Construction of Multibubble Solutions for the Critical GKDV Equation

Full Family of Flattening Solitary Waves for the Critical Generalized KdV Equation

“…We recall that F(s, y) = λ 3 4 σ (s) and λ(s)y < − 1 4 σ (s) ⇒ y < −cs, and then using (2.2), (2.5) (with κ 0 = 1 2 ), we deduce that, for s large enough,…”

“…(We also refer to [7] for the construction of exotic solutions in other contexts.) The article [5], where a class of flattening bubbles is constructed for the energy critical wave equation on R 3 , is particularly related to our work. More precisely, W being the unique radial positive solution of W + W 5 = 0 on R 3 , it is proved in [5] that for any |ν| 1, there exist global (for positive time) solutions of ∂ 2 t u = u + |u| 4 u such that u(t, x) ∼ t ν/2 W (t ν x) as t → +∞; the case 0 < ν 1 corresponds to blow-up in infinite time, while 0 < −ν 1 corresponds to flattening solitons.…”

“…To complement Theorem 1.1, we prove in Sect. 5.6 that the solutions do not behave as solutions of the linear Airy equation ∂ t v + ∂ 3 x v = 0 as t → ∞ (non-scattering solutions). We claim that the restriction β ∈ ( 1 3 , 1) in Theorem 1.1 corresponds to the full range of relevant exponents.…”

“…which, together with (3.19), concludes the proof of (3.25). Note that in the case where y > −2|b| −γ , we get from (3.19) and the choice γ < 1 that λ(s)y + σ (s) > 3 4 σ (s), so that (3.26) follows from (2.2) and (2.5).…”

Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrödinger system

“…See e.g. Lemma 2.7 in [4] for a detailled argument in the case of the (gKdV) equation, and Lemma 7 in the present paper for the corresponding estimates on the time derivatives of the parameters. For technical reasons, one can fix zero initial conditions on γ 1 , γ 2 as in (2.11), but the initial conditions on σ 1 , σ 2 , β 1 and β 2 have to depend on a parameter σ ∞ to be fixed later by a topological argument.…”