutions, which satisfy global dispersive bounds. Unlike the higher equivariance classes, here we expect solutions to move arbitrarily far along the soliton family; however, we are able to provide a time dependent bound on the growth of the scale modulation parameter. We also show that within the equivariant class the ground state is stable in a slightly stronger topology $X \subset \dot H^1$. |
| title | Near soliton evolution for $2$-equivariant Schrödinger Maps in two space dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.16973 |