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arXiv · 2404.13980

Asymptotic stability of solitons for near-cubic NLS equation with an internal mode

Guillaume Rialland
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Abstract

We consider perturbations of the one-dimensional cubic Schr\"odinger equation, of the form $i \, \partial_t \psi + \partial_x^2 \psi + |\psi|^2 \psi + g( |\psi|^2 ) \psi = 0$. Under hypotheses on the function $g$ that can be easily verified in some cases (such as $g(s) = s^\sigma$ with $\sigma >1$), we show that the linearized problem around a small solitary wave presents a unique internal mode. Moreover, under an additional hypothesis (the Fermi golden rule) that can also be verified in the case of powers $g(s) = s^\sigma$, we prove the asymptotic stability of the solitary waves with small frequencies.

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Guillaume Rialland. 2024-04-22. Asymptotic stability of solitons for near-cubic NLS equation with an internal mode. https://arxiv.org/abs/2404.13980

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