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

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We construct in this paper global (for t ≥ 0) and bounded solutions u(t) for the nonlinear Schrödinger equation i ∂ₜ u + Δu + |u|ᵖ⁻¹ u = 0, t ∈ R, x ∈ Rᵈ in mass sub-critical cases (1 < p < 1 + 4/d) and mass super-critical (1 + 4/d < p < d+2/d-2) such that u(t) decomposes asymptotically into two solitary waves with logarithmic distance u(t) - eⁱ γ⁽ᵗ⁾ ∑ₖ₌₁² Q(· - xₖ(t))ₕ₁ → 0 and |x₁(t) - x₂(t)| 2 t, ast → + ∞. 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 has been shown in [14].