Existence of multi-solitary waves with logarithmic relative distances for the NLS equation

Nguyễn, Tiến Vinh

doi:10.1016/j.crma.2018.11.012

Partial differential equations

Existence of multi-solitary waves with logarithmic relative distances for the NLS equation
[Existence d'ondes solitaires multiples avec distances relatives logarithmiques de Schrödinger non linéaires]

Présenté par : Jean-Michel Coron

Nguyễn, Tiến Vinh ¹

¹ CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau, France

Comptes Rendus. Mathématique, Tome 357 (2019) no. 1, pp. 13-58.

Résumé
Abstract

On construit des solutions au problème de la propagation de deux ondes solitaires avec distance logarithmique de Schrödinger non linéaire,

i \partial_{t} u + Δ u + | u |^{p - 1} u = 0, t \in R, x \in R^{d},

dans le cas d'une masse souscritique

1 < p < 1 + \frac{4}{d}

et d'une masse surcritique

1 + \frac{4}{d} < p < \frac{d + 2}{d - 2}

, autrement dit,

u (t)

, qui satisfait

{‖ u (t) - e^{i γ (t)} \sum_{k = 1}^{2} Q (\cdot - x_{k} (t)) ‖}_{H^{1}} \to 0

| x_{1} (t) - x_{2} (t) | \sim 2 \log (t) quand t \to + \infty,

où Q est l'état fondamental. La distance logarithmique est liée à l'interaction forte entre ondes solitaires.

Dans le cas intégrable ( $d = 1$ et $p = 3$ ), l'existence d'une telle solution est connue par la méthode dite d'inverse scaterring (E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Physica D 25 (1987) 330–346 ; T. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69). Le cas d'une masse critique $p = 1 + \frac{4}{d}$ introduit un comportement spécifique lié à l'explosion, qui a été étudié précédemment par Y. Martel et P. Raphaël (Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér. 51 (2018) 701–737).

We construct 2-solitary wave solutions with logarithmic distance to the nonlinear Schrödinger equation,

i \partial_{t} u + Δ u + | u |^{p - 1} u = 0, t \in R, x \in R^{d},

in mass-subcritical cases

1 < p < 1 + \frac{4}{d}

and mass-supercritical cases

1 + \frac{4}{d} < p < \frac{d + 2}{d - 2}

, i.e. solutions

u (t)

satisfying

{‖ u (t) - e^{i γ (t)} \sum_{k = 1}^{2} Q (\cdot - x_{k} (t)) ‖}_{H^{1}} \to 0

and

| x_{1} (t) - x_{2} (t) | \sim 2 \log t, as t \to + \infty,

where Q is the ground state. The logarithmic distance is related to strong interactions between solitary waves.

In the integrable case ( $d = 1$ and $p = 3$ ), the existence of such solutions is known by inverse scattering (E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Physica D 25 (1987) 330–346; T. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69). The mass-critical case $p = 1 + \frac{4}{d}$ exhibits a specific behavior related to blow-up, previously studied in Y. Martel, P. Raphaël (Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér. 51 (2018) 701–737).

Reçu le : 2018-11-08
Accepté le : 2018-11-19
Publié le : 2018-12-13

DOI : 10.1016/j.crma.2018.11.012

Affiliations des auteurs :

Nguyễn, Tiến Vinh ¹

¹ CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau, France

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     title = {Existence of multi-solitary waves with logarithmic relative distances for the {NLS} equation},
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Nguyễn, Tiến Vinh. Existence of multi-solitary waves with logarithmic relative distances for the NLS equation. Comptes Rendus. Mathématique, Tome 357 (2019) no. 1, pp. 13-58. doi : 10.1016/j.crma.2018.11.012. http://www.numdam.org/articles/10.1016/j.crma.2018.11.012/

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