Multisolitons for the cubic NLS in 1-d and their stability

Herbert Koch; Daniel Tataru

doi:10.1007/s10240-024-00148-8

Article

Multisolitons for the cubic NLS in 1-d and their stability

Herbert Koch ¹; Daniel Tataru ¹

¹

Publications Mathématiques de l'IHÉS, Volume 140 (2024), pp. 155-270

Abstract

For both the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set $𝐌_{N}$ of pure $N$ -soliton states, and their associated multisoliton solutions. We prove that (i) the set $𝐌_{N}$ is a uniformly smooth manifold, and (ii) the $𝐌_{N}$ states are uniformly stable in $H^{s}$ , for each $s > - \frac{1}{2}$ .

One main tool in our analysis is an iterated Bäcklund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove a multisoliton from a multisoliton state (the soliton removal map). The properties and the regularity of these maps are extensively studied.

Received: 2021-10-29
Accepted: 2024-03-06
Online First: 2024-04-29
Published online: 2024-12-07

MR Zbl

DOI: 10.1007/s10240-024-00148-8

Author's affiliations:

Herbert Koch ¹; Daniel Tataru ¹

¹

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     author = {Herbert Koch and Daniel Tataru},
     title = {Multisolitons for the cubic {NLS} in 1-d and their stability},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {155--270},
     year = {2024},
     publisher = {Springer International Publishing},
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     volume = {140},
     doi = {10.1007/s10240-024-00148-8},
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     zbl = {1559.35329},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-024-00148-8/}
}

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Herbert Koch; Daniel Tataru. Multisolitons for the cubic NLS in 1-d and their stability. Publications Mathématiques de l'IHÉS, Volume 140 (2024), pp. 155-270. doi: 10.1007/s10240-024-00148-8

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