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

Abstract (VO)
Résumé

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).

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

et

| 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).

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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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

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