On the Bäcklund transform and the stability of the line soliton of the KP-II equation on $\mathbb R^2$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913615336964096 |
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| author | Pompili, Lorenzo |
| author_facet | Pompili, Lorenzo |
| contents | We study the Miura map of the KP-II equation on $\mathbb R^2$ and the resulting Bäcklund transform, which adds a line soliton to a given solution. This work aims to develop a complementary approach to T. Mizumachi's method for the $L^2$-stability of the line soliton, which the potential for generalization to multisolitons. We construct the Bäcklund transform by classifying solutions of the Miura map equation close to a modulated kink; this translates into studying eternal solutions of the forced viscous Burgers' equation under distinct boundary conditions at $\pm\infty$. We then show that its range, when intersected with a small ball in $|D_x|^{1/2} L^2(\mathbb R^2)\cap L^2(\mathbb R^2)\cap \langle{y}\rangle^{0-}\!L^1(\mathbb R^2)$, forms a codimension-1 manifold. We prove codimension-1 $L^2$-stability of the line soliton in the aforementioned weighted space as a corollary, providing the first stability result at sharp regularity. The codimension-1 condition in the range of the Bäcklund transform is an intrinsic property, and we conjecture that it corresponds to a known long time behavior of perturbed line solitons. The stability is expected to hold without this condition, as in Mizumachi's works. Finally, we show the construction of a multisoliton addition map for $(k,1)$-multisolitons, $k\geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12530 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Bäcklund transform and the stability of the line soliton of the KP-II equation on $\mathbb R^2$ Pompili, Lorenzo Analysis of PDEs We study the Miura map of the KP-II equation on $\mathbb R^2$ and the resulting Bäcklund transform, which adds a line soliton to a given solution. This work aims to develop a complementary approach to T. Mizumachi's method for the $L^2$-stability of the line soliton, which the potential for generalization to multisolitons. We construct the Bäcklund transform by classifying solutions of the Miura map equation close to a modulated kink; this translates into studying eternal solutions of the forced viscous Burgers' equation under distinct boundary conditions at $\pm\infty$. We then show that its range, when intersected with a small ball in $|D_x|^{1/2} L^2(\mathbb R^2)\cap L^2(\mathbb R^2)\cap \langle{y}\rangle^{0-}\!L^1(\mathbb R^2)$, forms a codimension-1 manifold. We prove codimension-1 $L^2$-stability of the line soliton in the aforementioned weighted space as a corollary, providing the first stability result at sharp regularity. The codimension-1 condition in the range of the Bäcklund transform is an intrinsic property, and we conjecture that it corresponds to a known long time behavior of perturbed line solitons. The stability is expected to hold without this condition, as in Mizumachi's works. Finally, we show the construction of a multisoliton addition map for $(k,1)$-multisolitons, $k\geq 1$. |
| title | On the Bäcklund transform and the stability of the line soliton of the KP-II equation on $\mathbb R^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.12530 |