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arXiv · 1912.10889

Traveling waves of the quintic focusing NLS-Szeg{ö} equation

Ruoci Sun
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Abstract

We study the influence of Szeg{ö} projector $Π$ on the L 2 --critical one-dimensional non linear focusing Schr{ö}dinger equation, leading to the quintic focusing NLS-Szeg{ö} equation i$\partial$ t u + $\partial$ 2 x u + $Π$(|u| 4 u) = 0, (t, x) $\in$ R x R, u(0, $\times$) = u 0. This equation is globally well-posed in H 1 + = $Π$(H 1 (R)), for every initial datum u 0. The solution L 2-scatters both forward and backward in time if u 0 has sufficiently small mass. We prove the orbital stability with scaling of the traveling wave : u $ω$,c (t, x) = e i$ω$t Q(x + ct), for some $ω$, c > 0, where Q is a ground state associated to Gagliardo-Nirenberg type functional I ($γ$) (f) = $\partial$ x f 2 L 2 f 4 L 2 + $γ$ --i$\partial$ x f, f 2 L 2 f 2 L 2 f 6 L 6 , $\forall$f $\in$ H 1 + \{0}, for some $γ$ $\ge$ 0. The ground states are completely classified in the case $γ$ = 2, leading to the actual orbital stability without scaling for appropriate traveling waves. As a consequence, the scattering mass threshold of the focusing quintic NLS-Szeg{ö} equation is strictly below the mass of ground state associated to the functional I (0) , unlike the recent result by Dodson [6] on the usual quintic focusing non linear Schr{ö}dinger equation.

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Ruoci Sun. 2019-12-23. Traveling waves of the quintic focusing NLS-Szeg{ö} equation. https://arxiv.org/abs/1912.10889

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