Comptes Rendus
Partial differential equations
Existence of multi-solitary waves with logarithmic relative distances for the NLS equation
Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 13-58.

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

itu+Δu+|u|p1u=0,tR,xRd,
in mass-subcritical cases 1<p<1+4d and mass-supercritical cases 1+4d<p<d+2d2, i.e. solutions u(t) satisfying
u(t)eiγ(t)k=12Q(xk(t))H10
and
|x1(t)x2(t)|2logt,ast+,
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+4d 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,

itu+Δu+|u|p1u=0,tR,xRd,
dans le cas d'une masse souscritique 1<p<1+4d et d'une masse surcritique 1+4d<p<d+2d2, autrement dit, u(t), qui satisfait
u(t)eiγ(t)k=12Q(xk(t))H10
et
|x1(t)x2(t)|2log(t)quandt+,
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+4d 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).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.11.012

Tiến Vinh Nguyễn 1

1 CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau, France
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Tiến Vinh Nguyễn. Existence of multi-solitary waves with logarithmic relative distances for the NLS equation. Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 13-58. doi : 10.1016/j.crma.2018.11.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.11.012/

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