Full family of flattening solitary waves for the critical generalized KdV equation

Abstract

For the critical generalized KdV equation ∂tu+∂x(∂2xu+u5)=0 on R, we construct a full family of flattening solitary wave solutions. Let Q be the unique even positive solution of Q′′+Q5=Q. For any ν∈(0,13), there exist global (for t≥0) solutions of the equation with the asymptotic behavior

u(t,x)=t−ν2Q(t−ν(x−x(t)))+w(t,x) where, for some c>0,

x(t)∼ct1−2ν and ∥w(t)∥H1(x>12x(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.