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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.) (English. French summary) Zbl 1406.35367

Summary: We construct 2-solitary wave solutions with logarithmic distance to the nonlinear Schrödinger equation, \[ \mathrm{i} \partial_t u + \Delta u + | u |^{p - 1} u = 0,\quad t \in \mathbb{R},\;x \in \mathbb{R}^d, \] 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 \[ \biggl\| u(t) - \mathrm{e}^{\mathrm{i} \gamma(t)} \sum_{k = 1}^2 Q(\cdot - x_k(t)) \biggr\|_{H^1} \rightarrow 0 \] and \[ | x_1(t) - x_2(t) | \sim 2 \log t, \quad \text{as } t \rightarrow + \infty, \] 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, Physica D 25, 330–346 (1987; Zbl 0634.35080); T. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”, Sov. Phys., JETP 34, No. 1, 62–69 (1972)]. The mass-critical case \(p = 1 + \frac{4}{d}\) exhibits a specific behavior related to blow-up, previously studied in [Y. Martel and P. Raphaël, Ann. Sci. Éc. Norm. Supér. (4) 51, No. 3, 701–737 (2018; Zbl 1403.35280)].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions
35B44 Blow-up in context of PDEs
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