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

Abstract

For the critical generalized KdV equation ∂ t u + ∂ x (∂ 2 x u + u 5) = 0 on R, we construct a full family of flattening solitary wave solutions. Let Q be the unique even positive solution of Q + Q 5 = Q. For any ν ∈ (0, 1 3), there exist global (for t ≥ 0) solutions of the equation with the asymptotic behavior u(t, x) = t − ν 2 Q t −ν (x − x(t)) + w(t, x) where, for some c > 0, x(t) ∼ ct 1−2ν and w(t) H 1 (x> 1 2 x(t)) → 0 as t → +∞. Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data. This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation. This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.